Symmetry and Complexity the Spirit and Beauty of Nonlinear Science

SYMMETRYANDCOMPLEXITY THESPIRITANDBEAUTYOFNONLINEARSCIENCE

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WORLD SCIENTIFIC SERIES ON

NONLINEAR SCIENCE

Series A

Vol. 51

Series Editor: Leon O. Chua

SYMMETRY AND COMPLEXITY THE SPIRIT AND BEAUTY OF NONLINEAR SCIENCE

Klaus Mainzer University of Augsburg, Germany

We World Scientific NEW JERSEY · LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · BANGALORE

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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SYMMETRY AND COMPLEXITY The Spirit and Beauty of Nonlinear Science Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-192-7

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Preface

Symmetry and complexity determine the spirit of nonlinear science. The expansion of the universe, the evolution of life and the globalization of human economies and societies lead from symmetry and simplicity to complexity and diversity. The emergence of new order and structure means symmetry breaking and transition from unstable to stable states of balance. It is explained by physical, chemical, biological, and social self-organization, according to the laws of nonlinear dynamics. Atoms and molecules, stars and clouds, organisms and brains, economies and societies are only some examples of dynamical systems. Thus, symmetry and complexity are the basic principles of a common systems science in the 21st century, overcoming traditional boundaries between natural, cognitive, and social sciences, mathematics, humanities and philosophy. This book treats the essence of my scientific work that can be described by a kind of dialectical triad. In a first step, I published a comprehensive treatise on Symmetries of Nature in 1988 (English: 1996). Early studies in geometry and space-time had inspired my interest in mathematical invariance and universal laws. As many other scientists, philosophers, and artists, I was also fascinated by the beauty of mathematical symmetries. But, physical, chemical, and biological symmetries are broken by natural processes, leading to the observed complexity and diversity of the world. Therefore, I studied the foundations of nonlinear dynamics and, as a second step, published the book Thinking in Complexity. The

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Complex Dynamics of Matter, Mind, and Mankind in 1994 (4th enlarged edition 2004). My interest in complexity dates back to my Ph.D. thesis on the foundations of constructive mathematics and computational degrees of complexity in 1973. In many other books and articles, I enlarged the applications of nonlinear systems to computer science, cognitive science, and social science. After thesis and antithesis, this book is the synthesis of Symmetry and Complexity. It also connects my life-long love of music and art with nonlinear science. Research is always realized in a network of cooperation and communication. Therefore, I want to thank some colleagues, friends and institutions. Leon O. Chua (Department of Electrical Engineering & Computer Sciences, University of California, Berkeley) invited me to publish a book in his series on Nonlinear Science. He is also the editor of the International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. As a member of the editorial board of this journal, I have the opportunity to get an interdisciplinary overview of worldwide explorations in nonlinear science. Symmetry and complexity are also topics of international and cultural integration. International translations of my books underline the interest in symmetry and complexity beyond all cultural boundaries. I would like to thank the European Academy of Science (Academia Europaea in London) who invited me for a lecture on symmetry and complex systems in January 2005. Furthermore, I would like to thank the Leibniz-Community of German Research Institutions and the Japanese Research Institute of Integration Science for their kind invitation on a lecture on complex systems in October 2004. Thanks also to Hermann Haken and Wolfgang Weidlich (Institute of Theoretical Physics, University of Stuttgart), J¨ urgen Mittelstraß (Department of Philosophy, University of Constance), Martin Quack (Laboratory of Physical Chemistry, ETH Z¨ urich), and Alwyn Scott (Department of Mathematics, University of Arizona) who have inspired and supported my work. Last but not least, I would like to thank Dominik B¨ osl, Michael Hochholdinger, Jutta Janßen, Tobias Jung, Paul Williams (Univer-

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sity of Augsburg) and Lakshmi Narayan (World Scientific Publishing) for preparing the publication. Augsburg and Munich, September 2004

Klaus Mainzer

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Contents

Preface

v

Introduction

1

1.

Symmetry and Complexity in Early Culture and Philosophy 23 1.1 Cultural and Cosmic Harmony . . . . . . . . . . . . . 23 1.2 Cultural and Cosmic Diversity . . . . . . . . . . . . . 48

2.

Symmetry and Complexity in Mathematics

63

2.1 Symmetry and Group Theory . . . . . . . . . . . . . . 64 2.2 Symmetry Breaking and Bifurcation Theory . . . . . . 83 2.3 Complexity, Nonlinearity and Fractals . . . . . . . . . 99 3.

Symmetry and Complexity in Physical Sciences

107

3.1 Symmetry in Physics . . . . . . . . . . . . . . . . . . . 110 3.2 Symmetry Breaking and Phase Transitions . . . . . . 147 3.3 Complexity, Attractors and Dynamical Systems . . . . 158 4.

Symmetry and Complexity in Chemical Sciences

171

4.1 Symmetry in Chemistry . . . . . . . . . . . . . . . . . 171 4.2 Symmetry Breaking and Chirality . . . . . . . . . . . 184 4.3 Complexity, Dissipation and Nanosystems . . . . . . . 190 ix

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Symmetry and Complexity in Life Sciences

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5.1 Symmetry in Biology . . . . . . . . . . . . . . . . . . . 199 5.2 Symmetry Breaking and Evolution . . . . . . . . . . . 210 5.3 Complexity and Biodiversity of Life . . . . . . . . . . 223 6.

Symmetry and Complexity in Economic and Social Sciences

239

6.1 Symmetry, Social Balance and Economic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 239 6.2 Symmetry Breaking and Socio-economic Transitions . . . . . . . . . . . . . . . . . . . . . . . . 248 6.3 Complexity and Sociodiversity of Globalization . . . . 259 7.

Symmetry and Complexity in Computer Science 7.1 Symmetry Dynamics 7.2 Symmetry Dynamics

8.

273

and Complexity in Information . . . . . . . . . . . . . . . . . . . . . . . . . 273 and Complexity in Computational . . . . . . . . . . . . . . . . . . . . . . . . . 284

Symmetry and Complexity in Philosophy and Arts

329

8.1 The Philosophy of Symmetry and Complexity . . . . . 329 8.2 The Beauty of Symmetry and Complexity . . . . . . . 357 References

389

Subject Index

425

Name Index

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Introduction

Long before any science, man was fascinated by symmetry. Symmetric ornaments seem to represent perfect order, beauty, and divine harmony. Symmetrical forms and symbols are to be found in art and architecture as well as in everyday useful objects and in the mythologies of religions. Symmetry is a multi-cultural phenomenon. It spans the human life world, technology, culture and nature, and thereby searches out a unity of the natural and human sciences. Until today, symmetries have been a theme of current interest in the natural sciences. Nobel prizes in physics have been awarded for research into the symmetry of the elementary particles and the universe. Questions of symmetry are discussed in chemistry and biology as well. The various basic laws of natural sciences are derivable from unitary mathematical structures of symmetry. Modern scientists often share with the Pythagoreans of Antiquity the belief in a cosmos ordered and in balance by the highest and most perfectly mathematical laws: in the beginning, there was symmetry and simplicity. But, actually, the world is neither always simple nor in static balance and harmony. There is also steady change, dynamics, random and chaos. Symmetry is locally and globally broken by phase transitions of instability in dynamical systems generating a variety of new order and partial symmetries with increasing complexity. The states of complex dynamical systems can refer to, e.g., atomic clusters, crystals, biomolecules, organisms and brains, social and economic systems. This book analyzes dynamical balance as dynamical

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symmetry in dynamical systems. Their beauty can be visualized by computer simulations of computational systems. Symmetry breaking and the emergence of new order and chaos is an interdisciplinary challenge of nonlinear science. Linear systems are simple and can be resolved into more simple components, the effects of which are treated separately: the whole is the sum of its parts. Nonlinear systems are complex with several interacting effects of their components: the whole is greater than the sum of its parts, leading to the emergence of new structures and sometimes chaos. What is the common link between symmetry and complexity? It is symmetry breaking as the origin of dynamics and variety of forms and systems in the world. Thus, symmetry and complexity are the spirit of nonlinear science. The first chapter deals with symmetry and complexity in early culture and philosophy up to the beginning of the modern natural sciences in the Renaissance. The first section directs the reader to the use of symmetry patterns in early cultures. In the mythologies of the nature religions, gods and goddesses determine natural forces. These are replaced in the Greek natural philosophy by symmetry models for the rational explanation of nature. The mathematical formulation of the concept of symmetry was a decisive prerequisite. In the Pythagorean quadrivium — geometry, arithmetic, music and astronomy — the harmony and proportionality (1#02!.  of nature became the central concern of a mathematical philosophy. Animated by technical, aesthetic or religious motives, symmetry remains a favorite theme of Antique-Medieval mathematics. In geometry, theorems about regular plane figures and regular solids of Euclidean space are proven. In arithmetic and music, laws of proportion and harmony are explored. In astronomy, spherical models are used to explain the phenomena of the heavens. But what would harmony be without dissonance and violation of symmetry? Presumably, without charm, and boring, since it would be the everyday, the normal. But in fact we are flooded with information from the outer world in which chaotic multiplicity and change are the probable. Order and lasting symmetries seem to be the im-

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probable. The centro-symmetric spherical models of Eudoxus and Aristotle soon come into conflict with discrepant observations of the sky, which right up to N. Copernicus require increasingly elaborate geometrical and kinematic assumptions in order to save the symmetry of the model. What remains are artificial and complicated models of planets which lose their credibility because — as Copernicus in his Platonic tradition still believes — only the simple can be real, the simple which underlies the multiplicity of phenomena. The history of Antique-Medieval astronomy offers a convincing case study for investigating the interplay of original assumptions of symmetry on the one hand, and the breaks of symmetry on the basis of new realizations on the other hand, as a basic pattern of the development of research, which will be repeated on into modern physics. But not only the Greek macrocosm is determined by symmetries. The Greek atomists form a contrast to the organic conception of nature, not even comprehending life as it is holistically given, but instead wanting to trace all existent things back to the aimless collision of the smallest indivisible building blockss (x-##+) in empty space. They design a simple and linear world. Plato’s natural philosophy introduces a mathematical model of the microcosm for the first time, explaining the elements by means of the geometric symmetry of regular bodies. Modern physicists feel in the Platonic tradition, when they try to explain the world’s fundamental equations by more sophisticated mathematical principles of symmetry. Contrary to the eternal symmetries of macro- and microcosm, the mesocosm of the human world seems to be characterized by steady change, development, growth and decay, birth and death. In Antiquity, the mesoworld between the celestial spheres and atoms remains extensively separate from mathematics, since, as Aristotle says, physics deals with motion and change; mathematics with the unchanging. Therefore only the eternally recurring circular motions of the heavens are formulated mathematically and are considered divine. Only simplicity can be mathematized. The complexity and variability of nature on earth is collected, ordered, and explained by means of various qualitative principles of the natural philosophy from

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the time of the Presocratics. Examples are the change from fluid to solid states, from cold to heat as well as the organic developments and the life of humans and animals from birth to death, or the growth and decay of plants, which are understood to be goal-directed processes. Obviously, Aristotle explains change and process in nature as state transitions leading to final attractors of development. Nature is seen as a great organism whose processes are tuned to each other in balance, but which can be destabilized by catastrophes. It is a holistic and qualitative view of a dynamical world that is later on mathematized by nonlinear science. Symmetry assumptions in the astronomy and natural philosophy of Antiquity were founded on plane and solid figure symmetries of Euclidean geometry: the circle, the sphere, and regular solids. But in order to understand the laws of nature of modern physics as assumptions of symmetry, it is necessary to investigate symmetry and complexity in modern mathematics (second chapter). It was first of all algebra and group theory from the end of the 18th century that created the basis for the rigorous, general mathematical definition of the concept of symmetry (the “automorphism group”), which found its first application in the crystallography and stereochemistry of the 19th century and then in almost all parts of modern natural science. The second chapter treats the discrete symmetries of the ornaments and crystals from the point of view of group theory. But historically the group concept was first applied in the algebraic theory of equations (“Galois theory”), which can also be used to solve construction problems of symmetries from Antiquity. The continuous groups of differential geometry (“Lie groups”) became important for modern physics. Mathematically, symmetry is characterized by a group of transformations that leave certain features of a system unchanged. A system characterized by a larger group of transformations has higher symmetry. For example, a circle has more rotational transformations bringing it back into itself than a regular polygon. The symmetry of a system is broken when new features emerge that are not invariant under at least some of its transformations. An example is

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a corner appearing on a circle. Broken symmetry is a crucial concept to understand emergent phenomena of dynamical systems. If dynamical systems become unstable, they can jump from one state to another one with a completely different pattern of behavior. Such instabilities in behavioral patterns are called bifurcations. A cascade of bifurcations can lead from order to chaos which involves broken symmetries. Since Poincar´e’s discovery of deterministic chaos, the mathematical theory of bifurcation had been a basic topic of nonlinear science. The transition from order to chaos is connected with the emergence of fractals. They opened new avenues to the nonlinear world of our everyday life, which Aristotle strictly excluded from mathematization. Obviously, Euclidean geometry is unable to describe the shape of a cloud, a mountain, a coastline or a tree, because they are no spheres, cones, or circles. B.B. Mandelbrot introduced the notion of a fractal to describe these bizarre shapes with a new geometry of nature. Fractals are characterized by structures that are self-similar on various scales. They do not change their appearance when they are enlarged or diminished to arbitrary size. Self-similarity is a kind of symmetry. Therefore, fractal geometry reveals the mathematical beauty of symmetries even behind the nonlinear world of change and chaos. In the third chapter symmetry and complexity are examined in physical sciences. In classical physics, symmetry is understood to be invariance of natural laws or physical equations with respect to continuous transformation groups. This establishes that a natural law is objectively valid — independently of changes in the position or the point in time of its examination by an experimenter or observer. Symmetries correspond to the freedom to choose the system of coordinates of the observer. All natural laws are invariant with respect to translation, rotation and reflection of the system of coordinates. The symmetries are global in the sense that natural laws are invariant with respect to equivalent transformations for all points in the space. With the concept of an inertial system, the principle of relativity in classical physics is defined by Galilean invariance.

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The joining of electricity, magnetism and optics succeeds mathematically in electrodynamics, which can be formulated invariantly to the Lorentz group. J.C. Maxwell had already predicted that light could be reduced to the electromagnetic field. In fact, the wave equations for the phase velocity of light were derived from Maxwell’s equations and confirmed experimentally by H. Hertz. This brought about, for the first time, a unification of the phenomena of nature in mathematical physics. Further, Maxwell’s electrodynamics constituted, for the first time, a physical theory for which the modern physical concept of symmetry could be expressed with precision. The electromagnetic field has both “global” symmetry in accordance with the Lorentz invariance — in which all space-time coordinates can be altered — and also “local” symmetry in the sense of a gauge field. Even in 1923 H. Weyl characterized the theory of the electromagnetic field as “the most perfect piece of physics that we know today.” The application of physical symmetry concepts is closely connected with mathematical developments in algebra and geometry in the 19th century. In 1872, F. Klein, in his well-known “Erlanger Program”, had characterized and classified various geometric theories by means of continuous transformation groups. Under the direct influence of F. Klein, E. Noether in 1918 expanded this program for physics and showed how physical conservation principles can be characterized by means of transformation groups and traced back to space-time symmetries. The mathematical variational and extremal principles are of central significance for the physical concept of symmetry. Historically they arise out of the background of Leibniz’s natural philosophy of pre-established harmony and are determined by the search for a coherent basic principle of nature. For the symmetry concepts of modern physics, first to be discussed is the Lorentz invariance of the force-free 4-dimensional Minkowski space of the special relativity theory, in which two observers have constant velocity relative to each other. It is a matter of global symmetry, since the transformations refer to all space-time coordinates. The local Lorentz invariance of the general theory of

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relativity has to fulfill much stricter requirements. Now the physical laws have to keep the same form even when every single point is transformed independently of all the others. This mathematical sharpening has the same significance as the physical postulate that two observers may also increase their speed relative to each other, that is, that gravitational forces come into play. A key concept of modern physics is to describe the introduction of fundamental forces mathematically by means of the transition from a global symmetry to a local symmetry. Relativistic cosmology applies the differentialgeometric theory of symmetrical spaces, which E. Cartan had developed in the twenties of the last century from the theory of spaces with constant curvature (B. Riemann, S. Lie, H. von Helmholtz et al.). The solutions to Einstein’s gravitation equation allow for varying symmetrical models, e.g. that the spatially homogeneous universe expands isotropically, or collapses, or oscillates. The Platonic belief in a macroscopically symmetrical cosmos on the whole is once more urgent, even if mathematically more complicated and no longer in the form of the ancient harmony of the spheres. In this context it is noteworthy that D. Hilbert derived the relativistic equations together with Mie’s electrodynamic equations from a variation principle independently of A. Einstein. This was the first attempt at a unification of the fundamental forces in modern physics, which, however, succeeded only later under the conditions of quantum mechanics. Mie’s “theory of matter” from 1912 is also an important document of nonlinear science. He suggested a nonlinear augmentation of Maxwell’s electromagnetic equations out of which elementary particles (e.g. the electron) would arise in a natural way. Although Mie’s theory was later on refuted by experiments, Einstein was deeply convinced that elementary particles must be represented by exact solutions of nonlinear partial differential equations founding a unified field theory of matter. Therefore, Einstein never accepted the quantum mechanical approach. Next to the relativity theory, quantum mechanics is the framework theory of modern physics. Recall first, the spherically symmetrical characteristics of the early atom models by which N. Bohr

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explained the discontinuous spectral lines of the chemical elements. According to these models the electrons move on fixed paths around the nucleus, like the ancient planets. By analogy to the development of the Aristotelian planet models, Bohr’s originally simple atom model must also be assimilated by means of certain artifices (in this case the quantum numbers) to the complicated relationships that reveal themselves for various elements in the laboratory. Thus the basic equation of quantum mechanics, the Schr¨ odinger equation, exhibits two kinds of symmetry. It can be assumed, at least approximately, that the electrons have a spherically-symmetrical potential energy for which no direction is distinguished, so that the corresponding Hamiltonian function is invariant towards the symmetry operations of the sphere. Further, electrons are indistinguishable (G.W. Leibniz: “indiscernibiles”) in the sense that it makes no difference for the Hamiltonian function whether the positions of the electrons are exchanged and permuted. This permutation symmetry is closely connected to Pauli’s principle of exclusion, according to which two electrons do not have the same quantum number. With reference to Leibniz’s principle of indistinguishability, Weyl also speaks of the Leibniz-Pauli-principle of symmetry. Mathematically, the states of quantum systems (atoms, electrons, etc.) can be represented by vectors of a Hilbert space. The symmetry (automorphism group) of the Hilbert space formalism of von Neumann’s quantum mechanics has been investigated especially by Weyl, E.P. Wigner et al. since the end of the twenties of the last century, and related to the unitary transformations of the Hilbert space. The space-time symmetries are determined by a subgroup that can be represented by the Galileo group of classical physics. But the decisive distinction from classical physics (and from the relativity theory) is that quantities in (von Neumann’s) quantum mechanics that can be used as measuring quantities (“observables”), are not exchangeable (“commutative”). This group-theoretic characteristic of quantum systems comes to expression in terms of measurement technique in that it makes a difference in what sequence quantities are measured. There is an aggravating disadvantage in von Neumann’s quantum

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mechanics in that no classical (“commutative”) observables are allowable. But how can the demonstrable existence of commutative quantities in the quantum realm, e.g. spin or rest masses, be understood (the problem of “superselection rules”, respectively “violation of the principle of superposition”)? How can the measuring process of quantum mechanics be described? It is an interaction between a classical measuring instrument and a quantum system. Hence, generalized formalisms of quantum mechanics (e.g. C∗ -algebra, quantum logic) have been developed which also admit classical observables and are determined by symmetry groups. Thereby, from the perspective of uniting natural-science theories, a framework is built in which classical systems, quantum systems (in von Neumann’s sense), generalized quantum systems and thermodynamic systems can be examined. The historical development of physical theories is defined by a step-by-step unification. I. Newton achieved the first great unification when he traced trajectories of free-falling or projected terrestrial bodies to the same conformity with celestial bodies. Next came Maxwell who based electricity, magnetism and optics on electrodynamics. Newton’s gravitation theory had to be replaced by Einstein’s general relativity theory, and Maxwell’s electrodynamics had to be enlarged by special relativity theory, and quantum mechanics by quantum field theories, especially quantum electrodynamics. The first step in that direction was made, already in 1928, by P.A.M. Dirac, when he predicted — with a relativistic quantum mechanical wave equation — an anti-particle to the electron (“positron”), which in fact was discovered in 1932. The breakthrough for the theory of the electromagnetic interaction of electrons, positrons and photons came at the end of the forties with the work of R.P. Feynman, J.S. Schwinger et al. The group of (unitary) transformations, which leaves the laws of this theory invariant, has the so-called U(l) symmetry. Physically these transformations correspond to a process in which a particle is transformed from one state into another without changing its identity. Thus an electron can go to another energy state by sending out a photon. The initial state

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and the final state do not differ in electrical charge, and the transitions between the two states by means of the emission of photons can be represented by an 1 × 1 matrix. Here we encounter an entirely new kind of symmetry that is no longer a matter of “external” space-time symmetries such as reflections, rotations, translations, etc. but instead of “inner” (intrinsic) symmetries of transformations of matter. Another inner symmetry is isospin symmetry establishing a connection between the nuclear particles proton and neutron and the nuclear forces. Both particles possess the same spin and almost the same mass, so that they — as W. Heisenberg recommended — can be conceived of as two possible states of a particle, the nucleon. Transitions from one state to another are described by means of the so-called SU(2) group. While neutrons and protons are the only particles with strong interaction which are stable for a long time, with today’s high-energy technology a multiplicity of very shortlived particles with strong interactions (“hadrons”) can be produced. This “zoo” of hadrons, which was discovered in the fifties of the last century more or less by chance, was finally derived from a unitary symmetry structure. Since then all hadrons are built up out of subelemental “quarks”. Their strong interaction can be described by means of a SU(3) symmetry. This theory is built up according to the model of quantum electrodynamics and is called quantum chromodynamics since the strong force does not come into play between electric charges but between so-called color charges of the quark. Today a fourth fundamental force of nature, weak interaction, is distinguished from gravitational energy and the electromagnetic and strong interactions. While gravitational energy and electric and magnetic phenomena are well-known through everyday experience, nuclear forces and weak interactions can be observed only by means of the modern technologies. Thus the weak interaction is responsible for the β-decay. This force proves to be especially critical for the discussion of left-right symmetry (“parity”) in nature. Namely, experiments at the end of the fifties of the last century confirm that for the weak interaction in the case of β-decay of 60 Co — in contrast

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to the other three basic forces — neither parity (P ) nor reversal of charge (C = charge) is a symmetry operation; rather, it is only the combination CP T with the operation T (T = time) for reversal of time (CP T theorem). After Weyl, already in 1918, had attempted a unification of the electromagnetic forces with gravitation, S. Weinberg, A. Salam, S. Glashow et al. succeeded in 1967 at uniting the electromagnetic and the weak interactions. Both forces derive from the so-called SU(2) × U(1) symmetry, which nevertheless is present only in extremely small spatial ranges and is broken already in spacings in the size range of the nuclear radii. While the interactions of gravitation and electromagnetism are spatially limitless and therefore are transmitted by massless particles (graviton, photon), the weak interaction (as well as the strong one) extends only for short distances. Therefore the breaking of the SU(2) × U(1) symmetry becomes observable when the intermediary particles (except for the photon) suddenly take on large masses. So far, the uniting of all four fundamental forces in one symmetry group has only been carried by assumption in mathematical models. Thus the SU(5) group, which is the smallest simple group which combines the SU(3) symmetry and SU(2) × U(1) symmetry, proves to be especially interesting for describing strong, weak and electromagnetic interaction. This theory predicts a tiny extension in which there are no fundamental differences between quarks and leptons or between the strong, weak and electromagnetic interactions, but instead only one kind of matter and only one fundamental force. In cosmic evolution the SU(5) symmetry would have existed a fraction of the first second after the big bang. The rest of the spatiotemporal evolution of matter consists of breaking of the basic symmetry and the appearance of partial symmetries with varying particles and fundamental forces — a cosmic kaleidoscope whose symmetries depend on spatial orders of magnitude and temporal developmental phases. Certain rules of conservation then become time-dependent, so that the decay of the proton is among the most spectacular prognoses of this theory and is sought after in expensive experiments. Finally, the superstring theory strives for a modern theory of the

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Platonic supersymmetry, in which all four fundamental forces are indistinguishable. The cosmic steps of symmetry breaking initiate the expansion of the universe. Thus, they determine the cosmic arrow of time breaking the symmetry of time. The cosmic distinction of a time direction seems to correspond to the second law of thermodynamics demanding increasing entropy for (isolated) dynamical systems until a maximal value of thermal equilibrium is reached. On the other hand, all fundamental laws of classical, relativistic, and quantum physics are invariant with respect to the reversal of time. But there is no contradiction: The thermodynamical arrow of time is a macroscopic feature of complex ensembles (e.g. gas) with elements (e.g. molecules) which still obey laws of interaction with invariance of time on the microlevel (microreversibility). Nevertheless, according to L. Boltzmann, the second law of thermodynamics indicates a direction of time from order to chaos, noise and decay of order. But there are not only collapses of stars to black holes, death of organisms, and dissolution of energy in the expanding universe, but also the birth of new stars and life. Since the ancient philosophers it has been a fundamental problem to understand how order arises from complex, irregular, and chaotic states of matter. There is no contradiction to the second law of thermodynamics that is restricted to isolated systems without any interaction with their environment. Modern thermodynamics describes the emergence of order by the mathematical concepts of nonlinear science. New dynamic entities emerge from phase transitions of complex dynamical systems with underlying partial differential equations (PDE). We distinguish two kinds of phase transition (self-organization) for order states: conservative self-organization means the phase transition of reversible structures in thermal equilibrium. Typical examples are the growth of snow crystals or the emergence of magnetization in a ferromagnet by annealing the system to a critical value of temperature. Conservative self-organization mainly creates order structures with low energy at low temperatures that are described by a Boltzmann distribution.

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Dissipative self-organization is the phase transition of irreversible structures far from thermal equilibrium. Macroscopic patterns arise from the complex nonlinear cooperation of microscopic elements when the energetic interaction of the dissipative (“open”) system with its environment reaches some critical value. The stability of the emergent structures is guaranteed by dynamical balance of nonlinearity and dissipation. Too much nonlinear interaction or dissipation would destroy the structures. A typical example is the laser light emerging from a complex system of nonlinearly interacting photons at a critical value of energy pumping. But even the flame of an ordinary candle is a simple example of nonlinear dissipative selforganization. The heat from the flame diffuses into the wax, vaporizing it at a rate required to provide fuel for the flame. The stability of the flame is being enabled by the dynamical balance of thermal diffusion and nonlinear energy release. The phase transitions of dissipative complex systems are mathematically described by nonlinear partial differential equations. The emergent structures are their solutions. In a more qualitative way we may say that old structures become unstable and break down by changing conditions (“control parameters”). On the microscopic level, the stable modes of the old states are dominated by unstable modes. They determine the macroscopic order and pattern of the system (“order parameter”). There are different final patterns of phase transitions corresponding to different attractors. Different attractors may be pictured with a stream, the velocity of which is accelerated step by step. A stream is a complex system of nonlinearly interacting molecules on the microscopic level, generating fluid patterns on the macroscopic surface. At a low degree of velocity a homogeneous state of equilibrium is shown (“fixed point”). At a higher degree the bifurcation of two or more vortices can be observed corresponding to periodic and quasi-periodic attractors. With increasing velocity we get a bifurcation tree of increasing complexity. Finally the complex order decays into chaos as final attractor of the fluid dynamics. These steps of phase transition mostly involve broken symmetries. Thus,

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generally speaking, the evolution of matter is caused by symmetry breaking. In a more mathematical way, the microscopic view of a complex system is described by the nonlinear partial differential equation of a state vector where each component depends on space and time and where the components may denote, e.g. the velocity components of a fluid or a temperature field. In a linear-stability analysis, we can distinguish the stable and unstable modes by eigensolutions of a corresponding eigenvalue equation. At critical values of a control parameter, the unstable modes of some few components can increase exponentially to macroscopic scale and dominate all the other stable ones. Thus, they become order parameters. The adaptation of the stable modes to the dominating unstable behavior is mathematically described by the fact that all terms of stable modes can be expressed by the few terms of unstable ones. Consequently, the millions of equations determining the single components of a gas, fluid or light on the microscopic level can be eliminated and replaced by some few macroscopic equations of order parameters: it is not necessary to know all microscopic states of a complex system, in order to understand its dynamics. Chemistry is the bridge between the microworld of atoms and the mesoworld of living organisms. The fourth chapter analyzes symmetry and complexity in chemical sciences. Symmetries, dissymmetries and asymmetries of molecular structures, orbits and crystals can be explained, using the methods of group theory. Chemical structures fascinate us with the beauty of their symmetries. Again, molecules can be considered as emergent entities, generated by the underlying rules of nonlinear partial differential equations. In this case, atomic elements organize themselves into molecules according to the laws of quantum chemistry. It thus becomes clear how, at a certain stage of development of matter, new structures of symmetry, dissymmetry and asymmetry occur which do not yet exist on the level of the elementary particles. For example, if one views a crystal in the atomic size realm, only the symmetry of the individual atoms becomes distinct. On the larger scale, the binding forces appear, breaking atomic

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symmetry but building up the new molecular symmetry of the crystal lattice. The old problem of left-right symmetry was already investigated in the 19th century for crystals in relation to polarized light, and led to stereochemistry. Chirality means molecular symmetry breaking of left–right with sometimes dramatic consequences for living organisms. The question arises if chirality is caused by a deeper cosmic symmetry breaking of parity violation with the emergence of weak forces or if it is a new phenomenon on the molecular level. On the level of nanomolecules, chemistry becomes a nonlinear complex science. These macromolecules are the building blocks of new materials. Gigantic chemical structures with beautiful symmetries emerge from the underlying rules of nonlinear interactions. Complex systems of the nanoworld and self-constructing materials are challenges of key technologies in the future. In the fifth chapter we consider symmetry and complexity in life sciences. In biochemistry the symmetry principles and their violations are today a widespread research field. The determination of a molecular chain direction is often important, for example, to reach an unequivocal gene coding for the DNA molecules. It seems to be characteristic that organisms prefer the middle realm of the transition from highest symmetry (e.g. crystal) to perfect chaos and randomness (e.g. gases). In the 19th century, L. Pasteur had advanced the thesis that dissymmetry was typical for life. We find this opinion reflected in literature in Thomas Mann’s “The Magic Mountain”: Hans Castorp, gazing at snow crystals, surmises: “Life shuddered before this exact correctness.” Indeed the dynamics of life processes can be described by means of symmetry breakings, as in the case of cell division. On the other hand, it is precisely living creatures, as selfreproducing systems, that display particular temporal developmental symmetries, which show themselves in the course of generations as the periodic recurrence of the cyclical course of growth of individuals. In today’s biology one also speaks of the “cell cycle” and the “hypercycle”.

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Morphological symmetries of plants and trees are striking to anyone. Movement in all directions in the isotropic medium of water is made possible by the central symmetry of many sea organisms, while the arrow form of the fish is expedient for a goal-oriented movement. Under specific environmental conditions symmetrical forms offer selective advantages, which have been imitated and further developed by modern technology (e.g. in building cars, airplanes and rockets). The bilateral symmetry of higher animals seems to solve the problem of optimal mobility with simultaneous balance of forces, while this organizational principle is followed only partially in the anatomy of the inner organs. Thus, although we have two lungs, we have only one left-leaning heart. In the macroscopic realm also it comes down to a layering and breaking of varying symmetries. The human brain is a remarkable complex organ with bilateral symmetry, but also local symmetry breaking. In the framework of complex systems the emergence of life is lawful in the sense of dissipative self-organization. Only the conditions for the emergence of life (for instance on the planet earth) may be contingent in the universe. In general, biology distinguishes ontogenesis (the growth of organisms) from phylogenesis (the evolution of species). In any case we have complex dissipative systems the development of which can be explained by the evolution of (macroscopic) order parameters caused by nonlinear (microscopic) interactions of molecules, cells, etc., in phase transitions far from thermal equilibrium. Forms of biological systems (plants, animals, etc.) are described by order parameters. Aristotle’s teleology of goals in nature is interpreted in terms of attractors in phase transitions. Phase transitions often involve broken symmetries and the emergence of new entities. Spencer’s idea that the evolution of life is characterized by increasing complexity can be made precise in the context of dissipative self-organization. It is well known that A.M. Turing analyzed a mathematical model of organisms represented as complex cellular systems. G. Gerisch, H. Meinhardt et al. described the growth of an organism (e.g. a slime mould) by evolution equations for the aggregation of cells. The nonlinear interactions of amoebas cause

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the emergence of a macroscopic organism like a slime mould when some critical value of cellular nutrition in the environment is reached. The evolution of the order parameter corresponds to the aggregation forms during the phase transition of the macroscopic organism. The mature multicellular body can be interpreted as the “goal” or (better) “attractor” of organic growth. Multicellular bodies, like genetic systems, nervous systems, immune systems, and ecosystems, are examples of complex dynamical systems, which are composed of a network of many interacting elements. Even the ecological growth of biological populations may be simulated using the concepts of nonlinear science. Ecological systems are complex dissipative systems of plants or animals with mutual nonlinear metabolic interactions with each other and with their environment. The symbiosis of two populations with their source of nutrition can be described by three coupled differential equations which were already used by E. Lorenz to describe the development of weather in meteorology. In the 19th century the Italian mathematicians A.J. Lotka und V. Volterra described the development of two populations in ecological competition. The nonlinear interactions of the two complex populations are determined by two coupled differential equations of prey and predator species. The evolution of the coupled systems has stationary points of equilibrium. The attractors of evolution are periodic oscillations (limit cycles). Evolution is obviously a process of phase transitions with symmetry breaking and the emergence of new molecular structures, organisms, species, and populations. Atomic elements organize themselves into molecules. Out of the nonlinear interactions of chemical molecules emerge the proteins acting as catalysts and enzymes in biochemical cycles. Biochemical cycles support the replications of biomolecules, underlying cellular reproduction. Cellular bodies are arranged as organs, which join together to form organisms. They go on to interact nonlinearly with each other to become components of species and the biosphere. Nervous systems and brains generate new patterns of behavior by learning and adapting to changing conditions of environment. Each of these hierarchical levels from molecules to

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the human mind is thought to be characterized by some order parameters obeying nonlinear rules. In this sense new locally stable entities emerge as solutions from nonlinear differential equations underlying each level in a dynamical hierarchy of life. In the sixth chapter we discuss symmetry and complexity in economic and social sciences. In the framework of complex systems the behavior of human populations is, again, explained by the emergence of order parameters generated by nonlinear interactions of human beings or human subgroups (states, institutions, etc.). Social or economic order is interpreted via attractors of phase transitions involving broken symmetries. Symmetry is understood as social balance and economic equilibrium. We distinguish the microlevel of individual decisions and the macrolevel of dynamical collective processes in a society. But people are not atoms or molecules determined by wellknown microscopic equations of nonlinear interactions. People are guided by their individual intentions, feeling, and thinking. In principle, we could consider their individual brain dynamics determined by some nonlinear partial differential equations. Intentions, feelings, and thoughts are new states (order parameters) of the brains, emerging from their nonlinear dynamics. But, until today, we have only had some very rough ideas of their underlying equations. Even if we had them, then their complexity would prevent us computing solutions for predicting individual behavior in the future. For practical reasons, a probabilistic description of the individual decision processes is preferred, neglecting individual details of behavior. It takes into account the trend forming influence of social macrovariables on the decision making of individuals. Examples are stock and flow variables in economy, social attitudes in sociology, or political opinions in politics. They are comprehended in socioconfigurations representing the collective macrostate of the whole social system at a certain time. Individual decisions and actions are considered as individual transitions of microstates (e.g. change of a political opinion before an election) that are described by probabilistic processes. So the individual freedom of decision and action is not restricted. The dynamics of a social system is modeled by the

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probabilistic transition rates of its social macrovariables. They are the constitutive elements for setting up a macroscopic equation of sociodynamics. It is a stochastic partial differential equation (master equation) describing the time-depending evolution of a probabilistic distribution function over the socioconfigurations of a social system. An example of application is symmetry and complexity of migration in a society. We can distinguish typical scenarios of migration like, for example, stable balance of ethnic groups in a society or isolation in ghettos with dangerous instabilities. Social symmetry is a basis for social peace. In the framework of complexity, scenarios of migration are new macrostates of sociodynamics, emerging from social phase transitions. They correspond to attractors of nonlinear sociodynamics such as stable fixed points, oscillations or chaos that are well-known from other applications of nonlinear dynamics. Economy opens deep insights into symmetry and complexity of sociodynamics. From a qualitative point of view, Smith’s model of a free market was already an example of a dynamical symmetry, emerging from phase transitions of economic self-organization. A. Smith underlined that the good or bad intentions of individuals are not essential. In contrast to a centralized economical system, the balance of supply and demand is not directed by a leader, but is the effect of an “invisible hand” (Smith), i.e. nothing but the nonlinear interaction of consumers and processors. Later on, the Lausanne school explicitly used mathematical terms of thermodynamics like, for example, equilibrium to describe economic balance. The recent interest of economists in nonlinear dynamics is inspired by the dynamics of globalization and the unstable attractors of oscillation and even chaos. Symmetry and complexity are not always obvious in the dynamic processes of the universe, life, and human society. During the last years, they could only be detected, computed, and visualized by the increasing power of high-speed computers. Thus, in the seventh chapter, we consider symmetry and complexity in computer science. Fractal geometry was a first example of computational application. Mandelbrot’s detection of the Mandelbrot set was a historical mile-

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stone in the history of computational mathematics. Its self-similar symmetries of complex fractals could only be illustrated in computer graphics. They reveal a hidden virtual world of mathematical beauty behind chaos. Phase transitions of nonlinear dynamics can be animated in visual computational processes. Sometimes their features were detected in computer experiments before they were analytically proved from the underlying equations. Computer experiments enlarge mathematical imagination and thought experiments with high technology. Multimedia and Internet deliver new tools of mathematical exploration. The traditional analytical approach of nonlinear science is sometimes replaced by computer experiments, because it is easier to find features of complex dynamical systems by computational visualization than by analytical solution of the underlying nonlinear equations. According to the principle of computational equivalence, every nonlinear dynamical system corresponds to an appropriate computational system. Prominent examples are cellular automata which, in principle, can simulate all kind of symmetries and complexities which have been considered in this book. Nevertheless, computer experiments are not mathematical proofs. The standards of mathematical rigor have not been changed since the days of Euclid and Plato. Cellular Nonlinear Networks (CNN) make it possible to prove exact indices of symmetry and complexity in nonlinear science. Furthermore, they are no longer only simulations on standard computers like cellular automata. In the age of miniaturization, they can be built as high-speed chips. Computational systems are not restricted to human technology. According to the principle of computational equivalence, any nonlinear dynamical system can be understood as a computational system. In this sense, atomic, molecular or cellular systems are computational systems with phase transitions as computational processes. They are coded by the rules of atomic, molecular or cellular interaction. Like all nonlinear complex systems, they can have chaotic or even random dynamics that cannot be forecast in the long run. In short, complex computational systems may not be computable, although

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we know all the basic laws of their locally interacting elements. Is the universe in the end nothing more than an expanding complex quantum computer, generating symmetries and symmetry breaking? In this case, phase transitions of matter are coding transformations of quantum information that is reduced to quantum bits as digital building blocks. Symmetry and complexity are explained by an ultimate duality of binary units and their superpositions — the yang and yin of the quantum universe. In the last eighth chapter we discuss symmetry and complexity in arts and philosophy. What means beauty of symmetry and complexity today? It means the beauty of a nonlinear world. The unity of the experience of nature, art and religion in the myths of early peoples is surely lost. The Antique unity of the mathematical teaching of harmony, natural philosophy and art has also dissolved since the Renaissance. In art history (e.g. in Classicism), to be sure, there was always resonance to the Antique conception of art. But it has the effect of reciting old texts; it reflects a merely partial taste, and it no longer mirrors the cosmos and its laws. Art and science in modern times have differentiated themselves from each other into distinct media of life experience. The varied multiplicity of contemporary artistic attempts corresponds entirely to the complexity of the modern life world. In contrast to the Antique-Medieval life world, whose aesthetic forms depended on the potentialities of their handwork, we live today in a civilization that is determined by industry, technology and science, which co-determine our action, thinking and sensibility. The “Bauhaus” of the twenties of the last century was an artistic movement of modernity that tried to develop a new world of form under the conditions of technology and industry. The community of handworkers and artisans in the cathedral associations of builders and artisans of the Middle Ages and the artist engineers of the Renaissance led W. Gropius to the thought of uniting art and technology again under the conditions of modern industrial society. Architects, painters, graphic artists, sculptors, form designers, etc. were to work in coordination and by division of labor, seeking forms that

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would provide practical and functional solutions to people’s needs, whether it was a matter of furniture, dishes, homes, office buildings, factories, streets or leisure facilities. The measure, the “logos,” of this art is the human being with his needs in the technical-industrial life world. In the age of globalization mankind is growing together under new technological, social and economic conditions. Cultural symmetry is a challenge in a world of cultural diversity. Cultural asymmetries are dangerous for a peaceful balance in the world. Symmetry means unity. In science unified theories are explained by mathematical symmetries. Are they only theoretical tools used in order to reduce the diversity of observations and measurements to some useful schemes of research or do they represent fundamental structures of reality? This has been a basic question of philosophy since the Antiquity. Empirical results of modern science confirm that symmetries are not only mathematical imaginations of our mind. They dominated the universe long before mankind came into existence: in the beginning there was a dynamical symmetry expanding to the complex diversity of broken symmetries. Phase transitions involve the emergence of new phenomena on hierarchical levels of atoms, molecules, life, and mankind. They have not been determined from the beginning, but depend on changing conditions that happen more or less randomly. It is a challenge of nonlinear science to explore their fascinating symmetry and complexity.

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Chapter 1

Symmetry and Complexity in Early Culture and Philosophy

Regular patterns and symmetries are used in all known cultures. They recur continually as ornaments on finery, cult objects, and everyday objects. Regular forms in crafts and architecture prove to be more stable, more economical in the use of materials, more distinct, simpler, easier to reproduce and to hand down to succeeding generations and — not least — of great aesthetic charm. Since the earliest times nature itself has manifestly been a model, evincing regularity in sundry forms and occurrences — from the minerals and plants, to the anatomy of living beings, to the regularly recurring stellar constellations. The old high cultures, as well as the still extant cultures of various ethnic groups, e.g. Asia, Africa, North and South America, use certain symmetrical patterns to give order to nature and their life-world. Modern natural science and technology were not the first to achieve this.

1.1

Cultural and Cosmic Harmony

Anyone who seeks out the Navajo Indians in the North American Southwest is astonished by the symmetry forms that govern their culture [1.1]. What is so striking, is not so much the regular patterns and ornamentations of their artful textiles or ceramics, but rather the Navajo’s use of symmetries in the ordering of their rituals and myths, in short, their “Weltbild” (image of the world). In this context “Weltbild” is to be understood literally, as the representation 23

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of their life-world and mythologies of nature, not as a philosophical doctrine that has been derived from particular principles. Here we have in mind the sand paintings of the Navajo, which represent a “Weltbild” that has a particular ceremonial intention. These paintings are produced from pulverized sandstone in red, yellow or white, the pigments of cornmeal, plant pollens or flower petals. Fig. 1 shows the sandpainting of the “rainbow people” who are meant to be mythological representations of rain and light. In the centrally-symmetric hub of the cosmos there is the source of life — water, bordered by four rainbow bands in the four directions of the heavens or the four directions of the wind. The four sacred plants — maize, beans, pumpkin and tobacco — grow out of the center. Two masculine (round-headed) and two female (angular-headed) rainbow people are situated behind each of the four rainbow bands. The enclosing circle represents the goddess of the rainbow who protects the life-world of the Navajo. Two flies serve as messengers or sentinels. This sandpainting displays an abundance of superimposed symmetries. The centrally-symmetric square has all the reflection symmetries of the diagonals and lateral bisectors. Therefore the center, with water as the basis of life, produces a statically resting effect, which is emphasized by its black color. However, the surroundings of this center display only rotational symmetries. Thus the foursome groups of rainbow people in the four directions of the sky can be joined to each other by quarter-turns of the circle around the center. The feather decorations and the outstretched arms have the effect of small directional arrows and provide an impediment to reflection symmetry at the diagonals and lateral bisectors of the square, which would otherwise convey an impression of resting stasis. Therefore the rainbow people travel around the center in the direction of the sun. This dynamic impression is further underscored by the rainbow goddess, whose arc, with its inscribed head and feet, has the effect of a torque vector. Therefore the message of this world picture is clear: the element water is at the center, and all natural and life processes revolve around it.

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Fig. 1.

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Symmetric sandpainting of the Navajo Indians [1.2]

In the world of the Navajo, symmetry does not have a separate aesthetic, religious or technical purpose. Their central concept is called “h´ ozh´o,” which is often translated as beauty, but cannot be separated from health, happiness and harmony [1.3]. The life and culture of the Navajo is based on a unity of experience that is expressed as “h´ozh´o.” “H´ ozh´o” is the intellectual concept of order, the emotional state of happiness, the moral value of the good, the biological condition of health and well-being and the aesthetic charm of balance, harmony and beauty — a projection of wishes, ideas and experiences which is found also in other cultures of America (e.g. Aztec world map).

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An Asian example is the “Weltbild” of the Jaina from the Indian sphere of influence. In addition there are miniatures of the Jaina K¯ alpasutra, which were not documented until the 15th and 16th century, but go back to very old sources. Represented here is the Samavasarana that the gods erect for every Jina [1.4]. It is a round or square space, surrounded by three circular walls with four gates to the regions of the world. The Jina sits in the center and meditates or preaches, magically quadrupled on lion thrones under a tree. The tranquility and composure that these images radiate is achieved formally by means of central symmetry and reflection symmetry. This impression is strengthened by the fourfold point reflection of the Jina at the midpoint of the four corners of the miniature. The inner wall consists of jewels and is decorated with pinnacles of rubies; the middle one is made of gold. In various cultures symmetry characteristics are used cabbalistically, i.e. with words or letters, in order to gain insights by means of geometrical arrangements and combinations. A noteworthy example of Indian cabbalistics is the Scr¯ıcakra. It consists of a diagram made of 43 triangles, called Meru (Fig. 2). It is surrounded by an 8-petaled lotus and a 12-petaled lotus, which are again enclosed by four circles. It is characterized by four T-formed structures at the sides of the outer square frame. Instructions are indicated for the two lotus blossoms, and most especially for the potential combinations inherent in the Meru diagram. Proceeding from the outside to the inside, one distinguishes a 14-pointed star, an outer 10-pointed star and an inner 10-pointed star. The center is a triangle which is also the structural principle of the diagram. It is interpreted either according to the nature mythology of the three Vedic lights — the moon, the sun and fire; or linguistically in accordance with the sounds of various syllables; or anthropologically in accordance with the trinity of thought, voice and body. Although no historical explanation can be cited, the Buddhist diagrams from India show a strong similarity to Chinese mirrors from the early Han period. These mirrors with their characteristic T-, Land V-formed corners (“TLV mirrors”) are interpreted unequivocally

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Fig. 2.

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Symmetric Meru diagram from India [1.5]

as cosmological. The animals of the wind directions (dragon, bird, tiger and a turtle with snakes twisted around it) are frequently portrayed with a swarm of legendary animals and demons. Later these mirrors are further developed into compass cards, which employ — along with the four cosmic animals — 12 cyclical animals (in analogy to the l2-day week), 28 star pictures as constellation figures, etc. The use of symmetries in China has a special charm. Very early on, they were interpreted in the framework of philosophy of nature. In the “Book of Changes” (the I Ching), from the 8th century B.C., four pairs of natural opposites — forces and elements such as heavenearth, fire-water, lake-mountain and thunder-wind — were symbolized by eight triagrams arranged according to reflection symmetry (Fig. 3a). They were also represented on coins (Fig. 3b). According to a later interpretation in the “Great Treatise” (Ta Chuan), these symmetries derive from the duality of light (—) and dark (– –), yang and yin. The use of it as an oracle book for divining decisions by

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Symmetry and Complexity

Fig. 3a.

Chinese symmetries of the I Ching [1.6]

Fig. 3b.

Chinese symmetries on coins [1.6]

combinations of yes (—) and no (– –) approaches contemporary ideas of information theory, but remains hypothesis. According to the Chinese conception, symmetry also had to do with rightful proportional relationships, which in a large central

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apparatus of state and government like the Chinese imperial realm had to be regulated in detail. There is much similarity here to the known mathematical documents from Babylon. Particular forms and proportional relationships are designated for cultic and ritual purposes. In Egypt pyramids served as monuments to the dead. In the Indian Salvasutras the symmetry of the Hindu altars was calculated with the use of Pythagorean number triads. In the astronomy of these cultures periodic celestial motions were registered and then viewed astrologically in connection to the course of lives on earth. But in Greek mathematics something happened that was completely new. Symmetries were made the systematic object of mathematical research. It is probable, to be sure, that the early Pythagoreans drew their basic mathematical knowledge from Egyptian and Babylonian sources. There, however, individual proportions remained related to technical-practical purposes. They were not based on proofs but, at best, determined by approximate reckoning procedures. Yet the Pythagoreans made the mathematical concept of harmony the central theme of their philosophy, which is based on geometry, arithmetic, music and astronomy. In Plato’s time a general mathematical doctrine of proportions was developed, and it remained the mathematical basis of the concept of symmetry until the beginning of the modern era. In all known cultures the circle is the symbol of perfection or of eternal recurrence. While it displays infinitely many symmetries resulting from random rotations and reflections at the diameters, the regular polygons inscribed in it possess a finite number of symmetries. If one connects the vertices of regular polygons with the center, one derives directional indicators that are useful for geodetic and astronomical orientation. The technical application as wheel was an important innovation of mankind. Appropriate connections of the vertices render aesthetically charming star patterns that are also often used as ritual symbols. In architecture, centrally symmetrical edifices still play a great role. Accordingly, the Pythagoreans set themselves the objective of constructing regular polygons with mathematical precision using the

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Fig. 4.

Euclid’s construction of the 15-angled polygram [1.7]

compass and the ruler. This, then, is not only a matter of nearregularities which are found approximately by trial and error and which could be altogether adequate for technical purposes. It has come to be a matter of mathematical symmetry, which is provable and exists independently of technical application and perception, as an ideal form, as Plato will later say. The Pythagorean doctrine of the regular polygons has been handed down in Book IV of Euclid’s “Elements”. It deals with the construction of the 3-, 4-, 5-, 6- and 15-angled polygons (Fig. 4). The symmetries of regular polygons were clearly of considerable practical interest. That is documented by the practical geometry of Medieval Arab mathematicians for craftsmen. The great astronomer Ptolemy recognized trigonometric significance of certain polygons and applied their construction for chord tables in astronomical calculations. The mathematical question: which regular polygons can be constructed with the compass and the ruler, was answered fully by the young C.F. Gauss at the end of the 18th century. He found that a regular n-sided polygon can be constructed with a compass

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and a ruler if, and only if, the uneven prime factors of n are differing k Fermat prime numbers pk = 22 + 1. Fermat’s prime numbers are po = 21 +1 = 3, p1 = 22 +1 = 5, p2 = 24 +1 = 17, p3 = 28 + 1 = 257, p4 = 216 + 1 = 65 537. Since 7 is not one of Fermat’s prime numbers, in principle a heptagon cannot be constructed with a compass and a ruler. Since the prime factors of 9 are not different, the same conclusion holds for the regular nonagon. This negative argument was absolutely new compared with the geometry of Antiquity. New also was Gauss’ positive deduction that it must be possible in principle to construct the regular 17-sided polygon with a compass and a ruler. But this conclusion was not reached by means of experimenting. Instead, the constructibility of the regular 17-sided polygon was predicted on the basis of the algebraic analysis of the problem. In 1832 F.J. Richelot constructed the regular 257-sided polygon. J. Hermes worked for ten years on the construction of the 65 537-sided polygon.

The geometric constructability of regular polygons is now reduced to the number-theoretical question of whether Fermat pk numbers can also be prime numbers for certain greater values of k. Such problems in number theory today depend extensively on the efficiency of modern computers. Therefore the symmetry of regular polygons is an “evergreen” which has been newly investigated in every phase of mathematical history — from the elementary constructions with a compass and ruler in Antiquity, via algebraic number — theoretical analyzes in modern times to the calculability problems of modern computers [1.8]. The star polygons, derived from the regular polygons, were investigated in the High Middle Ages, possibly reflecting a special aesthetic interest, such as that expressed in the lovely rosette windows of Medieval cathedrals. Especially well-known are the pentagram made from the pentagon as a secret sign of the Pythagoreans, and the Star of David made from the regular hexagon (Fig. 5). J. Kepler was also interested in the star polygons [1.9]. Regular polygons and star polygons can be realized optically by means of the reflections of a kaleidoscope. It is presumed that such an instrument was first described historically in the “Ars magna lucis et umbrae” (1646) of A. Kirchner, the great Baroque scholar, who tried to find the universal symmetries of the cosmos.

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Fig. 5.

Symmetries of star polygons

Along with the regular symmetry figures in the plane, the symmetrical bodies of space have fascinated human beings from of old. In pre-Greek times some of these bodies already had cultic and religious symbolic value because of their regular construction and their crystalline structure. The Pythagoreans were acquainted with the regular tetrahedron composed of four regular triangles, the cube composed of six regular squares and the dodecahedron composed of twelve regular pentagons (Fig. 6). A specimen of the dodecahedron made from steatite is extant from the Etruscan time (500 B.C.). But a complete derivation of all five possible regular solids was first handed down in the last (XIII) book of Euclid’s “Elements”, which dates back to the Greek mathematician Theaetetus (415–369 B.C.). Therefore the octahedron with eight regular triangles and the icosahedron with twenty regular triangles were probably also first constructed by Theaetetus (Fig. 6). The last theorem of Euclid’s “Elements”: that these are the only regular solids in Euclidean space, is already a significant mathematical insight. The proof is this: it is generally required of a regular polyhedron that all its corners, edges and surfaces be indistinguishable. Further, all surfaces should be regular polygons. This definition suffices to justify the five mentioned Platonic

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Fig. 6.

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The regular (Platonic) solids of Euclidean space

solids as the only regular bodies. First, a regular polyhedron will not possess any invaginated corners and edges. Since not all corners and edges could invaginate, some corners or edges would be distinctive — contrary to the definition. Therefore, also, the sum of the polygonal angles that come together at one corner must be smaller than 2π. Otherwise these polygons would lie in one surface and invaginating edges would go out from this corner. Further, at least three polygons must come together in one corner. Beyond that, for the sake of regularity all angles of the polygon must be equal. Therefore they must all be smaller than 2π/3. In the regular hexagon the polygonal angle amounts to an even 2π/3. Since the angles for n ≥ 3 increase in the regular n-angled polygon, only regular 3-, 4- and 5-angled polygons can be chosen as surfaces of regular polyhedra. In the case of the regular 4-angled polygon, the square, which has only right angles, no more than three squares can come together in a corner without exceeding the angle sum of 2π. In the case of the regular pentagon, no more than three pentagons can meet in a corner. A regular body is by definition already completely determined if the number of surfaces abutting in a corner and their number of corners is known. Therefore there can be, at the most, only a single regular polyhedron that is bordered by squares and similarly only one bordered by regular pentagons. By contrast, three, four or five equilateral triangles can come together in a corner since it takes six triangles to yield the corner angle sum 2π. The regular (equilateral) triangle can thus appear as a surface in three different polyhedra.

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Altogether, therefore, five possible regular polyhedra emerge:

number of

polyhedron

bordering polygon

corners

edges

surfaces

surfaces meeting at a corner

tetrahedron octahedron icosahedron cube dodecahedron

triangle triangle triangle square pentagon

4 6 12 8 20

6 12 30 12 30

4 8 20 6 12

3 4 5 3 3

Solid stars are examined by analogy to the star figures in the plane. To every regular solid a reciprocal one can be assigned which is enclosed by the planes of the polygon at every corner of the original polyhedron. For that reason the edges of the reciprocal polyhedron are centrally perpendicular to those of the original. Fig. 7 shows the octahedron as a reciprocal polyhedron to the cube and vice versa, and the reciprocal polyhedron of the regular tetrahedron as an equal tetrahedron. In nature the combination of the two reciprocal tetrahedra appears as the twin crystal. The semi-regular polyhedra do exhibit forms of solid bodies that were already familiar in the everyday as crystals, precious stones or building stones. These polygons are called semi-regular since each is bounded by various regular polygons. Systematic constructions of the solids were indicated by Kepler in his work “Harmonice mundi.” In many early cultures proportional relationships are described by means of numbers. But the Pythagoreans, as far as we know, were the first to want to base characteristics of harmony and symmetry on specific numerical relationships. With the Pythagoreans the tetraktys (quaternity) of the numbers 1, 2, 3, 4 occupies a special position since it “begets the number ten” arithmetically, forms a regular triangle geometrically, is assigned musically to the four strings of the lyre, namely Hypate, Mese, Paramese, and Nete, and their properties correspond to the harmonious sounds of the musical fourth (4:3), the

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Fig. 7.

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Regular solid (octahedron) and its reciprocal one

fifth (3:2) and the octave (2:1) [1.10]. In the Pythagorean conception, the harmony of nature is expressed in the unity of arithmetical, geometrical and musical proportions. Euclid calls such proportions “logos”” (λ ). In this sense the logos is the measure of all being. Pythagoras demonstrates in his music theory why the numbers 12, 9, 8, 6 are excellent. To that purpose he uses the monochord, an instrument with only one string, which is divided into twelve equally large intervals. It is possible, namely, to express in whole numbers half, two-thirds and three-quarters of the number 12, thus the shortened lengths 6, 8, 9 of the whole string 12, which correspond to the octave, fifth and fourth. The arithmetic, geometric and harmonic means constitute the three Pythagorean ratios called ) " i.e. proportional ratios of three magnitudes, the middle one being determined by the other two on the basis of proportion. At a later time further ratios are added, by means of equivalent formulations of the Pythagorean ratios and exchange of the component parts. The famous Golden Section, which Pythagoras is said to have taken over from the Babylonians or Zarathustra, and which was considered for centuries to be simply the aesthetic standard, came to our attention earlier in the pentagram of the Pythagoreans (Fig. 5). Each of its five lines divides each other one according to the Golden Section, i.e. the ratio of the whole line to the greater part is equal to the ratio of the greater part to

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the smaller part. The proof is an immediate consequence of similar triangles. There are many indications that precisely this symbol of the order of the Pythagoreans made their philosophy fundamentally questionable. What is under consideration here is the discovery of incommensurable straight-line proportions — presumably by the Pythagorean Hippasus of Metapontum in the 5th century B.C. — which is said to have set off a shock in Pythagorean circles. Ultimately this discovery called into question the assumption on which the philosophy of the Pythagoreans was originally based, namely that all proportions of magnitude could be expressed in ratios of whole numbers — like the harmonies on the monochord [1.11]. In this sense harmony and whole-number rationality coincide in the philosophy of the Pythagoreans. For that reason the discovery of proportions of magnitude that are not in the ratio of whole numbers also seemed to them to be an incursion of the irrational, which according to legend brought the punishment of the gods upon the discoverer.

Fig. 8.

Symmetry of the Golden Spiral [1.12]

Another application of the Golden Section is the Golden Rectangle being used for constructing the Golden Spiral (Fig. 8): in a Golden Rectangle the longer side is divided according to the Golden

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Section. We get a square that is separated from the Golden Rectangle and the remaining rectangle is again “golden,” and once more a square is separated off, etc. Two opposite corner points of the squares form points of intersection of the spiral in the manner of a clock dial. In fact, already in 1202 Leonardo of Pisa (“Fibonacci”) gave a number sequence whose values correspond to the rotation angles of the spiral and behind which, Leonardo conjectured, must lie a law of biological proliferation. He imagined that rabbits live as long as they like and that every pair produces a new pair every month, which in a month produces its first pair. The experiment begins, at the start of the first month, with a new-born pair. In the second month the same pair exists. In the third month there are 2 pairs, in the fourth 3 pairs, in the fifth 5 pairs, etc. If one designates the number of rabbit pairs in the n-th month as fn , one obtains the following table: n 0 1 2 3 4 5 6 7 8 9 10

fn 0 1 1 2 3 5 8 13 21 34 55

fn+1 /fn ∞ 1 2 1, 5 1, 6667 1, 6 1, 625 1, 6154 1, 6190 1, 6176 1, 6182

Later Kepler expressed the general law of this sequence, namely, f0 = 0, f1 = 1, fn + fn+1 = fn+2 . Kepler also noted that the quotients fn+1 /fn approach the Golden Section as n increases. However, this was first actually proven in the 18th century by R. Simon. J.W. von Goethe was yet to speak of the spiral effect in nature. Eighteenth-century biologists such as C. Bonnet (1754) were to try to identify the developmental law of the Fibonacci sequence in the coiling arrangement of blossoms and leaves (phyllotaxis) or in snail shells, leading up to the preoccupation of contemporary mathemati-

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cians with the computer-assisted computability of this (recursive) number sequence. This modern application reflects the age-old interest in a universal law of harmony and development stemming from the Pythagorean time and continuing into the present. Retroactively it is remarkable that the Antique doctrine of proportions provided not only the bases for laws of harmony of nature and aesthetics, but also for jurisprudence in human communities. Aristotle’s Nicomachean Ethics states: “This, therefore, is what is just: the proportional. And the unjust is the offence against the proportional. But the proportional is a middle way [1.13].” On the basis of the mathematical doctrine of proportions Aristotle made a thoroughly modern juridical distinction between distributive justice in public law and compensatory justice in civil law. When it is a matter of distribution of a common good — honors, offices, sums of money, etc. (today one would add the distribution of the tax burden) — the “commensurability” of the distributions and awards is determined by the proportions of the achievements, merits, diligence, etc., of the individual person. However, if it is a matter of compensating for damages in the contractual relations between people, “the law regards only differences in the degree of damages; it views the partners as equal.” Therefore Person a and Person b count as equal, i.e. a = b. The injustice that b exercises against a, e.g., by taking away good c, damages the equality, i.e. a − c < b + c. The compensation consists of the return of c, thus (a − c) + c = (b + c) − c. But that is nothing other than an arithmetical mediation that is generally provided by civil law. On the basis of the doctrine of proportion, therefore, there arises a unitary image of a proportionally well-ordered world of numbers, geometric magnitudes, musical harmonies and law. A question arises as to whether these mathematical harmonies fit the reality of nature. To answer that, we begin with a look at Antique-Medieval astronomy, which historically was the first discipline in the physical sciences to be mathematical and which constituted the fourth mathematical discipline of the Pythagorean quadrivium — along with arithmetic, geometry and music (the doctrine of harmony).

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In all early cultures we find astronomical knowledge that contributes to temporal and spatial orientation but is also meant to support astrological and religious interpretations of the world. Technical methods are developed for describing certain celestial phenomena in relation to the position of the observer. Among these are the rising and setting of the sun, moon and stars. Beginning ca. two millenia B.C., for example, Egyptian star calendars documented the heliacal rise of Sirius. The periodic celestial cycles helped determine the calendar and were brought into synchrony with annual occurrences in nature as, for example, the flooding of the Nile which insures the fruitfulness of the fields and thereby the basis for human life. The Oriental cult of resurrection and rebirth has its origin in the rising and setting of the stars and the fruitful periods on the earth that are linked to them, as well as the star mythology of the love and death of the gods Osiris (Orion) and Isis (Sirius). The Babylonian moon tables have an admirable exactitude, allowing laws of periodic courses to be inferred from them [1.14]. Along with determining the time by calendars, the Babylonian rulers were interested in horoscopes that were meant to determine their souls’ passage through life in the signs of the zodiac. This is not to be mistaken for the privatistic curiosity and horoscope credulity found among our contemporaries. The horoscope of the ruler was a matter of political interest: the stability and crises of the ruler, who represented the state, needed to be predictable and calculable. In the astronomy of the Maya, as well, exact observational tables were of primary interest, but so were the prophecies drawn from solar eclipses. In Chinese astronomy, spherical models of the celestial globes were developed very early. They were copied in mechanical models before the time of the Occidental celestial globes. Here too, astronomy was a matter of the national political interest, evidenced by the artful sundials and celestial globes in the Imperial Palace in Beijing or elsewhere. It was not priests or independent scientists, but high government officials in the imperial bureaucracy, who for centuries registered all celestial movements with painful exactitude [1.15]. The starry sky as image of the hierarchy of the imperial realm: in the

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center the pole of the sky like the emperor around whom everything turns. However, the Chinese possessed no mathematical theory like Euclid’s geometry, with its proofs, to give them a geometrical model for the exact explanation and derivation of their observations. It was Greek astronomy that first succeeded at that. There the movements of the planets were reconstructed in the movements of spheres, which were tuned to each other in artful proportions. The celestial harmony was the central theme of Pythagorean astronomy. It taught that each planet in its circular motion generates musical notes and that these sounds express a harmony of the spheres. Later the seven recognized planets were assigned to the seven strings of the lyre: “The heavens are harmony and number.” By the time of Plato the conviction prevailed that the cosmos is ordered in a centrally symmetrical manner, with the earth as a sphere in the center. Around it the whole sky turns to the right around the celestial axis, which goes through the earth. Sun, moon and planets turn to the left on spheres that have different distances from the earth in the sequence of moon, Mercury, Venus, sun, Mars, Jupiter, Saturn. The most external shell carries the sphere of the fixed stars. According to the Platonic-Pythagorean conception, the rotational periods are related to each other by whole numbers. There is a common multiple of all rotational times, at the end of which all the planets are exactly in the same place again. The motion of each one produces a sound so that the tones of the movements of the spheres jointly form a harmony of the spheres in the sense of a well-ordered musical scale. Geometry, arithmetic and aesthetic symmetries of the cosmos ring through the universe in a harmonious music of the spheres. Soon this emphatically symmetrical model of the cosmos was called into question by exact observations. A difficult problem was presented by the irregular planetary orbits, especially their retrograde movements. The irregularities in the sky were disquieting, especially for philosophers in the Pythagorean tradition, who were accustomed to comprehending the heavens — in contrast to the earth

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— as the realm of eternal symmetry and harmony. At this point the mathematician Eudoxus of Knidos suggested an ingenious solution. Namely, according to Eudemus, Plato posed this question to Eudoxus: By means of what regular, ordered circular movements could the phenomena of the planets be “saved,” i.e. kinematically explained [1.16]? This report formulated a new astronomical research program with the goal of a kinematic explanation for empirical data in a centrally-symmetrical model of the spheres that is presupposed a priori. Eudoxos suggested a planetary model with its spheres still positioned centrally around the earth. The retrograde movement of a planet is generated on the surface of an outer sphere by a combination with moving inner spheres of differently inclined axes. Because of the constant spacing of the spheres, the changing brightness of the planets could not be explained by Eudoxos. A greater exactitude in the reconstruction of observed curves was achieved when Apollonius of Perga (ca. 210 B.C.) recommended that the common center of the spheres be given up. But the spherical form of planetary movement and the equal speed of the spheres were to be retained. According to this proposal, the planets rotate uniformly on spheres (epicycles), whose imagined centers move uniformly on great circles (deferents) around the centerpoint (earth). By appropriately proportioning the speed and diameter of the two circular motions and by varying their directions of motion, it was possible to produce an unanticipated potential for curves, and these found partial application in astronomy from Ptolemy to Kepler also. The constant spherical symmetry of the models was therefore preserved, even if they no longer had a common center, but were distributed among various circle centers. The following examples from the epicycle-deferent technique show what a multiplicity of apparent forms of motion can be created by appropriately combining uniform circular motions [1.17]. This makes the Platonic philosophy more comprehensible in its view that behind the changes in “phenomena” there are the eternal and unchangeable forms. In Fig. 9 an elliptical orbit is produced by combining a defer-

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ent motion and an epicycle motion. Fig. 10 shows a closed cycloid. In this way, changing distances between planets and the earth can also be represented. In principle, even angular figures can be produced. In Fig. 11, when the epicycle diameter approaches the deferent diameter, an exact straight line results. Copernicus mentioned this construction in his book “De Revolutionibus.” Even regular triangles and rectangles (Figs. 12 and 13) can be produced by means of appropriate combinations of an epicycle motion and a deferent motion, if one changes the speed of the east-west motion of a planet that is moving on an epicycle with a west-east motion. If one lets the celestial body circle on a second epicycle whose midpoint moves on the first epicycle, one can produce a multiplicity of elliptical orbits, reflectionsymmetry curves, periodic curves (Fig. 14), and also nonperiodic and asymmetrical curves (Fig. 15). From a purely mathematical and kinetic standpoint, Plato’s problem of “saving the phenomena” is completely solved. There is no curve in observational astronomy that cannot be produced with almost any desired exactitude whatever, as a result of a combined epicycle-deferent motion. Indeed, looking at asymmetrical curves such as those in Fig. 15 leads one to add that even the trajectories of elementary particles that are captured on the photographic plates of contemporary high-energy physicists can be extensively reconstructed by the epicycle-deferent technique. In principle, therefore, Plato’s paradigm of symmetry in the sense of uniform circular motion (modified by Apollonius and

Fig. 9.

Elliptical orbit

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Fig. 10.

Fig. 11.

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Closed cycloid

Straight line produced by epicycle-deferent technique

Ptolemy) could influence the sciences right up until today. In any case it cannot be disproved by phenomenological description of curved paths. In particular, from this standpoint not only the reversal of the earth and the sun in the so-called Copernican revolution, but also Kepler’s change from circular to elliptical orbits,

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Symmetry and Complexity

Fig. 12.

Regular triangle produced by epicycle-deferent technique

Fig. 13.

Rectangle produced by epicycle-deferent technique

seem secondary, since both initiatives can be traced back to the combination of circular motions in accordance with the epicycle-deferent technique.

The decisive question in this case is, instead, which motions the planets “really” carry out, whether they are, in fact, combined,

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Fig. 14.

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Periodic curve produced by epicycle-deferent technique

Fig. 15. Nonperiodic and asymmetrical curve produced by epicycle-deferent technique

uniform and unforced circular motions that seem to us on earth to be elliptical paths, or whether they are in fact compelled to follow elliptical paths by forces. This determination, however, cannot be made geometrically and kinematically, but only dynamically,

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i.e. by means of a corresponding theory of forces, hence, by means of physics. But Antique-Medieval astronomy as the fourth discipline of the Pythagorean quadrivium is a purely geometric-kinematic discipline. The reasons that led to the abandonment of the Platonic symmetry program could, therefore, be understood only in connection with the Antique-Medieval philosophy of nature and the beginning of modern physics. Plato did not only propose a research plan for the macrocosm according to which the apparently irregular celestial phenomena were to be traced to unchangeable mathematical regularities and symmetries. In his dialogue “Timaeus,” he introduced the first synthesis of atomism and mathematical symmetry. The changes, mixings and separations on earth were traced to unchangeable mathematical regularities and symmetries of an atomic microcosm. The Democritean atoms were mathematically too unspecific and structureless for that. Moreover, in Empedocles’ four elements, namely fire, air, water, earth, a classification was at hand that was immediately accessible to experience. All the regular solids of Euclidean geometry (Fig. 6) were joined to the natural elements then on the basis of external features that seem arbitrary to us today: Fire was composed of the smallest and most pointed bodies, the tetrahedra; earth was composed of the most stable ones, the cubes. Air, composed of octahedra, and water of icosahedra, were assumed to be like two intermediate proportionals. The dodecahedron was used for the celestial sphere because of its similarity to the cube; therefore in the narrow sense it did not belong in Plato’s earth physics. In fact the dodecahedron, with its characteristic of the Golden Section, met the highest Greek requirements for symmetry and thus may have seemed especially appropriate as the celestial symbol. The regular bodies can be cut open along appropriate edges, and their surface elements can be unfolded as nets (Fig. 16). One can easily see how two tetrahedra with a common edge can be formed from the net of the octahedron, and two octahedra and a tetrahedron or five tetrahedra can be formed from the net of the icosahedron.

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Fig. 16.

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Symmetries of Platonic physics

From the point of view of philosophy of nature one could speak here of a kind of chemical analysis and synthesis [1.18]. If one characterizes the Platonic elements as F (fire), A (air), W (water), E (earth), then the obvious “chemical formulas” are 1 A = 2 F and 1 W = 2 A + 1 F = 5 F. Plato consciously avoided the Democritean designation “atom” for his elements, since they can, after all, be decomposed into separate plane figures. Thus tetrahedra, octahedra and icosahedra √ consist of equilateral triangles with sides 1, 2 and 3, while the regular rectangles of the cubes, when they are bisected, yield right-angled √ triangles with side lengths 1, 1 and 2 (Fig. 16). A consequence is

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that “fluida” like water, air and fire can combine with each other whereas a solid made of earth building blocks, because of its different triangles, can only be converted into another solid. However, the significance of this initiative was essentially unappreciated until modern times, if one disregards sporadic mention by a few Neoplatonists. With the rise of crystallography and stereochemistry, the Platonic core idea became a successful mathematical research program for explaining crystals and atomic and molecular compounds by means of a concept of symmetry (an expanded one) and for making new phenomena predictable and subject to empirical re-examination. A high point up to now in this development is modern elementary particle physics. Heisenberg made this observation about it: “. . . The elementary particles have the form Plato ascribed to them because it is the mathematically most beautiful and simplest form. Therefore the ultimate foot of phenomena is not matter, but instead mathematical law, symmetry, mathematical form.” In Antiquity and the Middle Ages Plato’s mathematical atomism gained little reception. The basic problem, for his successors, in his geometric theory of matter was already evident in the dialogue Timaeus. How are the functions of living organisms to be explained? The suggestion that certain corporeal forms are as they are in order to fulfill certain physiological purposes (e.g. the tunnel shape of the gullet for assimilation of food) cannot, in any case, be derived from the theory of regular solids. In addition, the idea of explaining the changing and pulsating processes of life on the basis of the “rigid” and “dead” figures of geometry, must have seemed thoroughly unnatural, speculative and far-fetched to the contemporaries of that time. Contemporaries of our time still have difficulties understanding the detour that today’s scientific explanations take through complicated and abstract mathematical theories. 1.2

Cultural and Cosmic Diversity

In our everyday-world we experience complexity and diversity, change and movement, chaos and random, and no eternal symme-

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tries. Platonic astronomers tried to reduce the irregular and complex planetary orbits as they were observed to regular and simple movements of spheres. For our everyday-world, this trial seems to be hopeless. Since the presocratic philosophers it has been a fundamental problem of natural philosophy to discover how order arises from complex, irregular and chaotic states of matter. What the presocratic philosophers did was to take the complexity of natural phenomena as it is experienced back to “first origins” , “principles” or a certain order. Let us look at some examples. Thales of Miletus (625–545 B.C.), who is said to have proven the first geometric theorems, is also the first philosopher of nature to believe that only material primary causes could be the original causes of all things. Thales assumes water, or the wet, as the first cause. His argument points to the observation that nourishment and the seeds of all beings are wet and the natural substratum for wet things is water. Anaximander (610–545 B.C.), who is characterized as Thales’ student and companion, extends Thales’ philosophy of nature. Why should water be the first cause of all this? It is only one of many forms of matter that exist in uninterrupted tensions and opposites: heat versus cold and wetness versus dryness . . . Therefore Anaximander assumes that the “origin and first cause of the existing things” is a “boundlessly indeterminable” original matterr (– ) out of which the opposed forms of matter have arisen. Accordingly we have to imagine the “boundlessly indeterminable” as the primordial state in which matter was boundless, without opposites, and, therefore, everywhere of the same character. Consequently, it was an initial state of complete homogeneity and symmetry. The condition of symmetry is followed by symmetry breaking, from which the world arises with all its observable opposites and tensions: The everlasting generative matter split apart in the creation of our world and out of it a sphere of flame grew around the air surrounding the earth like the bark around a tree; then, when it tore apart and bunched up into definite circles, the sun, moon and stars took its place [1.19].

The ensuing states of matter that Anaximander described in his cosmogeny were therefore by no means chaotic; instead they were

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determined by new partial orders. The fascination with Anaximander increases when one reads his early ideas of biological evolution. He assumes that the first human beings were born from sea animals whose young are quickly able to sustain themselves, as he had observed in the case of certain kinds of sharks. A century later searches were already being made for fossils of sea animals as evidence of the rise of humans from the sea. The third famous Milesian philosopher of nature is Anaximenes (†525 B.C.), who is thought to have been a companion of Anaximander. He regards change as the effect of the external forces of condensation and rarefaction. In his view, every form of matter can serve as basic. He chooses air : And rarefied, it became fire; condensed, wind; then cloud; further, by still stronger condensation, water; then earth; then stones; but everything else originated by these. He, too, assumed eternal motion as the origin of transformation. What contracts and condenses matter, he said is (the) cold; by contrast, what thins and slackens is (the) warm [1.20].

Thus Anaximenes assumes external forces by which the various states of matter were produced out of a common original matter and were transformed into each other. Heraclitus of Ephesus (ca. 500 B.C.), “the dark one”, as he was often called, is of towering significance for our theme. His language is indeed esoteric, more phrophetic than soberly scientific, and full of penetrating metaphors. He took over from Anaximander the doctrine of struggle and the tension of opposites in nature. The original matter, the source of everything, is itself change and therefore is identified with fire: The ray of lightning (i.e. fire) guides the All. This world order which is the same for all was created neither by one of the gods nor by one of the humans, but it was always, and is, and will be eternally living fire, glimmering and extinguishing according to measures [1.21].

Heraclitus elaborated further on how all states of matter can be understood as distinguishable forms of the original matter, fire. In our time the physicist Heisenberg declared: At this point we can interpose that in a certain way modern physics comes extraordinarily close to the teaching of Heraclitus. If one substitutes the word

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“fire”, one can view Heraclitus’ pronouncements almost word for word as an expression of our modern conception. Energy is indeed the material of which all the elementary particles, all atoms and therefore all things in general are made, and at the same time energy is also that which is moved . . . Energy can be transformed into movement, heat, light and tension. Energy can be regarded as the cause of all changes in the world [1.22].

To be sure, the material world consists of opposite conditions and tendencies which, nevertheless, are held in unity by hidden harmony: “What is opposite strives toward union, out of the diverse there , and the struggle makes arises the most beautiful harmony everything come about in this way.” [1.23] The hidden harmony of opposites is thus Heraclitus’ cosmic law, which he called “logos”

”(

).

What happens when the struggle of opposites comes to an end? According to Heraclitus, then the world comes to a final state of absolute equilibrium. Parmenides of Elea (ca. 500 B.C.) described this state of matter, in which there are no longer changes and motions in (empty) spaces. Matter is then distributed everywhere equally (homogeneously) and without any preferred direction for possible motion (isotropically). It is noteworthy that infinity is thought to be imperfection and therefore a finite distribution of matter is assumed. In this way Parmenides arrived at the image of a world that represents a solid, finite, uniform material sphere without time, motion or change. The Eleatic philosophy of unchanging being was, indeed, intended as a critique of the Heraclitean philosophy of constant change, which is put aside as mere illusion of the senses. And the later historical impact of the Eleatic philosophy in Plato appears in his critique of the deceptive changes that take place in sensory perception in contrast to the true world of unchangeable being of the ideas. But from the point of view of philosophy of nature, the world Parmenides described was not necessarily opposite to the teaching of Heraclitus; in his cosmogeny it can be understood entirely as a singular end state of the highest symmetry. After water, air and fire were designated as original elements, it was easy to conceive of them as raw materials of the world. Empe-

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docles (492–430 B.C.) took that step and added earth as the fourth element to fire, water and air. These elements are free to mix and bind in varying proportions, and to dissolve and separate. What, now, according to Empedocles, were the enduring principles behind the constant changes and movements of nature? First there were the four elements, which he thought arose from nature and chance e( ), not from any conscious intention. Changes were caused by reciprocal effects among these elements, that is, mixing and separation: “I shall proclaim to you another thing: there is no birth with any of the material things, neither there is an ending in ruinous death. There is only one thing: mixture and exchange of what is mixed.” [1.24] Two basic energies were responsible for these reciprocal effects among the elements; he called them “love”” ( ) for in analt attraction and “hatred”” ( is an ) for repulsion. Thereogy Chinese ogy in the yin-yang dualism of Chinese philosophy. Empedocles taught a constant process of transformation, i.e., combination and separation of the elements, in which the elements were preserved. He did not imagine these transformation processes to be at all mechanical (as the later atomists did), but rather physiological, in that he carried over processes of metabolism in organisms to inanimate nature. In his medical theories, equilibrium is understood to be a genuinely proportional relationship. Thus, health means a particular balance between the opposite components and illness arises as so on as one of them gets the upper hand. If we think of modern bacteriology with its understanding of the antibodies in the human body, then this view of Empedocles proves to be surprisingly apt. Anaxagoras (499–426 B.C.) advocated what was in many regards a refinement of his predecessors’ teaching. Like Empedocles he developed a mixing theory of matter. But he replaced Empedocles’ four elements with an unlimited number of substances that were composed of seed particles ( ) or equal-sized particles . They in were unlimited in their number and their num smallness, i.e. matter was assumed to be infinitely divisible. The idea of a granulated continuum comes forceably to mind. Anaxago-

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ras also tried to explain mixtures of colors in this way, when he said that snow is, to a certain degree, black, although the whiteness predominates. Everything was contained in each thing, and there were only predominances in the mixing relationships. More distinctly than some of his predecessors, Anaxagoras tried in his philosophy of nature to give physical explanations for the celestial appearances and motions that were described only kinematically in the mathematical astronomy of the Greeks. So in his cosmology he proceeded from a singular initial state: a homogeneous mixture of matter. An immaterial original power, which Anaxagoras called “spirit” ”( ), set this mixture into a whirling motion which brought about a separation of the various things depending on the speed of each of them. Earth clumped together in the middle of the vortex, while heavier pieces of stone were hurled outward and formed the stars. Their light was explained by the glow of their masses, which was attributed to their fast speed. Anaxagoras’ vortex theory appears again in modern times with R. Descartes, and then in more refined form in the Kant–Laplace theory of the mechanical origin of the planetary system. In modern natural sciences atomism has proved to be an extremely successful research program. In the history of philosophy the atomic theory of Democritus is often presented as a consequence of Heraclitus’ philosophy of change and Parmenides’ principle of unchanging being. The Democritean distinction between the “full” and the “empty,” the smallest indestructible atoms and empty space, corresponded to Parmenides’ distinction between “being” and “not-being.” Heraclitean complexity and change was derived from distinguishable reconfigurations of the atoms. Empty space was supposed to be homogeneous and isotropic. ), their positionn (h Atoms differ in their form ( ), and configura The their diverse configurations erial combi in material combinations. configuration of the atoms for the purpose of designation is compared with the sequence of letters in words, which has led to the presumption that atomistic ideas were developed only in cultures with phonetic alphabets. In fact, in China, where there was no

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phonetic alphabet but instead ideographic characters, the particle idea was unknown and a field-and-wave conception of the natural processes prevailed. The Democritean atoms move according to necessity in a constant whirll (* < or * < ). Here, by contrast with later Aristotelian notions, motion means only change of location in empty space. All phenomena, all becoming and perishing, result from combinationn ( ) and separationn ( ). Aggregate states of matter, such as gaseous, liquid or solid, are explained by the atoms’ differing densities and potentialities for motion. In view of today’s crystallography, the Democritean idea that even atoms in solid bodies carry out oscillations in place is noteworthy. Plato developed an internally consistent mathematical model by which various aggregate states and reciprocal effects of substances could be explained if one accepted his — albeit more or less arbitrary — initial conditions for interpretation of the elements. Nevertheless, as mentioned before in Chapter 1.1, even contemporaries of our time have difficulties to believe in mathematical symmetries behind the complexity and diversity of the apparent world. This is where Aristotelean physics begins [1.25]. Aristotle formulated his concept of a balance or “equilibrium” in nature chiefly on the basis of the ways in which living organisms such as plants and animals function. The process and courses of life are known from everyday experience. What is more obvious than to compare and explain the rest of the world, which is unknown and strange, with the familiar? According to Aristotle, the task of science is to explain the principles and functions of nature’s complexity and changes [1.26]. This was a criticism of those philosophers of nature who identified their principles with individual substances. The individual plant or the individual animal was not simply the sum of its material building blocks. Aristotle called the general, which made the individual being what it was, form m ( Í*@H). What was shaped by form was called matterr (à80). Yet form and matter did not exist in themselves, but were instead principles of nature derived by abstraction. Therefore, matter was also characterized as the potentiall (*b 2, only 5 regular (“Platonic”) polyhedra exist in 3dimensional space. Moreover, if we consider, additionally, the finite number of proper rotation groups around a center in space, we find only three new groups which leave unchanged or invariant (i) the regular tetrahedron, (ii) the cube or the octahedron, and (iii) the dodecahedron or icosahedron, respectively. For case (ii), inscribe an octahedron into a cube in such a way that the corners of the octahedron meet the corresponding sides of the cube at the centerpoints of the six square surfaces. Conversely, a cube can also be inscribed into an octahedron (Fig. 21). Then compare the analysis of the corners, edges and surfaces of the Platonic solids in the table of Fig. 6. Every rotation that turns the cube back into itself also leaves the octahedron invariant and vice versa. Therefore, the group for the octahedron is the same as for the cube. Analogously, it can be shown that the dodecahedron and the icosahedron are described by means of the same group. The regular solid that corresponds to the regular tetrahedron is the tetrahedron itself. This gives us three groups of proper rotations — group T of the tetrahedron, group W of the cube or octahedron and group D of the dodecahedron or icosahedron, with 12, 24 and 60 operations respectively. Corresponding to Euclid’s uniqueness of the Platonic solids in space, it can be shown that groups Cn (n = 1, 2, 3, . . .), Dn (n = 2, 3, . . .), T , W and P are the only proper rotation groups in

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space. This list has to be supplemented by the number of improper rotations in space, analogously to the plane rotation groups. An improper rotation in space is nothing but a rotation-reflection, i.e. the combination of a reflection and a rotation around an axis that is perpendicular to the mirror. A rotation-reflection can also be grasped as a rotation inversion, i.e. as a combination of a point reflection or inversion at the center O (which brings every point P back to P  on the extension OP  of line P O with P O = OP  ) and a rotation around an axis through the reflection point. By analogy to the situation in a plane, one can ask which of the finite point groups of motions leave space lattices invariant. In the 2dimensional case there are 10 point groups. In the 3-dimensional case one obtains 32 crystal classes that are of considerable significance for crystallography [2.10]. The corresponding groups to the 17 ornament groups in three dimensions are the 230 discrete groups of movements with three independent translations. As a whole, all the groups were first described by the Russian crystallographer Fedorov (1890), and also independently by the German A. Schoenflies (1891) and the Englishman W. Barlow (1894) [2.11]. The first 65 consist of proper movements. The simplest group contains only translations. The remaining 64 contain in addition rotations and screw axes, i.e. combinations of translations and rotations. Of these, 22 appear as 11 socalled enantiomorphic pairs, which are mirror images of one another, i.e. one member of the pair contains a left-, the other a right-handed screw. Examples in nature include the left- and right-rotating quartz, known since the early day of mineralogy. If one drops these practically important distinctions, one obtains only 54 cases, consequently altogether 219 groups. The remaining 165 groups contain, in addition to proper movements, also improper ones, such as reflections, rotation-reflections and glide reflections.

In mathematics the symmetry of figures or bodies is determined by the group of those mappings of them onto themselves (“automorphisms”) that let them unchanged (“invariance”). This idea can be generalized for all kinds of mathematical structures that are defined by axioms or theories. The old definition of “geometry” in the sense of “measuring the earth” became inadequate to cover specializations in research as early as the 19th century. Only with F. Klein’s

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“Erlanger Program” of 1872, with the concept of “geometric invariants” which remain unchanged with metric, affine, projective or topological transformation groups, among others, did it become possible to organize the various directions of research into a hierarchy of theories. Coordinate transformations of course played a large role in analytical geometry as long ago as the 17th and 18th centuries. In 2-dimensional analytical geometry, for example, geometric expressions concerning points of a plane are translated into analytical expressions of coordinate values, so that functions x and y for each point P correspond to the real values x = x(P ) and y = y(P ). The transformations x = ax+by +e and y  = cx+dy +f can be summarized as



 x ab x e = + y cd y f whereby the matrix

ab cd

must be orthogonal with



ab cd


 ac 10 = bd 01

The Cartesian geometry of the plane then consists of those expressions and characteristics that are invariant under these transformations. Since the successive application of Cartesian transformations in turn leads to more of the same, they form a group that clearly characterizes the invariant properties of Cartesian geometry. Examples of Cartesian transformations are rotation, reflection, and inversion, which can be represented by corresponding matrices.

According to Klein, the investigation of a geometric theory generally consists of the following algebraic problem: “There is a manifold, and in it there is a transformation group; we must investigate the elements belonging to the manifold with regard to those characteristics which are not changed by the transformations of the group.” In short: “There is a manifold, and in it there is a transformation group. Develop the invariant theory relating to the group.” [2.12] F. Klein distinguishes similarity transformations as key concept for Euclidean geometry, because there, figures can be enlarged or reduced arbitrarily without changing shape. Consider a triangle, for example, which can be arbitrarily enlarged or reduced without changing the angles. Only in Euclidean geometry is the group of motion a genuine subgroup of the similarity group.

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Absolute geometry arises from Euclidean geometry, without the acceptance of Euclid’s parallel postulate, according to which, through a given point, there is only a single line parallel to a given straight line. We move from absolute geometry to Euclidean or non-Euclidean geometry by adding either Euclid’s parallel postulate or one of the non-Euclidean versions. On the other hand, the parallel line plays a central role in affine geometry. For Euclid, the affine theorems are those that remain unchanged after parallel projection from one plane into another. Analytically, the affine geometry of the plane can be characterized by transformations with an invertible matrix. In contrast to affine geometry, there is no parallelism in projective geometry. Nor do length or angular measurements play any role. The historical origin of projective geometry is the problem of perspective. Each transformation from one figure to another by central and parallel projection or a finite series of projections is called a projective transformation. The projective geometry of the plane or of the straight line consists of the totality of those geometric properties that remain valid and unchanged over any number of projective transformations of the figures to which they relate. In contrast, metric geometry consists of the system of those geometric properties that relate to the sizes of figures, and remain invariant only under rigid motions. The most general of all geometries is topology, which is characterized by the group of continuous transformations. As an example of characteristics that are left invariant under transformations, let us consider polyhedrons. A polyhedron is called a simple polyhedron when its surface can be continuously deformed into a spherical surface, i.e. simple polyhedrons do not have “holes,” like a torus, for example. The Euler formula for the simple polyhedron is thus: E − K + F + 2 for the number of corners E, the number of edges K and number of surfaces F . We can easily verify this formula for the Platonic bodies, for example, but it covers a great deal more than just the polyhedrons of metric geometry with straight edges and plane surfaces. It even remains valid if we imagine the surface of a regular polyhedron made of rubber, which can be deformed arbi-

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trarily, as long as it is not torn. That is because only the number of corners (points), edges (lines) and surfaces is important for this formula. Length, surface area, linearity, cross ratio and other concepts of metric, affine or projective geometry are not left invariant under topological transformations. To explain the relationship between continuous groups and the concept of symmetry let us first consider a simple example. A rotation of a plane coordinate system around its origin in a counterclockwise direction by an angle θ can be considered a symmetry of the plane, because it leaves the relationships between distance and angle invariant. These rotations form a group. If, for example, the rotation by the angle θ1 is followed by an additional rotation by the angle θ2 , then the result is a rotation by the angle θ1 + θ2 . It can easily be verified that this rule of composition satisfies the group axioms. For example, the rotation by the angle 0 can be used as the unit element I. If σ1 is the rotation by the angle θ, and σ2 is the reverse rotation by the angle 2π − θ, then σ1 σ2 = I = σ2 σ1 . The group of rotations is continuous, since it is a function of a continuous parameter θ. The discrete groups of regular polygons are embedded in the continuous group of the circle. It describes the perfect symmetry of the circle that so fascinated ancient and medieval philosophers and scientists. As a result of a suitable composition of continuous rotation and stretching, we get a continuous rotation and stretching, by means of which the logarithmic spiral can be generated. The Swiss mathematician J. Bernoulli was so fascinated by its symmetry that he had the inscription “Eadem mutata resurgo” chiselled on his tombstone in the Basel Cathedral. In fact, this motto expresses the symmetry of the logarithmic spiral, since by means of continuous rotation and stretching, it can be transformed into itself. The Golden Spiral (Fig. 8) is an approximation of the logarithmic spiral. Like the spiral in the plane, the circular helix can be introduced in space by continuous helical motion, the technical application of which was discovered as long ago as Archimedes. The methods of group theory in geometry were an initial application in the natural sciences of the modern mathematical concept

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of symmetry. They were also based on earlier, visual and human notions of symmetry in art. But in the history of mathematics, the algebraic and number theoretical origins of the group theory concept of symmetry are older than the geometric approaches. They go back to the 18th century, and are related to the equation theories of J.L. Lagrange, Gauss, N.H. Abel and E. Galois in particular. The application of group theory in geometry is a question of distinguishing the symmetry characteristics of figures and bodies by invariance in relation to groups of transformations, for example, rotations, translations or reflections. It was the brilliant idea of Galois to also characterize the solutions of equations by characteristics of symmetry, which remain unchanged under specified transformations (“permutation group”), to thereby obtain information on solutions and the solubility of equations. Galois used this theory to answer basic problems of equation theory, and his achievement is thus the culmination of a development that stretches far back into Antiquity. On the other hand, his group theory methods are revolutionary, even independent of equation theory, and stand at the beginning of modern structural mathematics, which was also fundamental for physics. The analytical formulations of geometry since Descartes have followed analytical mechanics since J. d’Alembert and Lagrange, among others. Problems of motion in physics were translated into equations of motion, which in general have the form of differential equations. Under some side conditions, therefore, the solution of motion problems in physics meant the solution of differential equations. A transfer of the Galois Program from algebraic equations to differential equations was therefore also of interest in terms of physics. Then it was possible, to a certain extent, to determine the characteristics of the solutions to these equations, i.e. including the solutions of corresponding problems of motion, by means of symmetry observations. In 1874, Lie began to classify continuous transformation groups. He designated a group continuous, if all of its transformations are generated by “an infinite number of repetitions of infinitesimal transformations”. Lie’s theory of continuous groups became the “Galois theory” of differential equations when he used it in 1888 to character-

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ize their solutions of equations [2.13]. The theory of continuous and finite groups was investigated in the 1890s in France by H. Poincar´e and E. Cartan, among others, attracted a great deal of attention in physics (in particular with the theory of relativity). The differential geometry of Gauss, Riemann, Cartan and others form the basis for the symmetries of Einstein’s theory of relativity. Using the example of Gauss’ theory of surfaces [2.14], we shall first compile several illustrative results of differential geometry. The coordinate system on a surface xi (u1 , u2 ) which is generated by the curves u1 = const., and u2 = const., is called a Gaussian coordinate system. Curves on the surfaces (e.g. distances on the curved surface of the earth) with a < t < b can now be described by surface coordinates u1 = u1 (t), u2 = u2 (t) and by spatial coordinates xi = xi (u1 (t), u2 (t)). Since partial differentiation gives ∂xi du1 ∂xi du2 dxi = + dt ∂u1 dt ∂u2 dt such a curve has the arc length: 

b

s= a



b

= a

  3

 dxi 2  dt dt i=1 3 i=1

∂xi ∂xi ∂u1 ∂u1



du1 dt

2 +2

∂xi ∂xi ∂xi ∂xi du1 du2 + ∂u1 ∂u2 dt dt ∂u2 ∂u2



du2 dt

2 1 2

dt .

If, according to the Ricci calculation for Greek letter indices µ, ν, we accept summation over the indices of the surface coordinates, and for the Latin indices summation of the spatial coordinates, we get the abbreviated notation 

b

 gµν

s= a

duµ duν dt dt dt

with the metric coefficients gµν =

∂xi ∂xi , ∂uµ ∂uν

where gµν = gµν (u1 , u2 ) is only a function of the surface points, and the direction of the arbitrarily selected surface curves is expressed in terms duµ /dt.

The surface metric ds2 = gµν duµ duν is a positive-definite quadratic differential form, also called first fundamental form, which is generated by the (Euclidean) scalar product in the tangential plane

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of the surface point in question. It is invariant under well-defined coordinate transformations that are also continuously differentiable in each direction. The decisive factor is the Gauss’ assumption that in any small area of the surface, a “local” Euclidean coordinate system (y1 , y2 ) can be found, in which the distance from (y1 , y2 ) and (y1 + dy1 , y2 + dy2 ) can be measured by the Pythagorean metric ds2 = du21 + du22 . There are length-preserving (with invariance of the curve length), conformal (with invariance of the angle) and areapreserving (with invariance of the surface area) transformations. In cartography, for example, the Mercator projection is drawn by means of a conformal, but area-distorting mapping, while a Lambert projection is drawn with an area-preserving transformation. According to Gauss’ theorema egregium, the curvature of a surface can be determined solely by the metric coefficients gµν and their derivations, i.e. the curvature is a function only of the intrinsic geometry of the surface and not of the surrounding space. Therefore, for length-preserving transformations (bending of the surface), the Gaussian curvature of the surface points is preserved. Gauss’ assumption of local Euclidean coordinate systems is also a function of intrinsic metric characteristics of the surface, and not of the surrounding space. The overall curvature of a surface patch, according to the theorem of Gauss and O. Bonnet, is related in a simple manner to the total lateral curvature of its edge. Geodesic lines as the shortest and straightest connections between points on a surface found applications both in geodesy and in mechanics. The results of the Gaussian surface theory can be generalized without restriction to the n-dimensional surfaces. Riemann made a broader generalization, by expanding the intrinsic geometry of the 2-dimensional surfaces to n-dimensional differentiable manifolds whose metrics are no longer induced by embedding in a surrounding Cartesian space and its Euclidean scalar product [2.15]. Here, rather, a fundamental tensor gµν is specified for µ, ν = 1, . . . , n with the positive-definite metric ds2 = gµν duµ duν . Analogous to the 2-dimensional Gaussian surfaces, it is assumed for n-dimensional Riemannian manifolds M , that a Euclidean coordinate system y1 , . . . , yn with Pythagorean metric ds2 = dy12 + · · · + dyn2 can be found locally (“in the infinitely small neighborhood of a point P of M ”), which can be represented in the “tangential space” TP of M in point P .

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The 2-dimensional Gaussian surfaces can generally have different curvatures varying at different points. Mathematically, the curvature of a surface at one point can be defined as a function which is dependent only on the metric coefficients gµν , with µ, ν = 1, 2. Without going into any further mathematical detail here, it is also possible to define a curvature tensor for n-dimensional Riemannian manifolds which is a function only of the fundamental tensor gµν with µ, ν = 1, . . . , n [2.16]. While additional visualizations can be combined with the term “curvature” for the Gaussian curved surfaces in Euclidean space, this heuristic approach fails completely for Riemannian manifolds with an arbitrary number of dimensions. But even in the 2-dimensional case, the Gaussian surface curvature is determined by the “intrinsic” metric characteristics, and is therefore not always identical to the visual curvature that relates to the surrounding space. A trivial example is the surface of a cylinder, which is “visually curved,” but which has the same Gaussian curvature as the Euclidean plane. By generalizing the special 2-dimensional Riemannian manifolds with constant curvature, symmetrical spaces can be studied which are characterized by homogeneity and isotropy. Leibniz had previously identified homogeneity as a symmetry characteristic of space, because, in this case, all points of the space are indistinguishable. In a presentation entitled “The facts on which geometry is based” (1868), Helmholtz attempts to explain these symmetry characteristics of 3-dimensional physical space in terms of measurement techniques and physiology. He begins with a visually-based introduction of homogeneous point manifolds. Helmholtz, in his research into the sensory organs, investigated the manifolds of acoustic and color sensations as a physiologist and physician. Sounds can be defined by their continuously changing pitch and volume, i.e. as points of a 2-dimensional manifold. According to Helmholtz, colors can be generated as mixtures of three basic colors, the respective proportions of which are continuously changing, on the retina, i.e. they can be interpreted as points of a 3-dimensional manifold. The surface of the skin on the body is an

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example of a 2-dimensional manifold, whose “points of sensitivity” are characterized by different sensitivity to stimuli from location to location. How do these physiological manifolds differ from the known geometric manifolds, such as a plane? Helmholtz’s answer is that, on geometric manifolds, we can compare distances between points everywhere, i.e. we can freely move a rigid measurement body everywhere over the manifold. On the other hand, for example in the 2-dimensional acoustical manifold, there is no such comparability of distances between points everywhere: Two sounds of the same pitch and different volumes are not comparable to two sounds of different pitch but the same volume. The Euclidean plane is a homogeneous manifold in the sense of the free mobility of a rigid measurement body. But the surface of a sphere is homogeneous, if we select as the scale a segment of a great circle in “close” contact. The surface of an egg is a contrasting example. For example, if we plot a circle with the radius r at the small end and at the big end of the egg, the circumference U of the circle is smaller in the first case than in the second case. The circle cannot be moved freely, since the curvature of the surface of the egg is different at the small end and at the big end. Are there also homogeneous surfaces with negative constant curvature? An initial model was proposed by E. Beltrami (1835–1900). But among them, there is no shape as simple as the surface of a sphere, because they have singularities, beyond which the surfaces may not be continuously extended. The Helmholtz requirement for the “free mobility of a rigid measurement body” and the homogeneous manifold was explained and generalized n-dimensionally by Lie with his concept of the continuous group [2.17]. Lie proceeds from the assumption of an n-dimensional continuous and differentiable manifold M . Instead of a physically rigid body, there is a metric d(x, y) for points x, y from M . The physically free mobility of the rigid body is explained mathematically by one-to-one point transformations that leave the distances between points defined by the metric unchanged. They are therefore

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isometries or congruent mappings. They can occur, for example, as translations or rotations, and the operations can be performed one after the other. The geometric composition of movements corresponds algebraically to a composition of the transformations that satisfy the group characteristic. In particular, Weyl gave the following explanation of the Helmholtz–Lie requirement: The group of congruent mappings can transform any arbitrary point into any arbitrary point of the manifold, and with a defined point, can transform any arbitrary line direction in this point into any other line direction, with a defined point and defined line direction, any arbitrary surface direction running through it into any other arbitrary such surface direction, etc., up to the (n − 1)-dimensional direction elements. But if there is a point, a line direction running through it, a surface direction etc. passing through it up to an (n−1)dimensional direction element, apart from identity there is no congruent mapping which preserves this system of elements connected in this way, like point, line direction, etc. The decisive factor is then the mathematical proof that under this homogeneity requirement, only Euclidean, hyperbolic and parabolic manifolds are possible, i.e. only the three known types of geometries with constant curvature. Mathematically, this explanation makes it clear that for homogeneous manifolds, the free mobility of a “body” need not be required, but free mobility “in the infinitesimal”, in the sense of the indicated homogeneity requirement for the continuous group of congruent mappings. The homogeneous Riemannian manifolds are therefore not — as Helmholtz believed — based on a physical “fact”, but on a mathematical concept: the continuous isometry group. But Helmholtz’s visual and physical considerations were of major heuristic value in the discovery of this concept [2.18]. The Helmholtz–Lie idea of constructing homogeneous manifolds with the requirement of a continuous isometry group was generalized by Cartan. Finally, we should discuss his theory of symmetrical spaces, since it forms the mathematical framework for discussions of symmetry in modern (relativistic) cosmology.

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Cartan defines a “symmetrical space” [2.19] as a Riemannian manifold in which the reflection at any given point A is an isometric transformation. A reflection is defined as the assignment to a point M sufficiently close to A of that point M  which is obtained if the geodesic arc M A is constructed, and is changed so that M A and AM  have equal lengths. An isometry is defined as a coordinate transformation uµ → u ¯µ which leaves the form of the metric of the manifold invariant, i.e. for the metric coefficients g¯µν = gµυ . It is immediately apparent that the Cartan symmetry requirement is equivalent to the requirement that the Riemannian curvature is preserved at any given point A in relation to any plane element proceeding from A, if this plane element is displaced anyway in a parallel fashion. A parallel displacement of a vector from a point A to an infinitesimally adjacent point A results from the two successive reflections at A and at the center of the geodesic arc AA . If the reflection is an isometric transformation, the parallel displacement leaves the metric and thus also the curvature unchanged.

A symmetrical Riemann space allows a transitive group of rigid displacements. In this case, transitivity means that each point of the space can be transformed into any other point by an element of the group. For example, if A and B are two points sufficiently close together, then the product of the reflection at A and at the center of the geodesic arc AB is an isometry which transforms A into B. Infinitesimal isometries are of particular interest to physicists. A Riemannian space is called isotropic if it is isotropic in every point. The symmetries of quantum mechanics are investigated in the mathematical framework of Hilbert spaces and representations of discrete and continuous groups. A Hilbert space is nothing more than a complete linear space with a scalar product. In quantum mechanics, the states of particles are described by wave functions. Therefore the function spaces play a major role for the application in physics. The linear operators on Hilbert spaces are of interest in physics because they describe changes of the Hilbert spaces, i.e. in terms of physics, the changing states of the quantum systems. The operators are an important mathematical technique for the study of the symmetry characteristics of vector spaces. Symmetries in vector spaces are described by group transformations of the vectors (e.g. rotations, translations). In the vector space V , let us define a group of transformations G which transform the vectors from V into corresponding vectors from V . Let us also consider the func-

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tional space F (V ) with functions f , which are dependent only on the vectors x from V . In terms of physics, V can be visualized as the 3-dimensional coordinate space of a body, by means of which determined symmetry operations such as rotations around the origin can be executed. The functional value f (x) can be imagined, for example, as the temperature of a body at the position x. The question arises what temperatures the body has after the rotation. In response, we can say that a symmetry operation G in the vector space induces a transformation T (G) of the temperature function. In quantum mechanics, the functional space will be a Hilbert space with corresponding wave or state functions. The question then arises, what operators on the states of the quantum system are induced by symmetries of the coordinate systems. The definition of the linear operators by their matrices, i.e. the representation of groups by matrices is physically very important [2.20]. For example, to be able to use the symmetry of an abstract group in physics, the elements of the group must be quantified. That is what the matrices do. We cannot only investigate representations of symmetries in 3-dimensional spaces of classical physics, but also in functional spaces, which are a key concept in quantum mechanics. For (finite) groups, it can be demonstrated that all possible representations can be constructed of a finite number of representations that cannot be further reduced. Therefore, it is always sufficient to study these irreducible representations of a group. 2.2

Symmetry Breaking and Bifurcation Theory

The symmetry of dynamical systems is mathematically analyzed in the theory of differential equations. The reason is that the dynamics of a system, that is, the change of its states depending on time, is mathematically described by differential equations. They allow to compute the final states of a dynamical system as solutions of equations. In the following chapters, we shall see that the state of a system is a very universal concept which does not only refer to moving states of, for example, atoms or molecules in physical or

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chemical systems, but to states of, for example, cells in organisms or products in a market system, too. Thus, problems of symmetry and symmetry breaking in these areas are reduced to the symmetry of differential equations, that is, their invariance with respect to certain groups of transformations. A 1-dimensional system is described by a differential equation of the form dx/dt = f (x) with state x and time t. In differential calculus, dx/dt is the rate of change of x with respect to time t. We study linear and nonlinear equations of this form that are symmetric

Fig. 22a.

Fig. 22b.

Harmonic oscillator

Linearity of Hooke’s law

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Fig. 22c.

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Potential energy

with respect to certain transformations. But the symmetry can be broken by actually realized solutions. A simple example is a harmonic oscillator (Fig. 22a) with a mass attached to a spring resting on a table [2.21]. If the spring is neither stretched nor compressed, the mass will rest a steady-state position x = 0. If the spring is compressed with x < 0 or stretched with x > 0, there will be a force tending to increase or to decrease the elongation x of the mass. According to Hooke’s law the force f is proportional to the position x with a certain spring constant α that is, f (x) = −αx. The minus sign results from the fact that the elastic force tends to bring the particle back to its equilibrium position. Hooke’s law is a linear relationship between force and position, represented by a linear equation (Fig. 22b). The potential energy of a mass on a spring is U (x) = 1/2αx2 (Fig. 22c). In general, the potential energy is a function with the property dU (x)/dx = −f (x), that is, the negative expression of its rate of change is the force.

We consider an equation of motion, that is, the rate of change of position x of a particle under the action of a force, which has the form of a linear differential equation dx/dt = −αx. Obviously, this equation remains unchanged with resepect to the inversion x → −x, that is, the linear system is symmetric with respect to inversion. The potential energy (Fig. 22c) remains also invariant under this transformation U (x) → U (−x) = U (x). Its symmetric curve illustrates

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the full stability of the system. If the particle is displaced, its potential energy is brought to a certain point on the curve like a ball on the slope of a hill. In any case of displacement, the ball will fall back down the slope and, taking friction into account, come to rest at the bottom of the hill, the stable equilibrium point x = 0 of the system with dx/dt = 0. The situation changes for nonlinear 1-dimensional differential equations of the form dx/dt = f (x, α), depending on a changing control parameter α. In this case we can study the symmetry of equations, but also symmetry breaking by unstable equilibrium points. An example is the anharmonic oscillator [2.22] which contains a cubic term besides a linear term in its equation of motion dx/dt = −αx − x3 . The cubic term can be interpreted as a perturbation of the oscillator. For a changing control parameter with α < 0 and α > 0, we get different curves of potential energy (Fig. 23a–b). The equilibrium points are determined by dx/dt = 0. Under this condition, mathematical solutions of the equation can easily be found: We get only one solution x = 0 in the case of α > 0, but three solutions x = 0, √ √ x1 = − |α|, and x2 = + |a| in the case of α < 0. If the values of potential energy in Fig. 23a–b are interpreted as positions of a ball on the slope on a hill, then the ball rests in a stable position at x = 0 for α > 0 and for α < 0 at x1 and x2 , but in an unstable position at x = 0 for α < 0, where it can roll down the hill after tiny elongations.

Fig. 23a.

Potential energy of anharmonic oscillator

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Fig. 23b.

Fig. 23c.

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Potential energy of anharmonic oscillator

Bifurcation scheme of anharmonic oscillator

Again, the differential equation and the potential energy are symmetric with respect to inversion x → −x. But the symmetry is broken by actually realized solutions with changing control parameter. If the control parameter is gradually changed from positive to negative values, then the stable equilibrium position x = 0 becomes

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unstable at α = 0 and two new stable equilibrium points emerge. The development can be illustrated by the changing potential curve in Fig. 23a–b. If it is deformed from α > 0 to α < 0, it becomes flatter and flatter in the neighborhood of x = 0. Consequently, the ball rolls down the curve more and more slowly. At α = 0 two new symmetric valleys with stable equilibrium positions at the bottom emerge. The ball at x = 0 breaks the symmetry spontaneously and rolls down into one of them. In an actual system the potential energy settles into one of them in stable equilibrium. In Fig. 23c symmetry breaking is illustrated by a bifurcation scheme. The equilibrium coordinate xe is defined as a function of α. For α > 0, it is xe = 0, but for α < 0, xe = 0 becomes unstable (dashed line) and is replaced by two stable positions which are plotted by a solid fork. Thus, bifurcation mathematically only means the emergence of new solutions of equations at critical values. Actually, bifurcation and symmetry breaking is a purely mathematical consequence of the theory of nonlinear differential equations. But, bifurcations of final states as solutions of differential equations correspond to qualitative changes of dynamical systems and the emergence of new phenomena in nature and society that will be studied in the following chapters. The most important case of 1-dimensional nonlinear systems is the situation, when besides a stable point after changing control parameter a new couple of a stable and an unstable point emerge. The possibilities of final states is enlarged by 2-dimensional systems which are represented by 2-dimensional differential equations of the form dx/dt = f (x, y, α) and dy/dt = g(x, y, α). In this case, there are not only points on the x-axis as final stable or unstable states, but also points and closed curves (“limit cycles”) on the x–y plane. In dynamical systems, closed curves can be interpreted as periodic time-depending behavior. Thus, these equations are mathematical models for a large class of rhythms or oscillating systems like clocks and watches, but also oscillating electronic systems in technology, biological or even economic and social oscillations.

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An example is the anharmonic oscillator with a rotational symmetric potential energy [2.23]: The potential curve U (x) is rotated around the U -axis (Fig. 24a) with two equations of motion for polar coordinates r (radius) and ϕ (angle of rotation): dr/dt = f (r, α) is the nonlinear equation of the anharmonic oscillator and dϕ/dt = ω the equation of constant angular velocity of rotation. We can also use orthogonal coordinates x = r cos ϕ and y = r sin ϕ which span up a 2-dimensional x–y plane (Fig. 24b). The path of a ball along the valley is a circle. If the ball starts to roll down at the unstable center x = y = 0, it spirals away to the circle. Again, we observe symmetry breaking if the control parameter is changed from α > 0 to α < 0. Since the ball also ends up in this cycle, if it starts from the outer side, the limit cycle is stable.

In Fig. 24c the symmetry breaking of a limit cycle is illustrated by a bifurcation scheme. The equilibrium coordinates xe and ye are defined as functions of control parameter α. For α > 0, it is xe = ye =0, but for α < 0, xe = ye = 0 becomes unstable (dashed line) and is replaced by the stable limit cycle which is plotted by a rotated fork. This new form of bifurcation is called a Hopf-bifurcation. Limit cycles need not be circles, but can be other closed curves. In dynamical systems they describe a periodic time-depending behavior as final stable state. In 1- and 2-dimensional dynamical systems, stationary points and limit cycles are the only possible stable final states. Their bifurcation behavior were already discovered by Poincar´e (1885), and further analyzed by the American and German mathematicians G. Birkhoff and E. Hopf (1942) [2.24].

Fig. 24a.

Rotational symmetry of potential energy

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Fig. 24b.

Symmetry breaking of a limit cycle

fixed point attractor

limit cycle bifurcation point

Fig. 24c.

Symmetry breaking of a limit cycle (bifurcation scheme)

The possibilities of final states are dramatically enlarged and changed for systems with more than two dimensions. A 3dimensional system can be represented by 3-dimensional differential equations of the form dx/dt = f (x, y, z, α), dy/dt = g(x, y, z, α), and dz/dt = h(x, y, z, α). In this case, stationary points, limit cycles, quasi-periodic oscillations and the famous chaos emerge as possible

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solutions. How is their bifurcation behavior? Poincar´e suggested to reduce the complexity of a 3-dimensional state space to the wellknown dynamics in a 2-dimensional state space. His genius idea reminds us of Plato’s famous cave-allegory. According to Plato, human beings recognizing the real world are like prisoners in a cave who try to reconstruct the 3-dimensional real shape of objects outside the cave from their 2-dimensional shadows on a wall inside the cave. Actually, Poincar´e introduced a 2-dimensional plane (Poincar´e map) intersecting the orbits in the 3-dimensional state space transversally. These curves (trajectories) represent the time-depending developments of states in the 3-dimensional system. We analyze their intersecting points on the Poincar´e map (Fig. 25). Their sequential positions P0 = (x0 , y0 ), P1 = (x1 , y1 ), P2 = (x2 , y2 ), . . . with two coordinates xn and yn (n = 0, 1, 2, . . .) on the Poincar´e map describe discrete time dynamics which deliver sufficient information about the continuous dynamics in the 3-dimensional state space. As the interval of time between the sequential positions on the Poincar´e map are finite, the discrete time dynamics define a sequence of recurrence in the 3-dimensional system. Mathematically, it can be represented by 2-dimensional finite difference equations xn+1 = f (xn , yn ) and yn+1 = g(xn , yn ). At first, we study oscillations of limit cycles in a 3-dimensional space on a 2-dimensional Poincar´e map [2.25]. If the points P0 , P1 , P2 , . . . converge to a fixed point P on the Poincar´e map, then the cor-

Fig. 25a.

Poincar´e map and limit cycle

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Fig. 25b.

Poincar´e map and spiral

Fig. 25c.

Poincar´e map and torus

responding 3-dimensional dynamics lead to a limit cycle (Fig. 25a). The existence of a fixed point P is proved by the fact that the finite difference equations have a simultaneous solution xn+1 = xn , yn+1 = yn resp. xn = f (xn , yn ), yn = g(xn , yn ). In general, we can consider the case that xn+1 , xn+2 , . . . are different to xn and

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yn+1 , yn+2 , . . . resp. to yn until an iteration k with xn+k = xn and yn+k = yn is reached. An iteration k is called a cycle of order k. Obviously, Fig. 25a shows a cycle of order one with one fixed point. Fig. 25b is a cycle of order two corresponding to the three-dynamical dynamics of a spiral with two cycles. Another case is Fig. 25c where the points P0 , P1 , P2 , . . . converge to a closed curve C on the Poincar´e map. The corresponding trajectories of the 3-dimensional dynamics spiral on the 2-dimensional surface of a torus. An example of chaotic 3-dimensional dynamics is given by the three differential equations dx/dt = −y − z, dy/dt = x + ay, and dz/dt = bx − cz + xz with positive constants a, b, c and only a quadratic nonlinearity. These equations have two fixed points P0 with x0 = y0 = z0 = 0 and P1 with x1 = c − ab, y1 = b − c/a, and z1 = c/a − b. In the neighborhood of P0 we observe unstable chaotic behavior of the trajectories. They seem to be sometimes attracted and sometimes repelled by P0 , although they are caught in a bounded region of P0 . The orbits round about P0 are completely non-periodic without recurrence of any pattern. Further on, the trajectories depend sensitively on initial states. Only tiny changes of them lead to completely different developments of trajectories after a few steps. Fig. 26a shows the 3-dimensional R¨ ossler attractor of chaos for numerical integration with a = 0.32, b = 0.3, and c = 4.5. The 3-dimensional chaotic behavior can also be studied on a 2dimensional Poincar´e map intersecting the state space in Fig. 26a with coordinates, for example, y = 0, x < 0, z < 1 [2.26]. In a further reduction we can consider the position of the n + 1-th intersecting point on the Poincar´e map as function of the preceding position of the n-th intersecting point, that is a discrete iteration equation xn+1 = f (xn ). In the case of the R¨ossler attractor, we get a bell-like curve of sequential cutting points. But they are not given by an ordered sequence of points, but randomly and statistically distributed (Fig. 26b). The discrete iteration equation of the R¨ossler attractor is similar to the function xn+1 = f (xn , α) = 1 − αx2n . It generates an infinite

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Fig. 26a.

Fig. 26b.

Chaotic R¨ ossler attractor

Poincar´e map and R¨ ossler attractor

sequence of bifurcations which happen at well-defined critical values α1 < α2 < · · · < αn < · · · leading to cycles of higher order. Their periods double at each new bifurcation (Fig. 27). The critical values concentrate at a certain value α∞ . Beyond α∞ , orbits with infinite

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Fig. 27.

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Cascade of symmetry breaking: Periodic doubling bifurcations

periods emerge with chaotic non-periodic behavior. The discrete dynamics of period doubling bifurcations ends in a chaotic scenario. It is governed by a law of constancy, because the critical values αn converge to α∞ with a constant ratio, the so-called Feigenbaumconstant δ ≈ 4.669. The period doubling cascade is numerically calculated in the following table. It can easily be done by a pocketcomputer and is a first example of experimental mathematics: n (period of orbit = 2n )

bifurcation point

αn − αn−1

0 1 2 3 4 5 .. . ∞ (unperiodic behavior)

0.75 1.25 1.3680989394 1.3940461566 1.3996312384 1.4008287424 .. . 1.4011552

0.5 0.1180989394 0.0259472172 0.0055850823 0.0011975035 .. .

δ=

αn − αn−1 αn+1 − αn

4.233738275 4.551506949 4.645807493 4.663938185 4.668103672 .. . 4.669

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The tree of periodic doubling bifurcations in Fig. 27 is also a cascade of symmetry breaking [2.27]. If the control parameter α is gradually changed, then the locally stable equilibrium states become unstable and new locally stable equilibrium states emerge. The tree bifurcates into sequential forks with increasing complexity representing the whole scenario of possible developments of a dynamical system. The symmetry is broken by actually realized solutions with changing control parameter. Thus, the complex tree of sequential bifurcations illustrates the possible developments of a dynamical system, but the path of actually realized branches represents the factual history of the system. Locally stable states are not only points in a state space, but also periodic patterns of behavior on limit cycles (e.g. Fig. 24c and 25a–b), quasi-periodic patterns of behavior on a torus (e.g. Fig. 25c), or chaotic patterns (e.g. Fig. 26a–b). A cascade of bifurcations and symmetry breaking can lead to these types of locally stable states. An example is the Navier–Stokes equation of fluid dynamics. One way to stir water in a glass is with a cylindrical rod, the rotation of which is driven by a machine with increasing velocity. The speed of rotation is the control parameter of the equations. When it is increased, one can observe fluid patterns with increasing complexity from a homogeneous fluid mass at rest, in uniform rotation, and with wavy cells up to mild and chaotic turbulence. A first mathematical model for the state of the fluid is a velocity vectorfield in the fluid domain. At each point in this domain is drawn a vector representing the velocity of the particle of the fluid. The instantaneous state of a whole vectorfield can be regarded as a point of a state space. An orbit in the state space represents the temporal development of a state. In Fig. 28, the state space is shown as a vertical plane. The third dimension represents the control parameter, i.e. the speed of rotation. The composite space of the state space and the control parameter is called the bifurcation space of a dynamical system illustrating the whole complex dynamics. In Fig. 28, the fluid at rest is represented by a point at the origin of the bifurcation space. The next step is the fluid mass in uniform

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Fig. 28. Thom’s superspace (“Big Picture”) of bifurcation and symmetry breaking [2.29]

rotation. In the third step, we observe a moderate increase in the rate of stirring. The fluid motion has separated into a stack of ringshaped cells. In spite of the increasingly complex fluid motion, the vectorfield is still stationary. It is a point attractor of the corresponding state space. In the next step the cells’ boundaries have developed waves, and the wavy cells are slowly rotating around the central axis. The wavy vortex phenomenon is represented by a velocity vectorfield in the fluid domain. The pattern repeats every few seconds. The periodic change in the fluid velocity vectorfield indicates a periodic limit cycle in the corresponding state space. In the bifurcation space the periodic attractor emerge after a Hopf-bifurcation. In the fol-

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lowing step we observe mild turbulence. The complete rings are still wavy vortices, but they wave irregularly. The corresponding velocity vectorfield is rather complex and never returns to an exact copy of an earlier state. The wandering of the state point within the state space fills out with a thickened torus or some other chaotic attractor. The last step shows fully developed turbulence. The dynamical system has suffered further bifurcations, from one type of chaos to another. Depending on their control parameter the equations of dynamical systems can have different state spaces with different patterns of behavior, from point attractors, periodic limit cycles and quasiperiodic tori up to chaotic attractors. The state of a system is a point wandering along the orbits in a state space. Is there a universal superspace to represent all kinds of state spaces and dynamical systems? In 1972, the French mathematician R. Thom introduced a bigger picture of bifurcation theory in a further step of abstraction [2.28]. In Thom’s Big Picture, every kind of attractor in a state space at a certain value of the control parameter is represented by a single point. In Fig. 28, these steps of modeling abstraction are illustrated. On the basis, there are the velocity vectorfields of the observed fluid patterns. In a second step, we get the corresponding attractors in state spaces depending on increasing values of the control parameter. In a third step, the dynamics in the bifurcation space is transformed into Thom’s superspace. Each and every value of the control parameter specifies a copy of the state space having its own superdynamics. Each of these superdynamics becomes a single point in the superspace, generating an orbit with increasing control parameter. The starting superdynamics determines a point corresponding to the lower endpoint of this curve, further superdynamics determine sequential points on the curve up to the final superdynamics with a point corresponding to the upper endpoint of this curve. The bifurcation points correspond to intermediate points in the curve, shown at the intersection of the curve and a surface. There are infinitely many of these sheets that accumulate in the end with increasing complexity.

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The sheets of the superspace are parts of the so-called bad set. In 1937, the Russian mathematician A. Andronov proved that the bad set consists of dynamical systems that are not structurally stable. A vectorfield is called structurally stable if sufficiently small perturbations of it stretches or slides the state space only a small amount. Thom’s Big Picture analyzes the structure of the bad set within the superspace. It represents all kinds of unstable situations leading to bifurcations and symmetry breaking with new locally stable states of equilibria. In the following chapters, we will recognize that locally stable states of equilibria are connected with the emergence of new phenomena in nature and society. Thus, Thom’s superspace is a universal abstract model for all kinds of evolutionary processes in nature and society. According to Plato, there are eternal, invariant and stable mathematical structures (Plato’s ideas) behind the instability, change and dynamics of phenomena we observe in the world. Thom’s book “Structural Stability and Morphogenesis” (1975) seems to be in the Platonic tradition. 2.3

Complexity, Nonlinearity and Fractals

Bifurcation trees of nonlinear dynamics leading to chaos attractors have nevertheless a remarkable property of symmetry: They are selfsimilar. Self-similarity defines the geometry of structures in which a small part when expanded looks like the whole. If one cuts a limb off a tree, the resulting object will resemble the tree itself in miniature. If one cuts a branch off this limb, the shape of the resulting object will again be similar to the limb and to the entire tree. Contrary to mathematical trees with infinite bifurcations, self-similarity in natural objects cannot be observed for arbitrary miniaturization. But many objects in nature are self-similar with a limited scale. Examples are the branching system of the bronchi in the lungs or the networks of streams that flow into rivers. Self-similarity is not limited to objects with treelike geometry. Some types of clouds are also self-similar. In reality, it is worthwhile to consider self-similarity at least in a statistical sense. In this case,

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a structure is self-similar if its parts, on average, are similar to the whole. The shore of an ocean, for instance, has inlets or bays, which often contain similar inlets or bays of smaller size themselves, and so on. Thus, Mandelbrot asked the famous question: “How long is the coastline of Great Britain?” [2.30] If one follows up all sections with smallest length of scaling, its length seems to become infinite. This example reminded him of geometrical figures which were analyzed by the Swedish mathematician H. von Koch in the beginning of the last century. Koch’s snowflake (Fig. 29a) is an exact self-similar shape that consists of four copies of itself, each of which is one-third the size of the whole set. In sequential generations each copy is divided into three parts, and the construction of the equilateral triangle is recursively iterated. The recursion corresponds geometrically to the self-similarity of the pattern that arises with arbitrary scaling. The length of the emerging figure increases with 3 · 4/3 · 4/3 · 4/3 · . . . ad infinitum. With analogue recursive procedures, we can construct the Hilbert and Sierpinski curves which fill out a plane with self-similar pat-

Fig. 29a.

Recursive program of Koch curve

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Fig. 29b.

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Recursive program of Cantor set

terns in iterated steps of increasing density. They seem to be more than lines with dimension one, but less than planes with dimension two. Their dimension is a “fraction” between the integers one and two. The fractal dimension can be illustrated by the geometrical dimension D of similarity. For Euclidean objects of dimension D, the length, area or volume of an object with edge length ε is proportional to εD . For example, a square with edge length ε has the area ε2 , a cube has the volume ε3 . For self-similar objects, one way to measure the length, area or volume of an object is to count the number of self-similar copies. If there are N copies each with an edge length ε, then the length, area or volume of the object is related to its dimension: N is proportional to εD . Thus, one gets D ∝ log N/ log ε. For Koch’s curve, the number of self-similar copies is N = 4 and the edge length is ε = 3. Therefore, Koch’s curve has a fractal dimension of D = log 4/ log 3 ≈ 1.26, which differs from its topological dimension 1. Another example is the Cantor set (Fig. 29b). It consists of two copies of itself, and the length of each copy is one third the length of the whole set, separated by an empty region whose length is also one third that of the whole set. Again, sets are generated in sequential steps with increasing recursion depth. The Cantor set seems to dissolve into a dust of points. Although its topological dimension is zero, it has the fractal dimension D = log 2/ log 3 ≈ 0.63. Self-similar mathematical objects consist exclusively of smaller copies of themselves. Our procedure to calculate the dimension of

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a fractal object is only useful if we know the number N of selfsimilar copies and the size ε of the original relative to each copy. For practical applications (e.g. a map or picture of a fractal object or real objects of the 3-dimensional world) we need a procedure to estimate the fractal dimension. The following procedure is directly motivated by the definition of fractal dimension: In a first step all points in the object are covered with N (ε0 ) squares or cubes of edge length ε0 . This step is repeated with squares or cubes of edge length ε1 = ε0 /2, then with ε2 = ε1 /2, and so on. By doing this, we get a function N (ε) sampled at the values ε = ε0 , ε1 , . . . . In theory, the dimension D is defined by limε→0 N (ε) = k · ε−D with a constant k. In practice, D can be estimated as D ≈ (log(N (εi+1 )/N (εi )))/(log(εi /εi+1 )). But, of course, it is inappropriate to make the squares or cubes smaller than the cells or particles that are considered as building blocks of the object. Fractals are constructed in a recursive process of iteration. In order to follow their iterated steps of construction on a plane rather than on a line, Mandelbrot looked at complex numbers instead of real numbers. Complex numbers z = x + iy consist of an imaginary number i and real numbers x and y. According to Gauss, they can be represented as points on a plane which is defined by a cartesian coordinate system with the real part Re(z) = x as x-axis and the imaginary part Im(z) = y as y-axis. Recursive equations zn+1 = f (zn ) correspond to iteration processes z0 → z1 → z2 → · · · of sequences of points on the Gauss plane. An example is the functon zn+1 = zn2 + c with a complex constant c, which is named in honor of Mandelbrot. For c = 0 the number is squared at each iteration, generating the sequence of points z0 → z02 → z04 → · · · . There are three possibilities for the sequence, depending on z0 : In the first case, the numbers become smaller and smaller, and their sequence approaches zero. Zero is a fixed point and is called attractor of the point sequence. All points less than a distance of 1 from this attractor are drawn into it. They are in the basin A(0) of attractor 0. In the second case, the numbers become larger and larger, tending to infinity. Infinity is the attractor for this process. All points further

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than a distance of 1 from zero are drawn into it. They are in the basin A(∞) of attractor ∞. In the third case, the points are at a distance of 1 from zero and stay there. Their sequence lies on the boundary between the two basins of attraction A(0) and A(∞), which, in this case, is the unit circle around zero. For c = 0, say c = −0.12375 + 0.56508i, the sequence z0 → z1 → z2 → · · · has the choice between these three possibilities, again. But the inner attractor is no longer zero. The boundary is no longer smooth, but resembles a fractal closed coast line of an island (Fig. 30a). Infinity is a distinguished attractor for all c of the Mandelbrot function. The boundary of the basin A(∞) of attractor ∞ for c is called a Julia set Jc . The self-similarity of these sets was already known to the French mathematicians G. Julia and P. Fatou who studied them at the end of the First World War [2.32]. Julia sets depend on the choice of the complex number c. For, say c = −0.12 + 0.74i, the Julia set is no longer a single, deformed circle but consists of an infinite number of deformed circles in a connected set (Fig. 30b). The interior of this Julia set is not attracted by one fixed point like in the former case (Fig. 30a), but by a cycle of period 3.

Fig. 30a.

Basin of an attractive fixed point

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Fig. 30b.

Basin of an attractive cycle of period 3 [2.32]

Fig. 30c.

Mandelbrot set

With his Mandelbrot set, Mandelbrot found a principle to decide which kind of Julia set a choice of c implies. The Mandelbrot set M contains all points c of the complex Gauss plane whose sequences of points, according to the corresponding Mandelbrot func-

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tion, do not only converge to infinity. Consequently, it follows 0 ∈ M (0 is element of M ), because for c = 0 sequences of points with |zn | < 1 converge to the attractor 0, but it is 1 ∈ / M (1 is not element of M ), because for c = 1 all sequences of points converge to the attractor ∞. For example, 0 ∈ A(∞) because of 0 → 1 → 2 → 5 → 26 → 677 → · · · . In Fig. 30c, all points of the Mandelbrot set in the window −2.25 < Re(c) < 0.75 and −1.5 < Im(c) < 1.5 of the Gauss plane are colored black. Intuitively, one recognizes that the buds on the fractal boundary repeat the shape of the Mandelbrot set with self-similarity and arbitrary miniaturization. The Julia sets Jc of the Mandelbrot set M can have extremely fractal structures, especially if c is in the interior of a bud or a germination point of a bud or any other boundary point of a bud. They were studied in computer experiments revealing the fascinating beauty of fractal geometry (Chapter 8.2). In 1982, A. Douady and J.H. Hubbard proved that all Julia sets of the Mandelbrot set are connected [2.33]. A Julia set Jc as boundary of the basin A(∞) is connected, if not all sequences of points, according to the Mandelbrot function with c, converge to infinity, that is 0 ∈ / A(∞). Thus, the Mandelbrot set can also be defined as the set of all complex numbers c the Julia sets Jc of which are connected. The other Julia sets are Cantor sets and dissolve into dust of points. Fractals have great importance for bifurcation and chaos theory. Obviously, the period doubling bifurcation tree of nonlinear dynamics leading to a chaos attractor has a self-similar structure (Fig. 27). It can even be related to the Julia sets of the Mandelbrot set. As the period doubling bifurcation tree takes place on the real axis, the elements c of the Mandelbrot set are varied now as real parameter. Imagine a path in the c-plane which begins in M and terminates outside of M . When c crosses the boundary of M , it will decompose into tiny buds with more miniaturized buds on their boundaries, and so forth. The associated Julia sets seem to explode into a cloud of infinitely many points. Obviously, the budding of the Mandelbrot set corresponds to the bifurcations of the periodic doubling scenario in nonlinear dynamics.

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Philosophically, fractal geometry opens new avenues to understand our world. The real world is not built up by idealized Euclidean forms and bodies like, for example, perfect spheres, cubes or cones. Mandelbrot had the great vision that, at least in a statistical sense and with limited depth, fractality is characteristic for many observed phenomena of reality. Nevertheless, he defends a platonic view of the world, because symmetry in the sense of self-similarity is hidden behind the sometimes bizarre and fractal silhouette of its phenomena. In mathematics, fractal geometry delivers new methods to analyze attractors of nonlinear dynamics. If an attractor is a fixed point, the fractal dimension is zero. For a stable limit cycle the fractal dimension is one. But for many nonperiodic (“strange”) attractors we get a fractal dimension. In this case, the dynamics of a system converge neither to a fixed point nor to a closed curve but to a bounded region of a state space that is filled up by irregular and nonperiodic trajectories with self-similarity. On Poincar´e maps, there are clouds of points reminding the observer of the Cantor set. The principles of fractal geometry were proven by analytic and axiomatic mathematical methods. But the complexity of fractals could only be discovered by computer-assisted visualization, due to the increasing capacities of modern computers.

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Chapter 3

Symmetry and Complexity in Physical Sciences

The symmetries of the laws and theories of physics and of the natural sciences in general became clear only after the application of group theory methods in the 19th and 20th centuries. In particular, F. Klein’s “Erlanger Program”, according to which the objective validity of geometric laws is defined by their invariance under certain groups of transformations, turned out to be the key concept for the mathematical explanation of the objectivity and invariance of the laws and theories of physics. Lie’s continuous groups became a valuable resource for classical physics. In classical mechanics, invariance under groups of transformations leads to important consequences. If the Lagrange equations of a physical problem are invariant with respect to an n-parameter symmetry group in the Lie sense, the n conservation quantities can be indicated explicitly. Conservation theorems of physical quantities, which have a long tradition in the philosophy of nature, as in the case of the conversation of mass, can now be traced to space-time symmetries. These general relationships between symmetry groups and conservation quantities are later found in an analogous fashion in the theory of relativity and quantum mechanics. Important characteristics of the modern concept of symmetry were already clear in Maxwell’s electrodynamics: a unified theory explained different physical phenomena which had been considered completely unrelated as recently as in the late 18th century, e.g. electrostatic charge, the effect of a magnet on a compass needle and

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the light from a candle. But in terms of group theory, electrodynamics does not have the space-time symmetry of classical mechanics (“Galilean invariance”), but what is termed Lorentz invariance. Since Einstein’s theory of special relativity, Lorentz invariance also determines the space-time symmetry of relativistically corrected mechanics. This is a global symmetry, since the Lorentz transformations modify all space-time coordinates in the same manner. The occurrence of electromagnetic interactions of moving charges had already been explained in electrodynamics by means of a novel symmetry designated gauge invariance. This is an example of a local symmetry, which takes into account local changes of the electrical field and the resulting magnetic fields. The gravitational fields of Einstein’s general relativity theory are also described by the transfer from the global symmetry of the special relativity theory to local Lorentz invariance. In relativistic cosmology, the differential-geometric theory of symmetrical spaces is applied, which Cartan had developed from the theory of spaces with constant curvature according to Gauss, Riemann, Lie, Helmholtz et al. The Platonic belief in a cosmos symmetrical on a large scale becomes clear once again, although it no longer employs the image of the ancient harmony of the spheres. In addition to the theory of relativity, quantum mechanics is the framework of modern physics. The symmetry of the Hilbert space formalism of (J. von Neumann) quantum mechanics was investigated by Wigner, Weyl and others, in the late 1920s, and a relationship was established with the unitary transformations of Hilbert space. In contrast to classical and relativistic physics, measurable quantities (“observables”) occur which are not interchangeable (“commutative”) in terms of group theory, i.e. there is a difference in the measurements, depending on the sequence in which they are measured. The unification of special relativity theory and quantum mechanics started with Dirac’s relativistic quantum mechanics wave equation for the electron in 1928, and is currently being investigated in quantum electrodynamics. The electromagnetic interaction of quantum electrodynamics is also based on a local gauge invariance (U(1) symmetry), as are the strong interactions which occur, for example, between the

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atomic nuclear particles protons and neutron (which are examples of “hadrons”), and which are investigated in quantum chromodynamics (SU(3) symmetry). Overall, the transition from global to local symmetries turns out to be a key concept to describe the occurrence of fundamental forces of physics. In addition to gravitational force, electromagnetic and strong interactions, the fourth fundamental force is currently identified as weak interaction. This force is well-known from radioactive β-decay and once again brings up the problem of left-right symmetry (parity) in nature. In 1967, Weinberg, Salam, Glashow, among others, explained a unification of electromagnetic and weak interaction with local gauge invariance of SU(2) × U(1) symmetry. Under conditions that are correctly defined by this unified theory, cases of symmetry breaking occur and can be observed experimentally using the resources of modern high-energy technology. For a unification of strong, weak and electromagnetic interaction, the SU(5) group is the smallest single group which comprises the corresponding gauge groups of these interactions. This theory predicts a tiny dilation in which there is no fundamental difference between the elementary particles of this interaction, but only one type of matter and only one fundamental force. In the evolution of the cosmos, the SU(5) symmetry would have existed for a fraction of the first second after the Big Bang. The subsequent space-time evolution of matter then consists of breaking of the basic symmetry and the occurrence of sub-symmetries with different particles and fundamental forces — a cosmic kaleidoscope, the symmetries of which are a function of spatial orders of magnitude and temporal development phases. The goal of the superstring theory seems to be a modern version of the Aristotelian “materia prima” with supersymmetry, in which all fundamental forces are indistinguishable. Do symmetry, transformation, and invariance only illuminate the structure of our physical models or do they represent real structures of the world? In the framework of modern physics, the emergence of the structural variety in the universe from elementary particles to stars and living organisms is modeled by phase transitions and symmetry

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breaking of equilibrium states. In the present state of superstring theories and M-theory, we do not have a complete theory explaining the evolution of matter with its increasing complexity. The presocratic wondering that “there is something and not nothing” is not dissolved. But the theory of complex systems opens new insights into the emergence of new structures by self-organization. From a methodological point of view, the question arises, how can we detect attractors of pattern formation in an immense variety of measured data. Complex data mining and time series analysis are challenges of the current theory of complex systems [3.1]. 3.1

Symmetry in Physics

From the point of view of everyday perception, the symmetry of space and time, i.e. their homogeneity and isotropy, is by no means self-evident. While in Euclidean geometry, space is of the same condition everywhere and in all directions, unchanging and unlimited, our senses give us the impression of the inequality of directions, of the changeability of points in space, and of the limited nature of perceptions. While physics proceeds on the assumption of a time that is constant and uniform, stress and fear can make minutes “seem like an eternity,” and hours of happiness can pass “in a few seconds”. The arbitrary capabilities of movement and action of the body as a unit, and its ability to adopt any given orientation, promote the opinion that we can execute these same movements everywhere and in all directions, and that space can be imagined as having the same properties, unlimited and infinite, everywhere and in all directions. If we continue to change our orientation, e.g. by rotation around the vertical axis, these same changes of positions in space are repeated over and over. Thereby, not only does the uniformity become clear, but also the inexhaustibility, the unlimited repeatability and continuity of certain spatial perceptions become clear. For example, our spatial perceptions of course gradually approximate geometric space, but are unable to completely achieve it in this manner. Therefore, it is necessary to define geometric figures such as lines, points, planes,

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etc. as basic concepts of a geometric theory, for which the physiological phenomena of points and surfaces etc. or corners, edges, surfaces etc. which can be technically produced are only approximated realizations. But the homogeneous and isotropic space in which we can move bodies without restriction in all directions does not have any metric. Only with the additional requirement that spatial dimensions, i.e. lengths and angles, be measured with rigid measuring rods, do we move from homogeneous sensory space to metric geometry. In the next abstraction step, Euclidean geometry is transformed into Cartesian geometry, in which geometric shapes such as points, lines, etc. are designated by (real) numerical coordinates, ordered sequences of numbers, equations, etc. Only now can we define the symmetry group of (Euclidean) space R3 . This is the 6-parameter Euclidean group that consists of the 3-parameter translation group and the 3-dimensional rotation group. Time is experienced in the form of changing positions where bodies or (idealized) mass points are located. The 1-dimensional quantity of all space points through which a mass point passes is a geometric path line. The process of the successive, point by point and continuous generation of a curve can be represented geometrically by defining the curve by a function x = x(θ) with a real, continuous parameter θ ≥ 0, whereby the curve originates with the constant increase of θ from a point of origin x = x(0). Since the selection of the parameter is specified only with the exception of one-to-one and continuous transformations, we can also speak of topological time. It designates only the sequence of points of time without a metric. It can be realized by any given continuous movement of a mass point. In particular, therefore, a straight line can also be selected. On the continuum of topological time, a metric is defined by any selected movement process, e.g. a standard clock. For that purpose, it is specified that identical Euclidean segments (“Pythagoras”) on the path of a mass point are traversed in equal intervals of time. With regard to this metric time, we designate a movement uniform, if the path traverses equal distances in equal intervals of time.

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The symmetry of 1-dimensional Euclidean time T (= R) is defined by the one-parameter translation group. Events occur at a given position at a given time, i.e. they can be represented as points (x, t) with x ∈ R3 and t ∈ T . In this case, the direct product can also be written R3 × T for the 4-dimensional space-time in which the events occur. The symmetry of this space-time in which, in addition to spatial rotations and translations, there are only temporal shifts, is naturally minimal. Mathematically, it is a question of a 7parameter group consisting of the 6-parameter Euclidean group (with 3-parameter rotation group and 3- parameter translation group) and the one-parameter translation group of time. In this space-time, it is correct to say of two events (x1 , tl ) and (x2 , t2 ) that they are spatially separate, even if they occur at different times. Physical events are related to space-time systems satisfying Newton’s law of inertia (“inertial systems”): mass points move uniformly on straight lines if they are left to themselves, that is no force is exerted on them. The laws of classical mechanics are invariant with respect to the Galilean transformations of all inertial systems. Intuitively, the invariance or symmetry of laws means that they are universally true, at any time and everywhere in the world if an appropriate inertial system as reference can be found. In general, physical laws and theories can be characterized by transformation groups in the sense of F. Klein’s “Erlanger Program” for geometrical theories. In the case of classical mechanics, the Galileo group Ggal consists of the following transformations: (1) The transition from an inertial system Σ to a system Σ  shifted in space around the vector ai is given by the transformation xi = xi +ai and t = t. This space transformation is obviously a function of three parameters, namely the components of the space-time constant vector ai . (2) The transition from a system Σ with the coordinates xi to a system Σ  with a rotated coordinate system is given by the transformation xi = aik xk and t = t with orthogonal matrix aik alk = δil = aki akl and the Kronecker symbol δil = 1, if i = l and δil = 0 otherwise. In vector notation, it is also abbreviated

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r  = Rr with the orthogonal rotation matrix R. The rotations in 3-dimensional space are also a function of three independent parameters. (3) The transition from a system Σ to a system Σ  displaced by the constant time interval b is given by the transformation t = t + b and xi = xi , which is a function of one parameter, the constant b. (4) The transition from a system Σ to a system Σ  displaced in relation to it at a constant velocity vi is given by the transformation xi = xi + vi t. These transformations are a function of three parameters, namely the space-time constant components vi of the vector of the relative velocity of Σ compared to Σ  . From transformations (1)–(4) we can also indicate the most general form of a Galileo transformation, namely r  = Rr + v t + a and t = t + b, which is a function of a total of 3 + 3 + 1 + 3 = 10 parameters, namely 3 parameters for a, 3 parameters for R, 1 parameter for b and 3 parameters for v . The Galileo transformations, with reference to the sequential execution of transformations, form a continuous 10-parameter Lie group: Since the elements are a function of a, R, b and v , we can also write the general group element as σ = σ(a, R, b, v ) = (a, R, b, v ). The identity transformation is the identity element t = (0, 1, 0, 0) of the group. The group operation has the form σ  • σ = (a , R , b , v  )(a, R, b, v ) = (a + Ra + bv  , R R, b + b, v  + Rv ). Obviously, the sequential execution of two transformations from Ggal again results in a transformation from Ggal . For each group element σ = σ(a, R, b, v ) an inverse element σ −1 = −R−2 (a − bv), R−1 , −b, −R−1v ) can be indicated, which satisfies the requirement σ • σ −1 = σ −1 • σ = I.

Several interesting subgroups can now be identified in Ggal [3.2]. Corresponding to the transformations (1)–(4) there are the following subgroups: (1) the 3-parameter (Abelian) group GT of the space translations, (2) the 3-parameter group GD of rotations in space, (3) the 1-parameter (Abelian) group Gt of the time translations, (4) the 3-parameter (Abelian) group G0 of the pure Galileo transformations. Additional examples of subgroups are the Euclidean group GE = GT × GD , which all contain space translations and rotations, and the subgroup U = Gt × G0 from the time translations and the pure Galileo transformations. U is important from a group theory point of view, since it is the maximum abelian invariant subgroup of Ggal .

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It can thus be shown that the Galileo group Ggal is not “simple” in the sense that it cannot be broken down into other groups. It can be proven that the Galileo group has the product representation Ggal = (GD × GT ) × U . We say that it is the semi-direct product of an (Abelian) group U with the (semi-direct) product of an (Abelian) group GT with GD . It should now be noted that the Galileo group Ggal and thus the space-time symmetry of classical mechanics is significantly more complicated than the symmetry of the Lorentz group which we will encounter in electrodynamics and relativistic physics. In the 18th century mechanics, like geometry, was transformed into an analytical theory, in which it became possible to solve problems of physics as a result of the solution of certain differential equations. The high point of this trend came in 1788, with “M´ecanique analytique” by Lagrange, i.e. 100 years after Newton’s “Principia,” after d’Alembert and L. Euler, among others, had worked on an analytical description of mechanics. To bring the Newtonian equation of motion into the Lagrange form, let us consider as an example the motion of a mass point with (Cartesian) coordinates xk (k = 1, 2, 3), t, the inertial mass m under the influence of an external force F (xi , x˙ i , t) with i = 1, 2, 3. The Newtonian equation of motion is: F (xi , x˙ i , t) = m

d2 xk = m¨ xk dt2

Langrange transformed the Newtonian equation of motion into a general model which can be applied to systems of mass points with and without rigid connections, to rigid bodies and the deformable continuums. For that purpose, he introduces a function (Lagrangian function) which characterizes the physical system in question. The Lagrangian function expresses the fact that a physical system is determined by its kinetic and potential energy. The Lagrange equations of motion are the direct result. This approach can be generalized for systems with a finite number of degrees of freedom. A system with n degrees of freedom is characterized by a Lagrangian function

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L(qk , q˙k , t) with generalized coordinates qk (t) and generalized velocities q˙k (t). Its development over time is described by the Lagrange equations: ∂L d ∂L − = 0. ∂qk dt ∂ q˙k In this case, therefore, the physical events correspond to the solutions qk (t) of a system of second-order differential equations. The event which actually occurs corresponds to a special solution which is unambiguously defined by initial conditions for qk (0) and q˙k (0). In place of Newtonian causality, which spoke of “forces” as the cause of effects or phenomena, there is now a formal system of equations which unambiguously determine the development of a physical system over time in the configuration space defined by position and time coordinates. A physical system is then characterized only by the Hamilton function H(qk , pk , t), which is a function of the coordinates of position, momentum and time. The 2n-dimensional space defined by the n coordinates qk and the n momentums pk is designated phase space. Each point (qk , pk ) of the phase space corresponds to a state of the system in question. The development over time of the system states in the phase space is determined by the Hamilton equations of motion: ∂H ∂H q˙k = , −p˙ k = . ∂pk ∂qk There is the following formal difference between the Lagrange and Hamilton equations: Since δL/δ q˙k generally contains q˙i , the second derivations of qi in the Lagrange equations are by time. In contrast, the Hamilton equations contain only first derivations of pi and qi . However, there are now twice as many equations. Both formalisms, however, supply exactly the same results. In Hamiltonian formalism, it is a question of first-order differential equations which, together with the initial conditions for qk and pk at the instant t = 0, determine the causality of the system.

The Lagrangian and Hamiltonian formalism open new insights into the symmetric structure of classical mechanics. Our everyday

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experience tells us that in nature, there exist systems which, in spite of changing ambient conditions, change little or not at all, i.e. they remain constant. Even the presocratic philosophy of nature could be summed up in the dispute between Parmenides and Heraclitus, according to which, on one hand, the world is unchanging and eternal, and on the other hand is in constant flux. In the 18th and 19th century, laws of conservation for mechanical energy, momentum, angular momentum, etc., were proven in mechanics. Thus, the conservation of physical quantities seemed to be a universal principle of nature. This idea was made precise by the fact that laws of conservation could mathematically be derived from space-time symmetry. To define the conservation quantities of a physical system in general, we start with a mechanical system of mass points with the Lagrange function L, the motions of which are determined by Lagrange equations of form, with suitable initial conditions of the location and velocity coordinates. A physical quantity E = E(xk , x˙ k , t) is called the conservation quantity of the system, and remains constant for all paths xk (t) that d are solutions of the equations of motion, i.e. dt E(xk , x˙ k , t) = 0. On the basis of this definition, the conservation quantities are first integrals of the equations of motion. The knowledge of a law of conservation thus has the mathematical advantage that only a first-order differential equation needs to be solved, and no longer a second-order differential equation such as the Lagrange equation. Now, the relationship between laws of conservation and space-time symmetry comes in [3.3]. The initially surprising tracing of conservation quantities to invariance characteristics of space and time can be explained with reference to simple examples. The equation of motion of a particle of mass m which is moved in one dimension under the influence of a potential V (x) reads m¨ x = − dV dx . We now assume that V (x) is invariant under translations, i.e. V (x) is constant, independent of x. Then m¨ x = 0. By integration, it follows that mx˙ is a constant, that is the law of conservation of linear momentum mx. ˙

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The equations of motion of this particle in two dimensions are: m¨ x=−

∂V ∂x

and

m¨ y=−

∂V . ∂y

We now assume that the potential V (x, y) is invariant under rotation around the origin. If we replace the Cartesian coordinates x, y, with polar coordinates r, θ with the polar angle θ, then the potential V is independent of the polar angle θ. In this case, ∂V = 0. On the other hand, ∂θ ∂x ∂V ∂y ∂V ∂V = + ∂θ ∂θ ∂x ∂θ ∂y = −y

∂V ∂V +x , ∂x ∂y

so that it follows, on the basis of the equations of motion, that ∂V d = m(y¨ x − x¨ y) = m (y x˙ − xy) ˙ . ∂θ dt From the invariance under rotation it therefore follows that the quantity m(y¨ x− x¨ y), i.e. of the angular momentum around an axis through the origin perpendicular to the plane is also constant.

For motions of the particle in three dimensions, in which the potential is invariant under rotation around any axis in space, each component of the angular momentum is constant. Therefore, for a spherically symmetrical potential, the value and direction of the angular momentum are conserved. In general, the following is apparently true: if the characteristic functions L (or H) of a physical system are independent of the location coordinate qk , i.e. if ∂L/∂qk = 0, then it follows that p˙ k = 0, i.e. the corresponding momentum pk is constant. If L (or H) is independent of time, it follows that H is constant. The Hamilton function, describes the energy of a physical system. To summarize, therefore: the conservation of energy follows from the homogeneity of space. Under the influence of F. Klein’s “Erlanger program”, Noether [3.4] demonstrated how the 10 conservation quantities of mechanics follow from the invariance characteristics of the Lagrange function and Hamilton’s action integral in relation to the (infinitesimal) transformations of the 10-parameter Galileo group. This represents the

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application of a general mathematical law to the special physical case of mechanics. Noether’s theorem claims in general that, for a mechanical problem, there exist n conservation quantities if the equation of motion is invariant under an n-parameter continuous group of transformations of the (3 + 1)-dimensional space-time continuum. On the other hand, we already know that the space-time of classical mechanics is determined by the Galilean principle of relativity, i.e. the equations of motion of a closed system of mass points are invariant under the transformations of the 10-parameter Galileo group. On the basis of Noether’s theorem, therefore, the 10 conservation quantities of a closed mechanical system are fully defined. The mathematical context of the Noether theorem not only makes possible a complete determination of conservation laws of classical mechanics, but also of electrodynamics, relativistic physics, and, with appropriate modifications, applications in quantum mechanics. In the beginning of the 20th century, physicists were shocked by the fact that the symmetry of classical space-time seemed to be violated by electrodynamics. According to the Galileo group of classical mechanics, the velocities of inertial systems are added relative to one another. But in electrodynamics, the constancy of light demands that the speed of light is independent of the speed of the light source. Thus, there is no velocity faster than light that results from adding velocities of inertial systems to the velocity of light. In his famous paper on the electrodynamics of bodies in motion (1905), Einstein unified the space-time of classical mechanics and electrodynamics. Therefore, Einstein considered inertial systems satisfying the principle of the constancy of light. To guarantee the invariance with respect to coordinate transformations, both for the equations of Newtonian mechanics and also those of Maxwellian electrodynamics, Einstein replaced the Galilean transformations with the Lorentz transformations. Mathematically, Einstein’s space-time is represented in the Minkowskian geometry. In Minkowski’s presentation, space and time are combined into a 4-dimensional space-time M 4 with Cartesian

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space coordinates x, y and z, and the time coordinate t. For the sake of simplicity, the units are selected so that the speed of light equals one, i.e. c = 1. Then the units of length and time are commutative, whereby 1 sec = 299 792.458 m and 1 year = 1 light-year.

Fig. 31.

Space-time symmetry of special relativity

Bodies at the speed of light move in this 4-dimensional model on straight lines at 45◦ to the t-axis. In accordance with the Pythagorean theorem, they form the light cone t2 = x2 + y 2 + z 2 (Fig. 31). On account of the constancy of the speed of light, future or past events can only lie within the light cone (“timely events”). Mass particles move on straight lines (“uniformly”) or curves (“nonuniformly”) within the cone; photons move as massless particles of light on the surface of the cone. The distance OQ from the origin O to a point Q with coordinates t, x, y, z in the Minkowski world M 4 is OQ2 = x2 +y 2 +z 2 −t2 , which differs from the Pythagorean metric term in a 4-dimensional Euclidean geometry by the minus sign. If Q lies on the surface of the cone, then OQ = 0; if Q lies inside the surface of the cone, then OQ > 0. For our considerations of symmetry in the Minkowskian world M 4 , it is appropriate to replace the Galilean

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coordinates with the designations x0 = t, x1 = x, x2 = y, and x3 = z. The Lorentz transformations which leave the Minkowskian metric invariant have the form xα = a0α x0 + a1α x1 + a2α x2 + a3α x3 + aα , with α = 0, 1, 2, 3 [3.6]. One of the fundamental consequences of the Lorentz transformations is the rejection of Newton’s absolute time. The measurement of time is no longer identical in any inertial system, but depends on its velocity. Today, this is a well confirmed fact of measurement. Atomic clocks moving with high speed (e.g. by airplanes) are slower relative to resting ones. In accordance with Einstein’s summation convention, Lorentz transformations can also be written xα = aαβ xβ + aα with β = 0, 1, 2, 3. These transformations form the inhomogeneous Lorentz group or Poincar´e group Gpoi . For aα = 0, we get the homogeneous Lorentz group. The subgroup of space-time rotations characterizes the isotropy of the Minkowski world, while the invariance of the Minkowskian metric with respect to the subgroup of space-time translations expresses its homogeneity. The Minkowski world is also invariant with respect to spatial reflection and reversal of time. Intuitively, invariance with respect to a reversal of time means that if a natural event is filmed, and the film is run backwards, the natural event corresponds to a process that does not violate the natural laws. The laws of conservation are also consequences of the space-time symmetry of the Minkowski world. Specifically, the conservation of momentum results from the invariance with respect to spatial translations, the conservation of energy from the invariance with respect to temporal translations, the conservation of angular momentum from the invariance with respect to spatial rotations, and the conservation of the center of mass from the invariance with respect to uniform motions. These symmetries are global, i.e. the laws of nature are invariant with respect to them only if the same transformation is applied for all four points of 4dimensional space. Lorentz invariance therefore includes the general assumption that the same laws of nature as in a research laboratory apply everywhere, i.e. the laws of physics always have the same form in any given coordinate system, regardless of how these systems are displaced or rotated, as long as they move at a constant velocity

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relative to one another. The global symmetries of Minkowski space therefore give us the freedom to arbitrarily select our laboratory’s coordinate system in the context of the above-mentioned conditions. Like Galilean invariance, Lorentz invariance is now identified as a global symmetry, since the Lorentz transformation change all points of Minkowski space-time in the same manner, and thus leave the laws of special relativity theory unchanged without the occurrences of forces. For example, if two astronauts are moving at a constant motion relative to one another, the transformation that transforms their two coordinate systems into one another is identical for all points. An intensification occurs if the two observers can also accelerate relative to one another, i.e. move differently “locally” or vary from the “global” constant relative motion to one another. Under these conditions, it will be initially suspected that the two observers would not derive the same (“invariant”) laws of physics, because an accelerated observer seems to be exposed to forces, such as centrifugal force during rotation. In his general theory of relativity, Einstein formulated a connection between these accelerations and the gravitational forces of masses. There are many observations and experiments confirming variances from the flat Minkowski metric under the effect of strong gravitational fields. Historically, Eddington’s 1919 observation is worth noting, namely that beams of light from distant stars are deflected in the sun’s gravitational field, although according to special relativity theory they ought to follow the Minkowski cone. Thus, the shortest connection of a light beam with maximal speed between two points in a gravitational field is not a straight line in sense of Euclidean geometry, but a curved line the curvature of which corresponds to the strength of the gravitational force. Therefore, Einstein assumed a kind of differential geometry to describe the curved spacetime of gravitational fields. The Minkowskian geometry is reduced to local regions where gravitational fields are absent. Thus, Einstein demands that at least “locally,” i.e. in very small segments of space-time in which the gravitational field does not change, an in-

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ertial system can be selected, whereby the effect of gravitation is eliminated. Historically, this principle was preceded by Galileo’s observation that all bodies, regardless of their properties, fall toward the earth at the same acceleration (if we overlook air resistance). The equivalence of heavy and inert masses had been experimentally confirmed. We all have seen the pictures of astronauts in orbit, who experience weightlessness during free fall in the earth’s gravitational field. On the assumption of a curved space-time the equations of motion had to be reformulated. An equation is valid in a general gravitational field if the following requirements are satisfied: (1) The equation is valid without the effect of gravitation, i.e. in this case, it corresponds to the laws of the special relativity theory. (2) The equation is invariant in general (“covariant”), i.e. it retains its form (“form invariance”) with respect to general coordinate transformations of a curved manifold. A gravitational field is therefore described as a so-called pseudo-Riemannian manifold with local Minkowski metric (Fig. 32), which replaces the local Euclidean metric in a Riemannian manifold [3.7].

Fig. 32.

Space-time symmetry of general relativity

Actually, the laws remain invariant if we include the gravitational field in the corresponding equations. That is precisely what occurs in the relativistic equations of motion and gravitation. The “local” deviations from the global symmetry are therefore eliminated by the assumption of additional force fields. Therefore, the equations are

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said to have local symmetries. In this sense, the relativistic theory of gravitation arises from the transition from the global Lorentz invariance of flat Minkowski space-time to local Lorentz invariance of a curved space-time. In general, we are already defining a model that will turn out to be fundamental in physics. If certain physical laws are invariant, then invariance under local symmetry can be achieved by introducing new force fields. We can also say that local symmetry is connected with the emergence of new force fields [3.8]. In relativistic cosmology, the universe is described as a whole system. Everyone on earth can be convinced of the symmetrical characteristics of the universe on a cosmic scale, at least to some extent. If we, from the earth, observe the “starry heavens overhead,” the naked eye and the strongest telescope always see the same conditions, a more or less uniform distribution of matter visibly condensed into heavenly bodies. These observations led to a general cosmological postulate, according to which matter is, on the average, uniformly distributed over the entire universe (homogeneity) and its characteristics remain unchanged, regardless of the observer’s angle of sight (isotropy). In this sense, homogeneity must be understood in the sense that gases, for example, can also be called homogeneous: homogeneity does not relate to the universe in detail, but to cells having a diameter of 108 to 109 light years, in which individual irregular condensations of matter can occur in the form of galaxies. Historically, the cosmological postulate corresponds to modern experience beginning with the Copernican revolution. Mankind and the earth on which it lives do not occupy a special position in the universe. In the cosmological postulate of contemporary astronomers, Bruno’s grandiose vision becomes a reasonable working hypothesis. In 1929, E.P. Hubble discovered that the speed of the receding motion of the galaxies increases with the distance between a group of galaxies and its observer. He reported observing that the light from very distant galaxies was shifted to the red portion of the spectrum, i.e. to longer wavelengths. The basis of this explanation is the Doppler effect, according to which the wavelengths of light emitted by a moving light source seem longer to a stationary observer when

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the light source is moving away, and shorter when it is approaching. The redshift can therefore be used to measure the speed at which galaxies are moving away from an observer. The amount Z of the redshift is equal to the relative increase ∆λ of the recorded wavelength λ, i.e. Z = ∆λ/λ. From the redshift, Hubble was able to calculate the speed at which galaxies are receding, and thus the total speed at which the universe is expanding.

Fig. 33.

Space-time symmetry of the cosmological principle

To explain the symmetry assumptions of the cosmological principle mathematically, Cartan proposed a differential geometry of symmetrical spaces (cf. Sec. 2.1). In such a geometry, all space points must physically undergo the same development, and must be correlated for time so that to an observer, all points at a fixed distance from him appear to be in exactly the same stage of development. In this sense, the spatial state of the universe at each point in the future (+) and past (−) must appear homogeneous and isotropic to the observer P (Fig. 33). Geometrically, for example, let us provide our observer, located in the center of the Milky Way, with a standard system of coordinates. The direction of three spatial coordinates xµ can be defined, for example, by the lines of sight from the observer

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to typical galaxies. For the time coordinate t, we can select a “cosmic clock”, e.g. the radiation temperature of a black body which decreases in monotone fashion everywhere. Therefore, the “cosmic standard coordinate system” of an observer is defined by transformations xµ → x ¯µ , t → t¯ = t, in which the physical condition remains unchanged, e.g. form invariance applies for the gravitational potential and the energy-momentum quantity of matter. Mathematically, therefore, the universe is portrayed as a 4-dimensional space-time manifold whose 3-dimensional “spatial” sub-spaces are isotropic and homogeneous. That was the assumption of the “cosmological principle”. In terms of differential geometry, therefore, it was the assumption of an isometry group which — in purely mathematical terms — makes it possible for us to define the “cosmic” metric of the 4-dimensional universe [3.9]. In 1935/36, H.P. Robertson and H.G. Walker indicated the conventional standard form of this metric. It depends on a world radius R(t) which increases in the expanding universe with the time t. Up to this point, the geometric description of the universe follows exclusively from the cosmological principle. R(t) remains an unknown time-dependent function. To be able to verify the symmetry characteristics of the universe in terms of physics, the “radius” R(t) must be calculated. For that purpose, assumptions concerning the material characteristics of the universe are necessary, like those expressed in Einstein’s gravitational equations. Therefore the Robertson– Walker metric must be defined as the solution of the gravitational equations. On the assumption of the cosmological principle, i.e. the assumption of a homogeneous and isotropic universe, we get standard models for three possible values k = +1, −1 or 0, by means of which spatial curvature is defined. Mathematically, these standard models are described by the development R(t) of the universe by means of first-order differential equations which can be derived from Einstein’s gravitational equation. In each case, this is a 4-dimensional spacetime manifold whose 3-dimensional homogeneous “spatial” subspaces expand temporally isotropically.

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Fig. 34. models

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Cosmic expansion of the three homogeneous and isotropic standard

Fig. 34 illustrates these subspace expanding in time as “spatial cross sections” for the three possible cases of k. In the case k = −1, each spatial cross section is a 3-dimensional Lobachevski geometry L3 with negative curvature. For k = 0, the spatial cross sections are 3-dimensional Euclidean spaces. For k = 1, they are spherical or elliptical spaces. In each case, there is an initial singularity, in which the spacetime curvature is infinite. Cosmologically, this is designated the Big Bang. According to this theory, the universe initially expanded very rapidly, and then continued to expand somewhat more slowly. In the case k = 1, the expansion reverses to a collapse, which represents a new singularity. We then speak of a closed universe. For k = 0 or k = −1, the expansion continues, but more rapidly in the case k = −1. Once it has been formed, the universe remains in existence and unlimited in both cases. We can therefore also speak of an open universe.

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The cosmological principle and the theory of relativity no longer suffice to explain this regularity and symmetry of the universe. Modern cosmogony is merging with quantum mechanics and elementary particle physics into a theory of physics in which the evolution of the universe is explained by the step-wise emergence of elementary particles, atoms, molecules, etc. Against this background, it can then be shown how, in the individual phases of development, some of the currently known basic physical forces of strong, weak and electromagnetic interaction initially prevailed, until the current structure of the universe with its macroscopically predominant gravitational force arose. Modern cosmology therefore regards the universe as a gigantic high-energy physics laboratory requiring a unified theory of natural forces for its complete explanation. Up to this point, the global and local symmetries of relativity have been considered as invariance of classical and relativistic theories. Historically, however, the theory of relativity was also the impetus for new ideas on a unification of various physical theories and explanations for the emergence of matter. The first attempt at a unified theory of matter in the 20th century dates back to the physicist Mie [3.10]. In his 1912–1913 publication on the “Principles of a Theory of Matter,” he attempted to define a link between the existence of electrons and gravitation. The mathematician Hilbert, in his 1915 and 1917 publications on “The Principles of Physics” established a unified mathematical theory of matter, in which the approaches adopted by Mie and Einstein are taken into consideration Hilbert’s proposal is also of great methodological interest, since it applies to physics the axiomatic method which Hilbert had previously explored in mathematics [3.11]. He celebrates his derivation of Einstein’s gravitational equations and the Maxwell–Mie equations of electrodynamics as the greatest triumph of the axiomatic method. Mathematical elegance, methodical simplicity and beauty were for him the motivation for a unified theory of matter. Such a unified theory would be the secularized version of an ideal of natural philosophy, which since the days of Pythagoras had linked harmony and beauty to mathematical regularity. During that time, gravitation

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and electromagnetics were the only known physical forces. Thus, he seemed to suggest a kind of world-formula. Mie used a nonlinear augmentation of Maxwell’s electromagnetic equations out of which elementary particles (e.g. the electron) would arise in a natural way. A world-function was introduced as an energy functional depending upon electric field intensity, magnetic flux density, and electromagnetic potential. The world function satisfied relativistic invariance. A specific choice led to a static spherically symmetric electric potential. Mie got a spherical model for the electron with a certain radius and electric potential. The Lorentz invariance that was built into the theory permits this solution of his equation to travel with any speed up to the limiting velocity of light with relativistic effects (e.g. Lorentz contraction). Although Mie’s atomic theory was physically false, it inspired Einstein’s life-long belief that the emergence of elementary particles should be founded by solutions of nonlinear differential equations. In short, emergence of matter results from nonlinearity. But Einstein’s ideas of unification with symmetry and emergence with nonlinearity could only be successful in the framework of quantum mechanics. The first atomic models proceeded on the assumption of visual symmetry characteristics that recall the planetary models of Antiquity. Bohr’s atomic model of 1913 was the first to provide information about how the electrons are distributed around the nucleus on the shell [3.12]. According to Bohr, the hydrogen atom consists of one proton and one electron. The negative electron moves centrosymmetrically like a planet in the Aristotelian planetary model on an orbit of radius r without any loss of energy (i.e. it emits no radiation) at a linear velocity v around the positive nucleus. The orbit is stable, because the centrifugal force which acts on the electron is equal to the Coulomb attraction between the electron and the nucleus. The energy E of the electron in its orbit is composed of potential and kinetic energy. According to the energy equation, as a function of the radius r, all values are allowed from 0 (for r = ∞) to ∞ (for r = 0).

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Fig. 35.

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Central symmetry of Bohr’s atomic model

To make the model compatible with the atomic spectra, Bohr assumed a quantization requirement. He linked the orbital angular momentum mvr of an electron with the mass m with Planck’s constant in the equation mvr = n · h/2π. For the “principal quantum number” n, only whole numbers 1, 2,. . . may be used. For each value of n there is a centro-symmetrical orbit with a defined energy E, which corresponds to a discrete energy level of the atom (Fig. 35). The most stable state of an atom is the lowest energy state. Higher orbits or states are called excited. According to Bohr, transitions between different orbits are possible when the amount of energy corresponding to the energy difference between the states in question is either input (absorbed) or emitted in the form of electromagnetic radiation (photons). Bohr used this model to calculate a theoretical spectrum for hydrogen that is in good agreement with the measured spectrum. The further development of Bohr’s atomic model recalls the history of the Aristotelian planetary model. Even then, the original spherical symmetry had to be restricted on the basis of more accurate observations by additional ad hoc hypotheses, and thus adjusted to fit reality. When transitions in heavy atoms were investigated, Bohr’s model no longer sufficed.

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Thus, Bohr’s atomic model was expanded to include elliptical orbits. Elliptical orbits, in contrast to circular orbits, have two degrees of freedom, since they are defined by two semiaxes. To describe the atomic spectra by transitions between elliptical orbits, two quantum conditions are therefore necessary. The orbital quantum number k is used in addition to the primary quantum number n. To explain spectra of atoms with a plurality of electrons, k was replaced by the secondary quantum number l. The secondary quantum number l defines the orbital angular momentum of the electron. The magnetic quantum number m was introduced as the third quantum number, to define the inclination of the plane of an elliptical orbit in relation to an external magnetic field. In spite of this and other improvements (e.g. the introduction of the spin quantum number as the fourth quantum condition), the atomic models failed for the same reason the symmetry models of the planetary theory of Antiquity had to be given up. There was no physical theory to explain these models. In Bohr’s atomic model, electrons are imagined as particles moving like mass points on fixed orbits. But depending on the manner in which the test is conducted, electrons can also behave like waves. For example, the electron of the hydrogen atom can be understood as a spherical, stationary wave in the space around the atomic nucleus, whose maximum amplitude is described by a corresponding wave equation. The electron is thereby described by a wave function ψ(x, y, z) in the space coordinates x, y and z. This interpretation, promoted by L. de Broglie, led to Schr¨ odinger’s quantum mechanics, which proceeded on the basis of the wave model. In classical Hamiltonian mechanics, a closed system is described by a Hamiltonian function. An example is a pendulum that oscillates with different energies. We then say that the system is in different states. In an analogous manner, a hydrogen atom is in the fundamental state or in one of the many excited states. In Schr¨ odinger’s sense, the states of an atom are represented by wave functions. M. Born established a connection between the wave and particle image of matter, by suggesting that an electron wave must be interpreted from

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the standpoint of probability. Places where the (square of the) magnitude of the wave is large are places where the electron is more likely to be found. Places where the magnitude is small are places where the electron is less likely to be found. In general, therefore, the states of a quantum system are described by the functions of a Hilbert space, i.e. a functional space of functions (“states”) as elements. The quantum system must be represented by a Hamiltonian operator, i.e. a functional depending on wave functions as system states. In classical Hamiltonian mechanics, the development of a closed system over time is described by the Hamiltonian equations of motion. In quantum mechanics, the development of a state over time is given by the time-dependent Schr¨ odinger equation. Physical quantities such as position vector, momentum, angular momentum or energy must also be represented by operators (“observables”) of the corresponding Hilbert space. Mathematically, the operator approach is a significant difference of quantum mechanics to classical mechanics. It results from the probabilistic description and the wave-particle dualism of the quantum world. Consequently, symmetries of a quantum system are defined as invariant properties of the corresponding Hamiltonian operator that can be explained in terms of group theory [3.13]. Let us imagine the motion of an electron around an atomic nucleus. Without external influence, it would be assumed that the Hamiltonian operator of this system possesses full rotational symmetry. This symmetry can be reduced by external influences. If an atom is embedded in a crystal lattice, the rotational symmetry corresponding to the bonds of a finite number of surrounding atoms can be reduced to a finite rotational group. Further on, electrons are not distinguishable, so that their position can be permuted in any given manner. Mathematically, that means that the Hamiltonian operator characterizing an atom with n electrons is invariant with respect to the permutation group with n! permutations. Notable consequences of symmetries with which we have already become acquainted in classical physics and relativity theory are the laws of conservation of physical quantities. According to the probabilistic approach of quantum mechanics,

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an observable of a quantum system is said to be conservative if its mean value or expected value in each quantum state of the system does not vary with time. Quantum mechanics is also invariant with respect to space reflection. But does it possess time symmetry, like the equations of motion of classical mechanics? In classical terms, the Hamiltonian equations of position q and momentum p of a particle must be invariant with respect to the transformation t → −t, that means q  (t) = q(−t) and p (t) = −p(−t). That corresponds to the idea that all positions of the corresponding particles are reversed. For example, a particle which is ascending at the time t = 0 and reaches the highest point at time t = 1, is replaced by a particle which falls from the same height at time t = −1 and reaches at the ground at the time t = 0. Another example of classical and relativistic physics comes from astronomy. Planets surrounding the sun could run in both directions without violating the laws of motion. Actually, time reversal has never been observed in the macroscopic world.

a

b Fig. 36.

Time reversibility of planets

Mathematically, the invariance of the macroscopic equations of motion with respect to the transformation t → −t can easily be proved. In the Newtonian version of mechanics, the acceleration of a body is proportional to the interacting force. Acceleration is the time-depending change of velocity. Velocity is the time-depending change of position. Thus, acceleration is the time-depending change of the time-depending change of position, i.e. the second derivation of the position of the body with respect to time. That is the reason why

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time is squared in the Newtonian equation of motion. If the positive values +t of forward running time are replaced by the negative values −t of backward running time, then the square t2 = (−t)(−t) = (+t)(+t) remains unchanged. In quantum mechanics, position and momentum are replaced by corresponding operators. Instead of measurement values, there are expected values in specified states (wave functions) of the system. Wigner introduced a time-reversal operator that satisfies the time symmetry of the corresponding Hamiltonian operator equation. But time symmetry of quantum mechanics is not merely the result of formal requirements. In the collision process of elementary particle physics, the time reversal operator describes a permutation of the incoming and outgoing particles that can be confirmed in experiments. In Fig. 37, both diagrams a and b can be read two ways — either as electron–photon scattering, whereby the electron is represented by the solid arrow and the photon as the broken line (a), or as positron–photon scattering, whereby the positron is represented as a downward arrow (b).

a

b Fig. 37.

Time reversibility of elementary particles

So far, we have investigated the symmetries of a Galileo invariant quantum mechanics with slow velocities like in classical mechanics. But actually, elementary particles like electrons move with high speeds near to or even with velocity of light. The goal of a Lorentz

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invariant quantum mechanics, that is a combination of special relativity theory and quantum mechanics, is pursued in quantum field theory. The first historical example of a relativistic quantum field theory was quantum electrodynamics, that is the combination of Lorentz invariant electrodynamics with quantum theory. Dirac’s initial approach to a relativistic quantum mechanical wave equation of the electron in 1927/28 turned out to be heuristically fruitful, since it led to what at that time was the surprising prediction of an antiparticle of an electron e− with negative charge which differs only by the reverse charge e+ (“positron”) [3.14].

Fig. 38.

Particle-antiparticle symmetry

Dirac’s prediction was magnificently confirmed by the discovery of positrons in cosmic rays. Fig. 38 shows the bubble chamber photograph of an e− e+ pair generation resulting from the collision of a photon (γ-quantum) with an electron. (The paths of the charged particles are curved “left” and “right,” as a function of their charge, on account of the application of a magnetic field perpendicular to the plane of the image.) Obviously, Dirac discovered a new symmetry of matter: For electrons there are antiparticles, namely positrons. Mathematically, the particle–antiparticle symmetry is described by

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an operator (“charge conjugation”), which transforms a particle in its antiparticle with conjugated charge. The particle–antiparticle symmetry seems to be fundamental in the sense that for every particle there is an antiparticle. An antiparticle can only exist if, at the site of its generation, the corresponding antiparticle originates at the same time. It is destroyed if it encounters a particle with which it is annihilated. Energy thereby originates in the same quantity as must be applied during generation. According to current elementary particle theory, matter consists strictly symmetrically of a particle world and an antiparticle world, which are connected to one another by the origin and annihilation of their elements. On one hand, of course, this particle-antiparticle symmetry has been confirmed by the experiments of high-energy physics, but on the other hand the universe seems to consist of more particles than antiparticles. This asymmetry would be explained as symmetry breaking at the origin of the universe, which will be discussed later in more detail. The basic theme of quantum electrodynamics is the interaction of particles of matter (e.g. electrons) or wave fields of matter with electromagnetic fields. Besides electromagnetic force modern physics distinguishes further fundamental forces like strong, weak and gravitational force. In the theory of relativity we learnt that gravitation arises by transition from the global Lorentz invariance of flat Minkowski space-time to local Lorentz invariance of a curved spacetime. In the framework of quantum physics, all physical forces can be introduced by transition from a global to a local symmetry. According to Weyl, forces are interpreted as so-called gauge fields compensating local deviations of a global symmetry. This is a fundamental insight into the theory of physical forces. So, let us briefly recall the difference between global symmetry and local symmetry in general. For example, imagine a balloon which is covered by a grid of coordinates (Fig. 39a). If the balloon is rotated around its axis around the center of the sphere, its shape remains unchanged or invariant (Fig. 39b). This invariance or symmetry is global, since all points of the surface were rotated by the same angle. For a local symmetry, the

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a

b Fig. 39.

c

Global and local symmetry

sphere must also retain its shape when the points of the sphere are rotated independently of one another by different angles (Fig. 39c). But the surface of the sphere is thereby distorted, i.e. forces occur on the surface of the sphere between the points. We therefore frequently speak of dynamic (“local”) symmetry. The forces or force fields compensate for the local changes in symmetry and retain the overall symmetry of the system (“shape of the balloon”). Therefore, we could also speak of a “restoration of symmetry” after local changes. In electrodynamics a magnetic field compensates a local change of an electric field, i.e. the movement of a charged body, and preserves the invariance of electromagnetic field equations. A bird on a high-tension line survives by global symmetry. There are no local differences of potentials. If the bird has contact with the high-tension pole, there is a local difference and the bird is killed. In quantum electrodynamics an electromagnetic field compensates the local change of a material field (the phase deviation of an electronic field) and preserves the invariance of the corresponding field equations. Mathematically, the phase deviations of a wave function ψ of an electron are described by transformations ψ → eiα ψ with a (unitary) 1 × 1matrix of the phase factor eiα . So the electromagnetic force is defined by a local (unitary) U(1) symmetry group of transformations. Such

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a gauge group characterizes a physical interaction mathematically in terms of local symmetry. But the existence of a physical force is an empirical question which, of course, cannot be derived from an a priori demand of local symmetry. We are already familiar with electromagnetic interactions from everyday life. The emission of electromagnetic waves by an accelerated atom is familiar, for example, from radio antennas or X-ray machines. On the other hand, the weak interactions in atoms are much less frequently observed, e.g. in the β-decay of a neutron which is transformed into a proton with the simultaneous emission of an electron-antineutrino pair. Initially, it seems that weak and electromagnetic interactions have little in common. The weak force is approximately a thousand times weaker than the electromagnetic force. While electromagnetic interaction can act over a great distance, the weak force acts only at distances that are significantly less than, for example, the radius of the neutron. Radioactive decays are much slower than electromagnetic decays. In electromagnetic interactions (e.g. dispersion of an electron on a proton), in contrast to the βdecay, no elementary particles are transformed into other particles. The particles that participate in the weak interaction are called leptons (from the Greek word for “tender”), e.g. neutrinos, electrons or muons. One of the most exciting differences was discovered in the 1950s. While electromagnetic interaction is invariant with respect to spatial reflection weak interaction violates parity or left–right orientation in space. In contrast to the other fundamental forces of physics, the spin of elementary particle plays a major role in weak interaction [3.15]. It can be imagined roughly as its characteristic angular momentum. The spin of a particle is represented by a vector which is parallel to the axis of rotation. It cannot be increased or decreased, and in the case of the lepton is h/4π (abbreviated 1/2). Spin 1/2 particles can only assume two directions in space allowed by quantum mechanics: the spin is either in the velocity direction of the particle or in opposite direction.

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In this context, we also speak of the right and left-handedness (chirality) of the particle. For example, if you hold your right hand so that the four fingers point in the direction of rotation of the rotating particle, the right thumb points in the velocity direction. The handedness of an electron, for example, can be reversed experimentally, by decelerating it and accelerating it in the opposite direction, but without changing the spin. Since massless particles (such as neutrinos and their anti-particles) move at the speed of light, such a deceleration and related change of orientation is not possible. Thus, they always retain their chirality or their helicity. In 1956, the theoretical physicists T.D. Lee and C.N. Yang described experiments in which the leptons might prefer a specified helicity. In fact, the experiments (e.g. those of the experimental physicist C.S. Wu), showed that for weak decays, particles are emitted only left-handed and antiparticles only righthanded [3.16]. In concrete terms, the neutrinos which seem to have exclusively weak interaction occur only as a left-handed helix (νL ), and antineutrinos only as a right-handed helix (νR ). Only the left-handed helix portion (e− L ) participates in the β-decay of the neutron (analogously for the muon). On the other hand, the electromagnetic interaction is characterized by no helicity. Both helix portions of the electron participate equally. The neutrinos do not participate. But if we specify that the weak interaction is related to a weak charge, then only the left-handed particles and right-handed antiparticles have a weak charge, while right-handed particles and left-handed antiparticles are neutral for the weak interaction. Therefore the weak charge is not conserved if an electron changes its handedness during its motion. Only if the leptons had no mass, i.e. could not change their direction of motion and handedness, would a conservation law apply for the weak charge like the one for the electrical charge.

As noted above, only the left-handed helix portion e− L of the electron participate in the β-decay of the neutron. Furthermore, only the left-handed helix portion νL of the corresponding neutrino oc-

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curs in nature. It follows that the left-handed helix portion which participates in the β-decay of the neutrino can be combined in a twocomponent wave function, which is notated as a left- hand doublet:

 νL L= − eL With global SU(2) symmetry, the states of the two-component wave function L are changed everywhere, at the same time and in the same way. For a local SU(2) symmetry, three gauge fields must be introduced corresponding to the three group transformations generated. Mathematically, the SU(2) combination of the three gauge fields is notated in the following matrix [3.17]: e− L

νL

e− L

Wµ0

Wµ−

νL

Wµ+

Wµ0

The parity symmetry of a quantum system means that the Hamilton operator of the system is invariant with respect to the induced operator transformation P . Analogously, the time-reversal symmetry T and charge symmetry C of particle antiparticle means invariance of the corresponding Hamilton operator. The successive application of all three symmetry operations leads to a famous symmetry, which is known as the CP T theorem. According to this theorem, the Hamilton operator of a (Lorentz invariant) quantum system is invariant with respect to the combination of parity, charge and time-reversal transformation. For classical systems of physics, this result is trivial, since such systems are more or less invariant with respect to each individual transformation of this type. That is also true for the electromagnetic (and strong) interaction, but not for the weak interaction. From a left-handed particle, for example, the parity transformation P produces a righthanded particle which does not exist in nature. From a left-handed neutrino, however, the successive application of P and C makes a

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right-handed anti-neutrino which does indeed occur in nature. For the weak interaction, therefore, during the β-decay, the product CP is conserved, as is T , but not P . It should be noted that several experiments with the decay of K◦ mesons also indicate a violation of P C and T with the conservation of the total symmetry CP T . Elementary particle physics intends to unify the four physical forces in one fundamental force. In spite of their different features, electromagnetic and weak forces could already be unified by very high energies in an accelerator ring of CERN. It means that at a state of very high energy the particles of weak interaction (electrons, neutrinos, etc.) and electromagnetic interaction cannot be distinguished. They can be described by the same symmetry group U(1) × SU(2). There are three gauge fields of SU(2) symmetry of weak interaction and one gauge field of the U(1) symmetry of the electromagnetic interaction [3.18]. The complex variety of particles like hadrons (protons, neutrons, etc.) which interact with strong forces (e.g. atomic nuclear force) can be reduced to the so-called quarks with three degrees of freedom which are called “colors” red (R), green (G) and blue (B). A baryon is built up by three quarks that are distinguishable by three different colors. These three colors are complementary in the sense that a hadron is neutral (“without color”) to its environment. The color state of the hadron preserves invariance with respect to a global transformation of the colors. But a local transformation of a color state (i.e. a color change of only one or two quarks) needs a gauge field, in order to compensate the local change and to save the invariance (symmetry) of the whole hadron. Mathematically we have a local so-called SU(3) symmetry group of transformations [3.19]. After the successful unification of electromagnetic and weak interaction physicists try to realize the “big” unification of electromagnetic, weak and strong forces, and in a last step the “superunification” of all four forces. In the context of quantum field theory, the strong, weak and electromagnetic interactions have been traced to fundamental symmetry structures. The trend in recent physics to unify different theories using the ideas of symmetry is therefore con-

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Fig. 40.

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Unification of physical forces

firmed once again. This trend is illustrated in Fig. 40, which shows Newton’s unification of Kepler’s celestial mechanics and Galileo’s terrestrial mechanics into the theory of gravitation, and finally Einstein’s relativistic version, Maxwell’s unification of electricity and magnetism into electrodynamics, the relativistic version of quantum electrodynamics, its unification with the theory of weak interaction and the theory of strong interaction. The framework of these unifications is formed by gauge theories in which the physical forces are introduced by the transition from global to local symmetries. Mathematically, it seems to be easy to embed the characteristic gauge symmetries of the quantum forces into larger common gauge groups of transformations [3.20]. But after the grand

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unification of weak, strong and electromagnetic forces, gravitation must now also be included. In a supersymmetry of all four forces, therefore, general relativity theory of gravity would have to be unified with quantum mechanics of quantum forces. The usual application of general relativity is that of large, astronomical distance scales. On such distances relativistic theory of gravitation implies that the absence of mass means that space is flat. But, on the short distance scales of Planck’s constant, quantities like momentum and location, energy and time, start to fluctuate according to Heisenberg’s uncertainty principle. Although classical reasoning implies that empty space has zero gravitational field, quantum mechanics shows that on average it is zero, but that its actual value undulates up and down due to quantum fluctuations. J.A. Wheeler used the term “quantum foam” to describe the ultramicroscopic quantum fluctuations of space-time [3.21]. Obviously, the notion of a smooth spatial geometry of a relativistic universe is no longer true in the quantum world of short distances. But it is the level of Planck’s constant with the minimal Planck’s length of 10−33 centimeter (which means a millionth of a billionth of a billionth of a billionth of a centimeter) where gravitation and quantum forces have to be unified by symmetry laws. At the level of elementary particles, there is no chance to unify the quantum forces with relativistic gravitation. Therefore, it seems to be quite natural to assume a common material sublevel at Planck’s constant and Planck’s length where elementary particles of the quantum world and gravitation are generated. According to string theory, the elementary ingredients of the universe are not point particles. Rather, they are tiny, 1-dimensional filaments of Planck’s length like infinitely thin rubber bands, vibrating to and fro. Just as the different vibrational patterns of a violin string give rise to different musical notes, the different vibrational patterns of a string give rise to different masses and charges of elementary particles. The loops in string theory can vibrate in resonance patterns in which a whole number of peaks and troughs fit along their spatial extent (Fig. 41). More frantic vibrational patterns have more energy than less frantic ones. The greater the amplitude and the shorter the wavelength

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Fig. 41.

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Symmetries of strings [3.22]

of vibration are, the greater is the energy. According to special relativity, energy and mass are equivalent. The mass of an elementary particle is determined by the energy of the vibrational pattern of its internal string. Heavier particles have internal strings that vibrate more energetically, while lighter particles have internal strings that vibrate less energetically. According to general relativity, mass and energy determine gravitational properties. Thus, even the graviton as messenger particle of gravitational interaction between masses is generated by characteristic vibrational patterns of strings. String theory avoids the inconsistencies, divergencies, and infinities of quantities that have arisen with point particles in the framework of quantum gravitation. What appear to be different elementary particles can actually be considered as different notes on a fundamental string. In the Platonic universe, the harmonies of nature could be illustrated by the Pythagorean music of celestial spheres. According to string theory the vibrational patterns of fundamental strings orchestrate the harmonies of the universe. It is assumed that all kinds of symmetries, e.g. space-time symmetries, gauge symmetries of elementary particles, CP T -symmetry, originate from string theory. In a supersymmetry of unification, particles or vibrational patterns with different spins must be included [3.23]. For example, fermions are typically matter particles with half a whole odd number amount of spins, while bosons as typically messenger particles of interactions (e.g. photons of electromagnetic force or gravitons of gravitation) have a whole number amount of spin. A string theory with incorporated supersymmetry of fermions and bosons is called superstring theory. In this case, for each bosonic pattern of vibration there is

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a fermionic pattern and vice versa whose respective spins differ by half a unit. These supersymmetric pairs are called superpartners. Mathematically, they can be combined as doublets of bosons and fermions. A supersymmetry transformation describes the transformation of bosons and fermions with spin numbers next to one another, i.e. differing by half a unit. In a quantum field theory, a boson-fermion field would have to be described by a Lagrangian operator that is invariant with respect to supersymmetry transformations. But, as far as we know, supersymmetry can be incorporated into string theory in not one but five different ways. These five superstring theories are called the Type I theory, the Type IIA theory, the Type IIB theory, the Heterotic type O(32) theory, and the Heterotic type E8 × E8 theory with different symmetry groups of transformations. Each method results in a pairing of bosonic and fermionic vibrational patterns, but the details of this pairing as well as other properties differ. Are these theories alternative hypotheses, which must be decided by experiments, or should they be unified in another supertheory? Superstring theories lead to a surprising and dramatical change of generally accepted physical concepts of our universe: Their unification of forces at high levels of energy needs more spatial dimensions than the three familiar ones of our universe. Historically, the idea of further spatial dimensions dates back to the Polish mathematician T. Kaluza who, in 1919, suggested a unification of Einstein’s gravitational theory with Maxwell’s electrodynamics. In Kaluza’s unified theory, the equations pertaining to the three ordinary dimensions were essentially identical to Einstein’s field equations of gravitation. His extra equations associated with the new dimension were those of Maxwell’s electrodynamic force. In those days, the idea of a fourth spatial dimension that could not be observed was rather strange. But in 1926, the Swedish mathematician O. Klein, for the first time, assumed that the spatial structure of our universe may have both extended and observable dimensions as well as curled-up ones that cannot be observed because of their tiny size [3.24]. The fourth curled-up dimension was illustrated by tiny circles at every

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point in the familiar 3-dimensional space, like the circular loops of thread making up the pile of a carpet. In superstring theories, the unification of more forces means the need for even more dimensions. Again, for instance, two extra dimensions can be imagined as curled up in the shape of tiny spheres or torus which are tacked on to every point of the familiar extended dimensions. But, the unification of electromagnetic, weak and strong forces with gravitation requires the particular number of six extra dimensions, i.e. nine space dimensions and one time dimension in a superstring theory. The reasons are purely mathematically explained by the formalism of a superstring theory in order to avoid inconsistent and senseless concepts. Obviously, a curled-up 6-dimensional space cannot be illustrated like the cases of one or two extra dimensions. In 1984, it could be proven that the so-called Calabi–Yau spaces meet the conditions which are required by the six curled-up extra dimensions of superstring theories. These tiny six-dimensional Calabi–Yau spaces are assumed to be tacked on to every point of the familiar 3-dimensional space. The 10-dimensional superstring theories have extremely symmetric properties. Their supersymmetry forecasts the existence of superpartners which should be detected by appropriate accelerators of elementary particles. But, from a theoretical point of view, one of their main advantages is their power of explanation. Newton and Einstein developed their theories of gravity only because their observations of the world showed them that gravity exists, and that, therefore, it required a mathematical model describing gravitational interactions. On the contrary, string theory demands and forecast the existence of gravity even if nobody ever has observed gravitational effects. Like the other fundamental forces, they are derived from superior principles of symmetry. But what about the five possibilities of superstring theories? State of the art is that they share many basic features. For example, their vibrational patterns determine the possible mass and charge of particles, they require 10 space-time dimensions, and their curled-up extra dimensions satisfy the conditions of a Calabi–Yau space. But

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there are also differences. Type I string theory, for example, has open strings with two loose ends in addition to the closed loops of Fig. 41. In a first approach, the five superstring theories were thought of as being complete separate alternatives which must be decided by observation. But further insights have supported the idea that all of the superstring theories can be viewed as a single, all-encompassing framework. Kaluza’s idea of unification by extra dimensions makes it mathematically possible. The unified theory should have ten space and one time dimension. One additional spatial dimension allows for a synthesis of all five versions of the theory. The five established superstring theories are only approximations of an exact theory that is still unknown in all its details. This 11-dimensional theory has provisionally been called M-theory. The ultramicroscopic, extended nature of a string can be approximated by a structureless point particle, using the framework of point-particle quantum field theory. When dealing with short distance and high-energy processes where gravity and quantum forces are unified, this approximation can no longer be used. The quantum field theory that most closely approximates superstring theories in this way is the 11-dimensional theory of supergravity. A remarkable consequence of the 11-dimensional M-theory is that the theory of supergravity can also be incorporated into the network of common dualities with superstring theories. M-theory does not only contain vibrating 1-dimensional objects like (closed or open) strings, but also includes 2-dimensional vibrating membranes, 3-dimensional objects (so-called three-branes), and a host of other more dimensional ingredients (so-called p-branes). The unification of M-theory is supported by proofs of dualities which describe exactly the common features of the five superstring theories. They are mainly derived from their symmetry principles. M-theory seems to be the ultimate framework of the initial symmetric state of our universe. In the beginning, all of the spatial dimensions are curled up to their finite smallest possible extent of Planck’s length, but not to zero. The temperature and energy are high, but not infinite. In the hot environment of the early universe,

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all fundamental forces were indistinguishable and merged together in one unified force. M-theory and superstring theories avoid the inconsistencies of an infinitely compressed zero-size starting point of infinite energy, which was assumed by the relativistic standard model of cosmology. Perhaps, some day, the finite conditions of the early universe can even be tested in a future accelerator ring. At this beginning state of the universe, the spatial dimensions are completely symmetric and curled up into a multidimensional, Planck-sized nugget. How could it expand and generate the variety of structures emerging at successive steps of cosmic evolution? 3.2

Symmetry Breaking and Phase Transitions

In the beginning, there was symmetry. Thus, the observed variety and diversity of structures in nature could only emerge by a reduction of symmetry or symmetry breaking. Processes of symmetry breaking are well known from everyday life. A (mathematically perfect) egg has rotational symmetry and symmetry of reflection with reference to its longitudinal axis. If we stand it vertically on a plate, and leave it to its own devices, it rolls over on its side and remains lying in some direction: The symmetry of the egg relative to the vertical axis on the table is broken, although the symmetry of the eggshell remains intact. The symmetry breaking is spontaneous, since it was impossible to predict the direction in which the egg ultimately came to rest. In this case, the cause is the earth’s gravitation, which allows the egg to assume an energetically more favourable state: the symmetrical state relative to the vertical axis of the plate was energetically stable. If the early universe started in a state of high symmetry, it was rather featureless with only one force. More features emerged as the universe cooled and expanded. The universe underwent a series of cosmic phase (or state) transitions, in which the primal symmetry was successively broken and the gravitation, strong, weak and electromagnetic interactions successively froze out. Phase transitions with symmetry breaking can be illustrated in many physical systems

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by lowering their temperature. Water provides a simple example. A complex system of water molecules is a gas or steam above 100 degrees Celsius. In this state, the system has a high degree of symmetry, because the molecules can move completely freely without any distinction of direction. If the temperature is lowered below 100 degrees Celsius, water droplets emerge by passing through a gas–liquid phase transition, and the symmetry is reduced. If the temperature is further lowered down to 0 degrees Celsius, the system pass through a liquid–water/solid–ice phase transition that is connected with another spontaneous decrease in symmetry. In this state, the liquid water begins to freeze and turn into solid ice. Liquid water looks the same regardless of the angle from which it is viewed. In this sense, the system is rotationally symmetric. But solid ice has a crystalline block structure looking different from different angles. The phase transition has resulted in a decrease in the amount of rotational symmetry. At 10−43 seconds after the Big Bang, the so-called Planck-time, the temperature of the universe is calculated to be about 1032 Kelvin. As time passed, the universe expanded and cooled. Between the Planck time and a hundredth of a second, the unified theory forecasts phase transitions with symmetry breaking and emergence of new structures which are similar to the phase transitions of water. E. Witten proved that, within M-theory, the strengths of all four forces can be unified (Fig. 42) [3.25]. When the temperature of the universe, above 1028 Kelvin, was still high enough, then, according to quantum field theory, at least the three nongravitational forces merged together. A bifurcation between gravitation and a unified electro-weak-strong force had happened. As the temperature dropped below 1028 Kelvin, the universe underwent a phase transition in which the three forces crystallized out from their union in different ways. Therefore, the symmetry among the forces at higher temperatures was broken as the universe cooled. Only the weak and electromagnetic forces were still interwoven. At 1015 Kelvin, the universe went through another phase transition that affected the electromagnetic and weak forces. At this temperature, they too crystallized

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GUT

energy temperature

weak interaction

electroweak 14

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strong interaction

gravitation

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Planck-time established physics Fig. 42.

Bifurcation tree of cosmic symmetry breaking [3.26]

out from their previous, more symmetric union. A bifurcation tree of spontaneous symmetry breaking led to the emergence of the familiar physical forces during phase transitions of the expanding and cooling early universe. The standard model of symmetry breaking in quantum field theory uses a Yang–Mills theory of quantum forces and the so-called Higgs mechanism [3.27]. The historical Yang–Mills theory with gauge symmetry proceeds on the assumption of the unlimited range of all the forces it describes. But, except for the photons of the electromagnetic force and the gravitons of gravitation, no massless particles occur in nature, with which interaction could be transmitted over unlimited ranges. Therefore, the Higgs mechanism describes a procedure of spontaneous symmetry breaking which results in the desired massive gauge particles. For a unification of the SU(2)×U(1) symmetry, four gauge field quanta are necessary. According to the Higgs mechanism, three of them are required to become massive for weak interaction, while the fourth gauge particle is the photon of electromagnetic interaction, which is massless and transmitted with the speed of light.

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A unification of strong, weak and electromagnetic forces with SU(3), SU(2) and U(1) symmetries is mathematically described by a SU(5) symmetry. If this symmetry of strong-weak-electromagnetic force is spontaneously broken, then the intermediate particles of interaction (analogous to the gauge particles of the weak interaction) take on large masses. At first, only the strong SU(3) interactions with the quarks could be distinguished from the electro-weak SU(2)×U(1) interactions of the leptons. During further expansion of the universe with decreasing temperature, these symmetries were also broken, and the forces act in the manner currently observed. It would be obvious, for the confirmation of the SU(5) unification to generate the X-particles at high energy in the laboratory, as was done previously for the intermediate vector bosons of the SU(2) × U(1) unification. Since such a process requires energies which are approximately 13 orders of magnitude greater than the energies required to generate the SU(2)×U(1) vector bosons (approximately 100 GeV), this method of testing the theory remains hypothetical. Analogous to the β-decay of the weak interaction, we still hope to observe a virtual X-particle during an elementary particle process. Such a process is predicted by the SU(5) theory for the decay of the proton, when a quark is transformed into a lepton. During the first stage of symmetry breaking at about the Plancktime, three of the curled-up spatial dimensions are singled out for expansion, while all others retain their initial Planck-scale size. The question arises why only three of the space dimensions have expanded to observably large size. If we imagine two point particles moving with different velocity along a 1-dimensional line, they will sooner or later collide. If they are randomly rolling around a 2-dimensional plane, it is likely that they will never collide. In three or higher number of dimensions, a meeting of the two particles gets increasingly unlikely. An analogous idea holds if the point particles are replaced with loops of string, wrapped around spatial dimensions. The rapid expansion in three dimensions is explained by the so-called inflationary period of the universe. During a tiny window of time, around

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10−36 to 10−34 seconds after the Big Bang, the universe expanded by a colossal factor of at least 1030 . The quantum theory of the inflationary period assumes an early state of the universe with small size, but very high energy (“quantum vacuum”) that expands very rapidly to macroscopic dimensions driven by a repulsive force of the quantum vacuum state (“antigravity”) [3.28]. This cosmic phase transition allows one to explain some well-known properties of the observed universe such as the relatively homogeneous distribution of stars and matter. During the inflationary period, some tiny deviations from symmetry and uniformity would have been amplified until they were big enough to account for the observed structures of the universe. In the expanding universe the density of matter varied slightly from place to place. Thus, gravity would have caused the denser regions to slow down their expansion and start contracting. These local events led to the formation of stars and galaxies. According to A. Linde, the brief but crucial burst of inflationary expansion may not have been a unique event. Instead, the conditions for inflationary expansion may happen repeatedly in isolated regions, which then undergo their own inflationary expansions, evolving into new, separate universes. Linde suggests a multiuniverse, generating a never ending web of ballooning cosmic expanses. Separating universes mean bifurcation and symmetry breaking. Thus, a multiuniverse is the ultimate version of cosmic symmetry breaking: In the beginning there was endless symmetry breaking. Whereas we assume a consistent and uniform physics in our universe, this may have no bearing on the physical attributes in these other universes. The list of elementary particles and forces may be completely distinct from ours. If we assume superstring theories, even the number of extended dimensions may differ with different possibilities of interacting strings and particles. If our universe is not alone but is instead interwined with a fractal “multiverse”, along with many other bifurcating universes, then we could think about “interuniversal” routes between the universes. According to Heisenberg’s principle of uncertainty, quantum fluctuations could open short-lived wormholes in space-time. So, the laws of quantum dynamics make it at least conceivable that wormholes can be employed as links between separated and bifurcating universes.

Only upon cooling did the early symmetry of the universe break apart into increasingly partial symmetries, and the individual particles were crystallized in stages. For example, it would be conceiv-

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able in terms of physics that after the end of the first epoch, the original symmetry had dissolved into the local subsymmetry of gravitation and the symmetry of the other forces [3.29]. In the SU(5) symmetry of the weak, strong and electromagnetic forces, quarks and leptons are still being transformed. The intermediate particles of interaction play the most important role. Ultimately, the SU(5) symmetry decays into the SU(3) symmetry of the strong forces and the SU(2) × U(1) symmetry of the weak and electromagnetic forces. Quarks and leptons thus become identifiable particles. Ultimately, the SU(2) × U(1) symmetry also decays into the SU(2) subsymmetry of the weak interaction and the U(1) subsymmetry of the electromagnetic interaction. The atomic nuclei of electrons, neutrinos etc. become identifiable particles. Below certain temperatures that correspond to certain average distances, the coupling constants of the various interactions became distinguishable. As a result of related, gradual and spontaneous symmetry breaking, the universe became asymmetrical and manifold. Finally, we should note a few “fossils” of these past symmetry breakings. The parity violation, i.e. the preference for one direction in space, like that which occurs during the β-decay of the weak interaction, as explained above, is a relic of the SU(2) × U(1) symmetry breaking. An additional remarkable asymmetry of the current universe is the surplus of matter over antimatter [3.30]. This fact can now be understood as a consequence of the breaking of the SU(5) symmetry. After the collapse of the SU(5) symmetry, more quarks than antiquarks might have been formed during the exchange of the intermediate particles, because the quarks decayed somewhat more slowly than the antiquarks. Later, after matter and antimatter had been mutually annihilated, there was a small surplus of protons and electrons, from which the stars, the earth and life on earth evolved. In a future stage of development of the universe, however, this temporary “surplus” of matter, on which our existence depends, could disappear again, specifically if the protons are annihilated. Of course, given the estimated lower bound for the average life of a proton, which is 1031

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years, the decay period of matter seems enormously long, but it is still physically conceivable and experimentally verifiable. It is remarkable that with the SU(5) symmetry breaking, the discrete CP symmetry must also be violated. Of course, the CP symmetry says that all physical laws remain valid in a reflected world if, in addition to the reflection (P ), all the particles are replaced by their antiparticles (C). If CP symmetry were to be always valid, then from an initial equilibrium of matter and antimatter, a predominance of one of the two parts could never have developed. For each origin of a particle, there would then be an equally probable reflected process in which the corresponding antiparticle would be formed. As noted above, violations of CP symmetry can be experimentally verified during the decay of the K◦ meson. Early cosmic evolution was a first example of phase transitions in which matter transforms from one structural state to another. We already emphasized the analogies of the early cosmic phase transitions with the familiar condensation of gases and the freezing of liquids. The different order of the molecules on the microlevel is a cause of a new feature of the material on the macroscopic level. Condensation and freezing relate to a complex state of molecules and cannot be reduced to a single molecule. In this sense, phase transitions are connected with the emergence of new macroscopic features of a system. Consider, for example, a ferromagnet losing its magnetization, when it is heated beyond a critical value. Magnetization is a macroscopic feature that can be explained by changing the degrees of freedom at the microscopic level. The ferromagnet consists of many atomic magnets. At elevated temperature, the elementary magnets point in random directions. If the corresponding magnetic moments are added up, they cancel each other. Then, on the macroscopic level, no magnetization can be observed. Below a critical temperature, the atomic magnets are lined up in a macroscopic order, giving rise to the macroscopic feature of magnetization. In these kinds of phase transition, the emergence of macroscopic order was caused by lowering temperature, but by maintaining a flux of energy and matter through them. Familiar examples are living sys-

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tems like plants and animals that are fed by biochemical energy. The processing of this energy may result in the formation of macroscopic patterns like the growth of plants, locomotion of animals, and so on. But this emergence of order is by no means reserved to living systems. It is a kind of dissipative (irreversible) self-organization far from thermal equilibrium that can be found in physics and chemistry as well as in biology. Emergence of order seems to contradict the second law of thermodynamics. According to that famous law, closed systems without any exchange of energy and matter with their environment develop to disordered states near thermal equilibrium. The degree of disorder is measured by a quantity called “entropy.” The second law says that in closed systems the entropy always increases to its maximal value. For instance, when a cold body is brought into contact with a hot body, then heat is exchanged so that both bodies acquire the same temperature, i.e. a disordered and homogeneous order of molecules. When a drop of milk is put into coffee, the milk spreads out to a finally disordered and homogeneous mixture of milky coffee. The reverse processes are never observed [3.31]. In this sense, processes according to the second law of thermodynamics are irreversible with a unique direction of time. But the symmetry of time is only broken on the macrolevel without contradiction to the time reversibility of other physical laws (e.g. quantum mechanics). The second law of thermodynamics refers to macroscopic distributions of particles (e.g. molecules of a gas) and their time-depending development that is irreversible with high probability. On the microlevel, the equations of the particles are still reversible (microreversibility). Basically, the thermodynamic arrow of time is explained by the global expansion of the whole universe leading from states of symmetry to symmetry breaking and diversity with increasing entropy [3.32]. But, in a global sea of entropy, local islands of new order like, e.g. stars, planets and life emerge and disappear. How is that possible? The emergence of order is made possible by phase transitions of open systems interacting with their environment. In hydrodynamics,

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the formation of weather and turbulences are classical examples. The earth, warmed by the sun, heats the atmosphere from below. Outer space, which is always cold, absorbs heat from the outer shell of the atmosphere. The lower layer of air tries to rise, while the upper layer tries to drop. This traffic of layers was modeled in several experiments by H. B´enard. The air currents in the atmosphere can be visualized as cross-sections of the layers. The traffic of the competing warm and cold air masses is represented by circulation vortices, called B´enard cells. In three dimensions, a vortex may have warm air rising in a ring, and cold air descending in the center. Thus, the atmosphere consists of a sea of 3-dimensional B´enard cells, closely packed as a hexagonal lattice. A footprint of such a sea of atmospheric vortices can be observed in the regular patterns of hills and valleys in deserts, snowfields or icebergs. In a typical B´enard experiment, a fluid layer is heated from below in a gravitational field (Fig. 43a). The heated fluid at the bottom tries to rise, while the cold liquid at the top tries to fall. These motions are opposed by viscous forces. For small temperature differences ∆T , viscosity wins, the liquid remains at rest, and heat is transported by uniform heat conduction. The external control parameter of the system is the so-called Rayleigh number Ra of velocity, which is proportional to ∆T . At a critical value of Ra, the state of the fluid becomes unstable, and a pattern of stationary convection rolls develops (Fig. 43b). Beyond a greater critical value of Ra, a transition to chaotic turbulence is observed [3.33]. Another example from fluid dynamics is the flow of fluid round a cylinder. The external control parameter is the Reynolds number Re

(a) Fig. 43.

(b)

Phase transition and symmetry breaking of a B´enard experiment

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of fluid velocity. At low speed the flow happens in a homogeneous manner (Fig. 44a). At higher speeds, a new macroscopic pattern with two vortices appears (Fig. 44b). With yet higher speeds the vortices start to oscillate (Fig. 44c–d). At a certain critical value, the irregular and chaotic pattern of a turbulence flow arises behind the cylinder (Fig. 44e).

Fig. 44.

Phase transitions of fluid dynamics [3.34]

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(a)

(b) Fig. 45.

Phase transitions of a laser [3.35]

A famous example from modern physics and technology is the laser. A solid state laser consists of a rod of material in which specific atoms are embedded. Each atom may be excited by energy from outside leading it to the emission of light pulses. Mirrors at the end faces of the rod serve to select these pulses. If the pulses run in the axial direction, then they are reflected several times and stay longer in the laser, while pulses in different directions leave it. At small pump power the laser operates like a lamp, because the atoms emit independently of each other light pulses (Fig. 45a). At a certain pump power, the atoms oscillate in phase, and a single ordered pulse of gigantic length emerges (Fig. 45b). The laser is an example of macroscopic order emerging from phase transitions. With exchange and processing of energy, the laser is obviously a dissipative system. Phase transitions in dissipative systems generate a bifurcation tree with emerging structures of increasing complexity. In this context, the degrees of increasing complexity are defined by the increasing bifurcations that lead to chaos as the most complex and fractal scenario. Each bifurcation illustrates a possible branch of solution for the nonlinear equation. Physically, they denote phase transitions from a state of equilibrium to new possible states of equilibria. If equilibrium is understood as a state of symmetry, then phase transition means symmetry breaking being caused by fluctuational forces.

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Mathematically, symmetry is defined by the invariance of certain laws with respect to several transformations between the corresponding reference systems of an observer. The hydrodynamical laws describing a fluid layer heated from below (Fig. 43a) are invariant with respect to all horizontal translations. Nevertheless, these highly symmetric laws allow phase transitions to states with less symmetry. For example, in the case of a B´enard experiment, the heated fluid layer becomes unstable, and the state of stationary convection rolls develops (Fig. 43b). This phase transition means symmetry breaking, because tiny fluctuations cause the rolls to prefer one of two possible directions. Our examples show that phase transition and symmetry breaking is caused by a change of external parameters and leads eventually to a new macroscopic spatio-temporal pattern of the system and emergence of order. Obviously, thermal fluctuations bear in themselves an uncertainty, or more precisely speaking, probabilities. A particle that is randomly pushed back or forth (Brownian motion) can be described by a stochastic equation governing the change of the probability distribution as a function of time. Fluctuations are caused by a huge number of randomly moving particles. An example is a fluid with its molecules. Therefore, a bifurcation of a stochastic process can only be determined by a change of probabilistic distribution and stochastic symmetry breaking. In general, the emergence of structures in the universe is made possible by decreasing symmetry (“symmetry breaking”) and increasing complexity during phase transitions. 3.3

Complexity, Attractors and Dynamical Systems

Emerging structures and entities in nature correspond to solutions of nonlinear differential equations modeling the time-depending evolution of dynamical systems. Why is nonlinearity a (necessary) reason for the emergence of new structures in dynamical systems? In the case of a linear equation, the sum of two solutions is also a solution of the equation. This property is often expressed with the statement: the whole is equal to the sum of its parts. Thus, the task of finding

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the general solution for a linear differential equations can be broken up into a collection of simpler problems. The threads of causal developments can more easily be separated and sorted out. In nonlinear systems, such a resolution is not possible because the sum of two solutions is in general not a solution of the nonlinear equation: the whole is often greater than the sum of its parts. The whole is a new dynamic entity and not only a collection of elements. Thus nonlinearity becomes a source of emergent phenomena. An example of hydrodynamics is the emergence of solitary waves from the nonlinear dynamics of water waves. In an experiment, a solitary wave can be created in a water tank [3.36]. We consider a column of water that is accumulated at one end of the tank. Release of this water by lifting a sliding panel generates a solitary wave that travels to the other end of the tank with the same speed and amplitude. The traveling wave corresponds to an exact solution of a nonlinear differential equation modeling the wave dynamics in the tank. The equation depends on the speed of amplitude waves and their dispersion that is determined by, for example, surface tension and the density of water. The shape of the solitary wave is made possible, because its effects of dispersion are in balance with those of nonlinearity. The effect of dispersion is to spread out the energy of the traveling pulse and the effect of nonlinearity is to draw it together. Spreading out and drawing together of energy forms a causal loop that generates a new dynamic entity. Causal loops correspond to nonlinear dynamics that cannot be separated into their parts. It is their dynamic interaction that produces emerging phenomena. Solitary waves are examples of Einstein’s vision that the emergence of entities in nature can be explained by exact solutions of nonlinear differential equations. There is an arbitrary number of solitary waves with varying speeds and amplitudes which even undergo a collective collision. In this case, they leave the interaction region of space-time with the same speeds and amplitudes that they had upon entry. Therefore, solitary waves are now called solutions, emphasizing their particle-like character.

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Hydrodynamic solitary waves are examples of dynamical entities conserving their energy. But there are also solitary waves that are generated by a dynamic balance between nonlinear and dissipative effects. An example is the emergence of the flame of a candle. In a lighted candle, heat generated by the flame diffuses into the solid wax, causing the release of a vapor that carries energy upward into the flame. Combustion of the vaporized wax provides the heat. The closed causal loop between thermal diffusion and nonlinear energy release generates a traveling-wave, moving down the candle at a fixed speed. This kind of causal loop is an example of nonlinear reaction– diffusion process that cannot only be observed in physics, but also in chemistry and biology. Roughly speaking, we distinguish conservative (“closed”) and dissipative (“open”) dynamical systems. More precisely, conservative as well as dissipative systems are characterized by time-depending nonlinear differential equations depending on an external control parameter that can be decreased and increased to critical values. As we mentioned in previous sections, Hamiltonian-like equations can be used to characterize any conservative dynamical system. Its fruitful idea is to characterize a conservative system by a Hamiltonian function (or operator), which is the expression for the total energy (which is the sum of kinetic and potential energy) of the system in terms of all the position and momentum variables. The corresponding state spaces allow us the evolution of the dynamical systems in each “phase.” Thus, they are called phase spaces. For systems with n particles, phase spaces have 3n + 3n = 6n dimensions. A single point of a phase space represents the entire state of a perhaps complex system with n particles. The Hamiltonian equations determine the trajectory of a phase point in a phase space. Globally, they describe the rates of change at every phase point, and therefore define a vector field on the phase space, determining the whole dynamics of the corresponding system. It is a well-known fact from empirical applications that states of dynamical models cannot be measured with arbitrary exactness. The measured values of a quantity may differ by tiny intervals being

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caused by the measuring apparatus, constraints of the environment, and so on. The corresponding phase points are concentrated in some small regions of a neighborhood. Now, the crucial question arises if trajectories starting with neighboring initial states are locally stable in the sense that they have neighboring final states. In this case, similar initial states lead to similar final states. This assumption is nothing else than a classical principle of causality in the language of Hamiltonian dynamics: similar causes lead to similar effects. Due to a theorem of the French mathematician J. Liouville, the volume of any region of the phase space must remain constant under any Hamiltonian dynamics, and thus for any conservative dynamical system. But its conservation does not exclude that the shape of the initial region is distorted and stretched out to great distances in the phase space. We may imagine a drop of ink spreading through a large volume of water in a container. That possible spreading effect in phase spaces means that the local stability of trajectories is by no means secured by Liouville’s theorem. A very tiny change in the initial data may still give rise to a large change in the outcome [3.37]. Nevertheless, Liouville’s theorem implies some general consequences concerning the regions which can be displayed by Hamiltonian dynamics, and thus by conservative systems. The mathematical pendulum without friction is a perfect conservative system without any loss of energy. As there is no friction, moving the pendulum a little to the left causes it to swing back and forth indefinitely. The full trajectory in the state space, corresponding to this oscillating motion, is a cycle or closed loop around a vortex point of equilibrium which is not an attractor (Fig. 46b). If the system is not closed and the effects of friction are included as in physical reality, then the equilibrium point at the origin is no longer a vortex motion of the point (Fig. 46a). It has become a spiraling type of point attractor. As any motion of the pendulum will come to rest because of friction, any trajectory representing a slow motion of the pendulum near the bottom approaches this limit point asymptotically. In Fig. 46a, trajectories are attracted to a field point, and the volume of an initial area shrinks. In Fig. 46b, the trajectories rotate

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Fig. 46a.

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Fixed point attractor

Fig. 46b.

Vortex point of oscillation

around a vortex point, and the volume of an initial area is conserved. Thus, due to Liouville’s theorem, we can generally conclude that in any conservative system attracting points must be excluded. The effect of shrinking initial areas can easily be visualized for the trajectories of limit cycles, too. Therefore, limit cycles as attractors are also not possible in conservative systems for the same mathematical (a priori) reasons [3.38]. A further mathematical result of Hamiltonian (conservative) systems says that there are irregular and chaotic trajectories. In the 18th and 19th centuries, physicists and philosophers were convinced that nature is determined by Newtonian- or Hamiltonian-like equations of motion, and thus future and past states of the universe can be calculated at least in principle if the initial states of present events are well known. Philosophically, this belief was visualized by Laplace’s demon, which like a huge computer without physical limitations can store and calculate all necessary states. Mathematically, the belief in Laplace’s demon must presume that systems in classical mechanics are integrable, and, thus are solvable. In 1892, Poincar´e was already aware that the non-integrable three-body problem of classical mechanics can lead to completely chaotic trajectories [3.39]. About sixty years later, A.N. Kolmogorov (1954), V.I. Arnold (1963) and J. Moser (1967) proved with their famous KAM theorem that motion in the phase space of classical mechanics is neither completely regu-

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lar nor completely irregular, but that the type of trajectory depends sensitively on the chosen initial conditions [3.40]. Since Poincar´e’s celestial mechanics (1892), it was mathematically known that some mechanical systems whose time evolution is governed by nonlinear Hamiltonian equations could display chaotic motion. But as long as scientists did not have suitable tools to deal with non-integrable systems, deterministic chaos was considered as a mere curiosity. During the first decades of the 20th century, many numerical procedures were developed to deal with the mathematical complexity of nonlinear differential equations at least approximately. The calculating power of modern high-speed computers and refined experimental techniques have supported the recent successes of the nonlinear complex system approach in natural and social sciences. The visualizations of nonlinear models by computer-assisted techniques promote interdisciplinary applications with far-reaching consequences in many branches of science. In this scientific scenario (1963), the meteorologist Lorenz a student of the famous mathematician Birkhoff [3.41], observed that a dynamical system with three coupled first-order nonlinear differential equations can lead to completely chaotic trajectories. Mathematically, nonlinearity is a necessary, but not sufficient condition of chaos. It is a necessary condition, because linear differential equations can be solved by wellknown mathematical procedures (Fourier transformations) and do not lead to chaos. The system Lorenz used to model the dynamics of weather differs from Hamiltonian systems `a la Poincar´e, mainly by its dissipativity. Lorenz’s discovery of a deterministic model of turbulence occurred during simulation of global weather patterns. The differential equations describing the B´enard experiment (Fig. 43) were simplified by Lorenz to obtain the three nonlinear differential equations of his famous model. Each differential equation describes the rate of change for a variable x proportional to the circulatory fluid flow velocity, a variable y characterizing the temperature difference between ascending and descending fluid elements, and a variable z proportional to the deviation of the vertical temperature profile from its

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Fig. 47.

Strange attractor of Lorenz [3.42]

equilibrium value. From these equations, it can be derived that an arbitrary volume element of some surface in the corresponding phase space contracts exponentially in time. Thus, the Lorenz model is dissipative. This can be visualized by computer-assisted calculations of the trajectories generated by the three equations of the Lorenz model. Under certain conditions, a particular region in the 3-dimensional phase space is attracted by the trajectories, making one loop to the right, then a few loops to the left, then to the right again, etc. (Fig. 47). The paths of these trajectories depend very sensitively on the initial conditions. Tiny deviations of their values may lead to paths which soon deviate from the old one with different numbers of loops. Because of its strange image, which looks like the two eyes of an owl, the attracting region of the Lorenz phase was called a “strange attractor.” Obviously, the strange attractor is chaotic with fractal dimension. If the attractor is a point (Fig. 46a), the fractal dimension is zero. For a stable limit circle (Fig. 46b) the fractal dimension is one. But for chaotic systems the fractal dimension is not an integer. In general, the fractal dimension can be calculated

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only numerically. For the Lorenz model, the strange attractor has the fractal dimension D ≈ 2.06 ± 0.01. Phase transitions of complex dynamical systems are associated with the emergence of new structures. Mathematically, they are solutions of nonlinear equations corresponding to a bifurcation tree of different attractors. Thus, we distinguish a hierarchy of structures with increasing complexity from fixed point attractors, periodic and quasi-periodic limit cycles up to chaotic attractors. In former days of history, scientists would have postulated certain demons or mystic forces leading the elements of these systems to new patterns of order. In the mathematical approach of complex systems, the emergence of macroscopic patterns is explained by the interactions of their microscopic elements. Their sometimes mystic “self-organization” happens in critical situations of the system during phase transitions which can mathematically exactly be analyzed. We distinguish phase transitions in thermal equilibrium and in nonequilibrium (“dissipative”) systems. An example of a phase transition in thermal equilibrium is realized by a ferromagnet that passes from a disordered state into an ordered state of its elements if the system is cooled down below a critical point. To characterize the variation between the macrostates, L.D. Landau introduced order parameters as macroscopic variables whose values are finite in the ordered state and zero in the disordered state [3.43]. The ordered state occurs in low temperatures in which the system exhibits a certain macroscopic structure indicated by a finite order parameter. When the macroscopic structure is destroyed by the random motion of the elements at increased temperatures, then the order parameter will vanish. In the case of the ferromagnet, the order parameter is the average magnetization, which vanishes in a state above the Curie-point of temperature. Below the Curie-point, the atomic dipoles arrange in a regular pattern on the microlevel corresponding to the state of magnetization on the macrolevel. In this case, the ferromagnet is in an equilibrium state corresponding to a fixed point attractor. For the gas–liquid transition, the order parameter is the difference in the densities of the gas and the liquid.

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Landau distinguishes discontinuous and continuous phase transitions. In discontinuous phase transitions, the order parameter jumps at the transition point with a finite difference of the new and past state. In continuous phase transitions, the order parameter decreases continuously to zero and the difference between the two phases becomes infinitesimal small at the transition point. The transition of a system can be discontinuous or continuous under different conditions. For example, the condensation of gases is a discontinuous phase transition at low pressures and a continuous transition at the critical point. At one atmospheric pressure and 100 degrees Celsius, the densities of steam and water differ by a large factor. As the pressure increases, the density difference decreases and finally vanishes at the critical point of 217 atmospheres and 374 degrees Celsius. At pressures above the critical point there are no distinct gas and liquid phases. A ferromagnet is an example of continuous phase transition.

Pattern formation in dissipative systems can also be explained by the concept of order parameters [3.44]. We start with an old structure, for instance, a homogeneous fluid or randomly emitting laser. In an open system the instability of an old structure is caused by an increasing input of energy (e.g. increasing velocity of a fluid or increasing energy pumping of a laser [3.45]). The old structure breaks down and the system takes a new equilibrium point of stability which is associated with the emergence of a new macroscopic pattern of order. With increasing input of energy the system is driven to new bifurcations of equilibria with the emergence of new macroscopic patterns of order with increasing complexity from fixed points to limit cycles and chaos attractors [3.46]. That is the usual bifurcation tree of nonequilibrium dynamics in open (dissipative) systems (Fig. 48). How does the concept of order parameters come in? Close to an instability point we can distinguish between elements with stable and unstable behavior (modes) on the microlevel of the system. The few unstable modes grow to amplitudes of macroscopic scale and influence the stable ones. They become macroscopic order parameters dominating the whole macrodynamics of the complex system (Fig. 49). Thus, it is suffcient to analyze some few order parameters to understand the macrodynamics of a complex system with many elements.

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Fig. 48.

Fig. 49.

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Bifurcation and emergence of order

Self-organization and order parameters

Mathematically, the nonequilibrium dynamics of order parameters can be modeled by the well established method of a linearstability analysis (Fig. 49) [3.47]. On the microlevel, we start with a nonlinear equation dx/dt = F (x, α) + f (t) of evolution with a nonlinear function F of the microstates x = (x1 , . . . , xn ) of the elements and a control parameter α at time t. The function f (t) represents small stochastic forces with additional external effects on the sys-

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tem which will be ignored in the following. The behavior of stability is mainly determined by the reaction of a system to perturbations. Therefore, we consider any well-known state for a given value α0 of the control parameter and analyze the dynamic behavior of the system in the vicinity of an instability after a small shift of the control parameter from α0 to α. In this case, the behavior of the system correspond to a solution x(t) = x0 + w(t) with the stationary solution x0 (which means F (x0 , α) = 0) and a small deviation w(t) from the stationary solution x0 . In order to analyze the question of whether x0 remains a stable solution of the microscopic evolution equation or whether there evolves another dynamics after the shift from α0 to α, the solution x(t) = x0 + w(t) is inserted into dx/dt = F (x, α). In a next step, we expand the evolution equation F (x0 + w(t), α) into a Taylor series with respect to the deviation w(t). As F (x0 , α) = 0, we obtain the equation dw/dt = L(x0 , α)w + N (x0 , α, w) with the term L for all linear terms and N for all nonlinear terms with second and higher expansion terms. As long as the deviations w(t) are small, one can neglect the higher terms and only analyze the approximate linearized equation dw/dt = L(x0 , α)w. The problem of stability is now reduced to a linear equation which can be solved by elementary methods. By analyzing the corresponding eigenvalue equation, the linearstability analysis allows to distinguish between unstable and stable elements on the microscopic level. Their unstable modes grow exponentially with time so that the linearized equation of evolution becomes invalid. The stationary solution becomes unstable and the corresponding linearized term of deviation must be substituted by an equation distinguishing between the amplitudes of the unstable modes and those of the stable ones. While the amplitudes of unstable modes begin to increase exponentially, the stable modes decrease exponentially. The unstable amplitudes grow to macroscopic order and thus become order parameters of the system. H. Haken called this process a “slaving principle” because the stable modes are “enslaved” by the unstable modes. Mathemati-

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cally, the adaption behavior of the stable modes to the unstable ones can in a first approximation be described by a so-called adiabatic elimination. In this case, the stable amplitudes can be expressed by the unstable ones and in this sense eliminated. It is sufficient to analyze the macroscopic order parameters of some few unstable modes to understand the whole dynamics of the system. An immense reduction of complexity has taken place: instead of dealing with billions of microscopic equations for all molecules in a fluid or atoms and photons in a laser it is now sufficient to treat the equations for a few macrovariables or order parameters characterizing collective patterns of a fluid or light modes and collective atomic behavior in a laser. The mathematical formalism of linear-stability analysis and adiabatic elimination can be generalized for huge classes of complex systems with nonequilibrium dynamics. Thus, order parameters are a universal instrument to model the emergence of complex structures in nature.

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Chapter 4

Symmetry and Complexity in Chemical Sciences

Physics, chemistry and biology have become intimately intertwined. Theoretical boundaries that historically were once assumed to exist between inanimate and animate world have turned out to be untenable. A unified theory of the natural sciences is beginning to develop, in which the classical disciplines investigate more or less complex subsystems having emerged in cosmic evolution. Examples of increasing complexity and decreasing symmetry are strings, elementary particles, atomic nuclei, atoms, molecules, crystals, genes and cells. On the scale of complexity, chemistry is a bridge between the microworld and macroworld. Chemistry is not just the science of electrons, atoms and molecules, but also of macroscopic objects such as crystals and gas clouds. Nevertheless, molecular structures are a key concept of chemistry with fascinating features of symmetry and complexity. Molecules have more or less symmetric and complex structures that can be defined in the mathematical framework of topology, group theory, dynamical systems theory and quantum mechanics. But symmetry and complexity are by no means only theoretical concepts of research. Modern computer aided visualizations show real forms of matter that nevertheless depend on the technical standards of observation, computation and representation. Their symmetries often have aesthetical qualities that seem to be inspired by the platonic idea of beauty. 4.1

Symmetry in Chemistry

Chemical symmetries depend on molecular structures. The molecular structure hypothesis states that a molecule is a collection of atoms 171

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linked by a network of bonds. Since the 19th century the molecular structure hypothesis has been a successful concept of ordering and classifying the observations of chemistry. But this hypothesis cannot directly be derived from the physical laws governing the motions of the nuclei and electrons that make up the atoms and the bonds. It must be justified that all atoms exist in molecules as separate definable pieces of the 3-dimensional (“real”) space with properties that can be predicted and computed by the laws of quantum mechanics. The well-known models of molecules with different information for a chemist are derived from the molecular structure hypothesis: (1) The 3-dimensional ball-and-stick model with balls for the atomic nuclei, sticks for the atomic bonds and their angles, (2) its 2-dimensional representation as structural formula, and (3) its 1-dimensional representation as linguistic name which can be derived from the structural formula. Graphic models are applications of mathematical graph theory that is a part of combinatorical topology. This mathematical theory became fundamental for chemistry, when in the midst of the last century the molecular structure of chemical substances were discovered [4.1]. Pasteur recognized that the relationship between symmetry of reflection and optical activity is not a function of the crystal structure of a substance. With certain water-soluble crystals, for example, the symmetry of reflection can be demonstrated both in the solid state and in the liquid state. Pasteur investigated tartaric acid and found a counterclockwise and a clockwise form, which are called L-tartaric acid and D-tartaric acid (D = dextro = right) respectively. He also isolated a third form of tartaric acid (mesotartaric acid), which cannot be separated into one of the other forms. To explain the optical activity, it was therefore necessary to investigate more fundamental structures than crystals or even molecules and the orientation of atoms. R.J. Ha¨ uy had already suspected that the form of crystals and their constituent components were images of one another. Pasteur therefore inferred the symmetric form of the crystal’s components from the crystal reflections [4.2]. Another important step was Kekul´e’s investigation of quadrivalent carbon atoms, for whose multiple bonds he also introduced a structural formula notation being still used in today’s organic chem-

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istry [4.3]. An essential advance occurred in 1864, when the Edinburgh chemist A. Crum Brown introduced his version of the graphic notation. Each atom was shown separately, represented by a letter enclosed in a circle, and all single and multiple bonds were marked by lines joining the circles. Crum Brown’s system is more or less the one in use today, except that the circles are now usually omitted. His notation was soon accepted everywhere, after some resistance from Kekul´e and others. Its acceptance was partly due to its success in explaining the strange fact that there are pairs of substances that have the same chemical composition, although their physical properties are different. The graphic notation made it clear that this is because the atoms are arranged in different ways in the different substances. This well-known chemical phenomenon is called isomerism, and in many cases there are more than two isomers with the same constitutional formula. In 1874, the great British mathematician A. Cayley wrote a paper “On the mathematical theory of isomers” inspired by the fusion of chemical and mathematical ideas. But the experiments of J.H. van’t Hoff and J.A. Le Bel were decisive for the assumption of a 3-dimensional molecular structure [4.4]. In 1874, independently of one another, they established a relationship between optical activity and 3-dimensional orientation of atoms. The initial example was the carbon atom, whose four valences were arranged in the form of a tetrahedron. A tetrahedral configuration with the carbon atom in the center makes possible the existence of two different arrangements being mirror images of each other (Fig. 50a). Tartaric acid has two carbon atoms which are each connected to the atoms or groups of atoms H, C, OH and COOH. For this combination, there are two arrangements (L- and D-tartaric acid) being mirror images of each other and one arrangement (meso-tartaric acid) which is reflective symmetric in itself (Fig. 50b). Van’t Hoff’s stereochemistry regarding the 3-dimensional structure of the atom must initially have appeared to a highly speculative idea, which betrayed a certain proximity to Platonic forms. Kekul´e may have been particularly adept at 3-dimensional visualization as a result of his prior study of architecture. Simultaneously with stereochemistry, geometry and algebra were also undergoing a fruitful

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Fig. 50a.

Symmetry of carbon atom with tetrahedral structure

Fig. 50b.

Symmetry of L-, D-, and meso-tartaric acid

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development [4.5]. Van’t Hoff’s success in experimental explanation and prediction made his geometry and algebra of the molecules soon a method accepted by chemists. But it lacked any definitive physical justification. At this stage of development, stereochemistry remains a successful heuristic approach meeting chemists’ need for a means by which they can visualize their structural analyses. From an experimental point of view the shape of molecules can be illustrated by an outer envelope of their electronic charge distributions. These representations are similar to the pictures of atoms that we can today obtain experimentally by the scattering of electrons in super microscopes or from the scanning tunnelling electron microscope. It is the distribution of charge that scatters the X-rays or electrons in these experiments. Thus, it is the distribution of charge that determines the form of molecular matter in 3-dimensional space. Mathematical methods of differential topology enable us to identify atoms in terms of the morphology of the charge distribution. The charge density p(r) is a scalar field over 3-dimensional space with a definite value at each point. Positions of extrema in the charge density with maxima, minima or saddles where the first derivatives of p(r) vanish can be studied in the associated gradient vector field ∇p(r). Whether an extremum is a maximum or a minimum, is determined by the sign of the second derivative or curvature at this point. The gradient vector field makes visible the molecular graph with a set of lines linking certain pairs of nuclei in the charge distribution. Local maxima of electronic charge distribution are found only at the positions of nuclei. This is an observation based on experimental results obtained from X-ray diffractions and on theoretical calculations on a large number of molecular systems. Thus, a nucleus seems to have the special role of an attractor in the gradient vector field of the charge density. In short: the topology of the measurable charge density defines the corresponding molecular structure. In the mathematical framework of dynamical systems theory the global arrangements of molecular forces can be represented by phase portraits with attractors as nuclei and trajectories representing the vector field. For example, Fig. 51 shows maps with nuclei and the symmetric structure of the ethylene molecule. Only those trajecto-

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ries are shown which terminate at the position of the nuclei. They define the basins of the nuclear attractors. In Fig. 51a only those trajectories are shown which terminate at the position of the nuclei. They define the basins of the nuclear attractors. Fig. 51b includes the trajectories terminating and originating at certain critical points (denoted by full circles) in the charge distribution. The pair of trajectories terminating at these critical points mark the intersection of an interatomic surface with the plane of the figure. The gradient paths originating at these critical points and define the bond paths are shown by the heavy lines. Fig. 51c shows a superposition of the trajectories associated with these critical points on a contour map of the charge density. These trajectories define the boundaries of the atoms in the nuclear graph. In general: the molecular graph is the network of bond paths linking pairs of neighboring nuclear attractors. An atom, free or bound, is defined as the union of an attractor and its basin. Atoms, bonds and structure are topological consequences of a measurable molecular charge distribution. In a next step, it is necessary to demonstrate that the topological atom and its properties have a basis in quantum mechanics. Topological atoms and bonds have a meaning in real 3-dimensional space. But this structure is not reflected in the properties of the abstract infinite-dimensional Hilbert space of the molecular state function. The state function ψ contains all of the information that can be known about a nuclear quantum system. From an operational point of view, there is too much and redundant information in the state function because of the indistinguishability of the electrons or because of the symmetry of their interactions. Some of it is unnecessary as a result of the two-body nature of the Coulombic interaction. Thus, there is a reduction of information in passing from the state function in the infinite-dimensional Hilbert space to the charge distribution function in real 3-dimensional space. But, on the other hand, we get a description of the molecular structure in the observable and measurable space. Quantum chemistry uses several mathematical procedures of approximation to achieve this kind of reduction [4.7]. A well-known approximation is the Born–Oppenheimer procedure allowing a sepa-

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(a)

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(b)

(c) Fig. 51.

Phase portraits with symmetric structure of the ethylene molecule

rate consideration of the electrons and nuclei of a molecule. We get the nuclear structure of a molecule beng represented by its structural formula. In order to distinguish the electrons as quasi-classical objects in orbitals, the Hartree-Fock method is sometimes an appropriate approximation for the electronic state function. The electronic charge density p(r, X) with the space vector r of an electron and the collection of nuclear coordinates X can be derived as the quantum mechanical probability density of finding any of the electrons in a particular elemental volume. In the case of molecules in station-

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ary states, the probability density is defined by the stationary-state function ψ(x, X) depending on the collection x of electronic space and spin coordinates and the collection X of nuclear coordinates [4.8]. This state function is a solution of Schr¨ odinger’s stationary state equation for a fixed arrangement of nuclei. The coincidence of the topological definition and the quantum definition of an atom in a molecular structure means that the topological atom is an open quantum subsystem of the molecular quantum system, free to exchange charge and momentum with its environment across boundaries which are defined in 3-dimensional real space. In this sense, symmetries of molecules referring to their topological structure are real forms of matter that can be calculated by quantum chemistry. Quantum chemistry and mathematical group theory are the modern bases of symmetry considerations in stereochemistry [4.9]. In quantum chemistry the symmetries of molecular systems are represented by the symmetries of the corresponding molecular Hamiltonian operators. In stereochemistry the structure of molecules is classified by the symmetry transformations of point groups. The symmetries of a free molecule (Fig. 52) can be completely defined by a few types of symmetry transformations. In general, the selection of the three coordinates axes x, y, z is arbitrary. The trivial symmetry transformation is identity I leaving each molecule unchanged. An additional symmetry element is the axis of rotation Cn around which a molecule can be rotated by the angle 2π/n without changing its position. Linear molecules, in which all atomic nuclei lie on a straight line (e.g. nitrogen N N or carbon monoxide C O), can be rotated around the connecting axis by arbitrarily small angles and have a continuous axis of rotation with infinite fold symmetry n → ∞. An additional symmetry element is the reflection CT on a plane in which the molecule does not change its position. For example, if the xy-plane is the plane of reflection, then replacing all the atomic z-coordinates by −z does not change the position of the molecule. Depending on the selection of the plane of reflection, a distinction is made between a vertical plane of reflection σn and a horizontal plane of reflection σh .

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Fig. 52.

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Symmetries of a free molecule

The next symmetry element is inversion in which a molecule remains unchanged during a reflection of all atomic coordinates (x, y, z) at the point of inversion to (−x, −y, −z). An additional symmetry element is rotary reflection Sn = σh Cn in which a molecule is first rotated by an angle 2π/n around the rotary reflection axis Cn and then reflected on the plane σh perpendicular to Cn through the center of the molecule, without changing its position. The remaining symmetry element is rotary inversion in which a molecule does not change its position in spite of rotation followed by inversion. It should also be noted that the compound symmetry transformations of rotary reflection and rotary inversion do not presuppose the partial transformations of rotation, reflection or inversion as symmetry elements of the same molecule. The symmetry transformations of a molecule, when executed one by one, produce symmetry transformations. In general, mathematical symmetries are definied by automorphisms that means self-mappings of figures or structures whereby

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the structure remains invariant (example: rotation or reflection of polygons in the plane). The composition of automorphisms satisfies the axioms of a mathematical group. So the symmetry of a molecular structure is defined by its group of automorphisms. There are continuous groups of symmetries (for instance, circles and spirals) and discrete groups (for instance, regular polygons, ornaments, Platonic bodies). On account of the finite number of combinations of symmetry elements, it is clear that there can only be a finite number of point groups. Thereby many different molecules can belong to the same point group, i.e. they can have the same symmetry structure. The classification of point groups also makes it possible to explain the relationship of optical activity and molecular structure in terms of group theory. According to Pasteur a compound had optical activity, if the molecule in question could not be made to coincide with its reflection. In that case, Pasteur spoke of dissymmetry [4.10]. Other terms are “enantiomery,” which in the Greek translation means opposite shape, or “chirality,” which alludes to the left and righthandedness of the reflective orientation. In terms of group theory, it is a matter of determining the elements of symmetry leading to optical activity. In general, (1) a molecule with any axis of reflection Sn cannot be optically active, and (2) a molecule without an axis of reflection is optically active. Point groups describe the symmetries of stationary molecules in the equilibrium state. Reduced symmetries may be present in the non-stationary case of translation, rotational motions, oscillations etc. Scalar characteristics such as mass, volume or temperature, which have only an amount but no direction, are apparently independent of the symmetry operations. But characteristics having not only an amount but also a direction can affect the symmetry. So far we have discussed the symmetries of the structures of molecular nuclei. What symmetries do determine the electron orbitals of the molecules? Molecular orbitals ψ are frequently introduced by approximation as linear combinations of the atomic orbitals χi , of the individual atoms of the molecule (Linear Combination of Atomic  Orbitals = LCAO method) with ψ = ci χi . Kekul´e’s famous ring i

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structure of benzene provides a clear example of orbital symmetry. The flat molecule C6 H6 consists of six carbon atoms which form a regular hexagon, and each of which is bonded to a single H atom. Each carbon atom has six electrons, two of which are in closed sshells, while the others are distributed into s and p orbitals. In Fig. 53a, one valence electron of the carbon is required to bond an H atom. Two valence electrons are required for the σ bond between the carbon atoms. The σ bonds are produced by a suitable mixing (hybridization) of the s, px and py atomic orbitals of the carbon. The fourth valence electron corresponds to the pz orbital, which is above and below the plane with its two dumb-bells, each perpendicular in the nodes of the carbon atom. The pz orbitals overlap with their respective neighbors and form a π bond. Fig. 51b shows a π orbital of benzene. In contrast to the σ bond, the π bond is weak, so that the π electrons can easily be influenced by extremal forces, and thus determine many of the spectroscopic characteristics of benzene. σ and π orbitals of benzene can be distinguished by their symmetry behavior in a reflection on the xy-plane. While σ orbitals ϕσ do not change their sign during the reflection z → −z and are therefore symmetric, antisymmetry occurs with the π orbitals ϕπ : ϕσ (x, y, −z) = ϕσ (x, y, z) ϕπ (x, y, −z) = −ϕπ (x, y, z) The system of π electrons offers a simplified way to calculate the energy levels of the benzene molecule. In the H¨ uckel model [4.11], we first consider π electrons, since it is assumed that the π molecule orbitals are significantly higher in energy than σ orbitals and can therefore be considered separately. Calculating ψ orbitals according to the LCAO method is therefore restricted in the H¨ uckel model to the atomic orbitals χi which form π molecular orbitals. That is another major simplification, of course, but one which has proven valuable in actual practice, e.g. in the calculation of benzene orbitals. The π orbitals of benzene are eigenfunctions of a Hamilton operator of the π electrons, which is invariant with respect to symmetry

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(a) Fig. 53.

(b) Symmetries of electron orbitals

operations of point group D6h of the regular hexagon with horizontal reflection σh . Physically, therefore, the potential energy of the n electrons is not changed when the benzene molecule is rotated, e.g., by 60◦ around the center. The H¨ uckel model and the orbital symmetries thereby assumed are also used to predict chemical reactions, as expressed in the Woodward–Hoffmann rules. One requirement is that the orbital symmetry is conserved during reactions, i.e., the symmetry of all occupied orbitals remains unchanged during the reaction with respect to each symmetry element shared by the reacting and resulting molecules [4.12]. In contrast to low-molecular chemistry, high-molecular or macromolecular chemistry is concerned with compounds which are composed of a great many atoms, and therefore have high molecular masses [4.13]. From the standpoint of symmetry, polymerizations are nothing more than polyadditions of monomers, the structural formulas of which form certain chains like those known from the frieze groups. These structural formulas recall the artful friezes in mosques, “structures of altogether unusual simplicity, unity and

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beauty” (W. Heisenberg) [4.14]. But with regard to the symmetries of the friezes of chemical structural formulas, we have to consider that these are only 2-dimensional projections of 3-dimensional structures. For crystal polymers in particular, X-ray diffraction spectra reveal stable conformations with well-defined symmetries. The significance of macromolecules in nature becomes clear when we investigate the structure and metabolism of living organisms. For example, their high molecular masses make it possible to construct solid and simultaneously flexible structures. On the other hand, their complex atomic structure makes it possible to regulate metabolic processes and to store information. From the standpoint of symmetry, proteins are of fundamental interest [4.15]. These are macromolecules of many amino acids of 20 different types in nature. Protein analysis shows that amino acids have an antisymmetrical carbon atom and occur only in the left-handed configuration in nature. If we investigate the 3-dimensional conformation of various amino acid units in the protein, we encounter a characteristic antisymmetry of the protein, in which the antisymmetry of its components is continued. L. Pauling, who detected a spiral structure in certain crystal protein fibers, called it α-helix. The α-helix consists of 18 monomer units on 5 revolutions each, which, among other things, are stabilized by intramolecular hydrogen bonds. One of the characteristic symmetry breakings of biopolymers is therefore the fact that proteins in nature form only left-handed spirals. Of course, reflections of the protein components also occur, but they cannot be fitted into

Fig. 54.

Antisymmetry of protein

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the molecular chains of proteins. Certain proteins differ from the regular helix structure. One well-known example is hemoglobin, whose stereochemical structure was reconstructed by the Nobel Prize winner M. Perutz, among others. To be precise, hemoglobin consists of a spherical protein (globin) and a complicated compound of iron (heme), which is not a protein. Hemoglobin is characterized by the double axis of rotation of its molecular chains [4.16]. X-ray crystallography now makes it possible to systematically analyze the symmetry structures of crystallized proteins and to explain them in terms of group theory. The mathematical structure of crystals is altogether independent of their physical or chemical interpretation. In biochemistry, atoms are not selected as structural units, but molecules. Since amino acids naturally occur in proteins only as left-handed configurations, certain symmetry elements of crystals requiring an equal number of left-handed and right-handed configurations, such as the plane of reflection, glide reflection and center of inversion, are a priori excluded. Mathematically speaking, from the 230 possible crystal groups only the first 65 ones with intrinsic movements remain. These 65 discrete intrinsic motion space groups in which biological macromolecules such as proteins can crystallize are therefore also called “biological” space groups [4.17]. They are used in the investigation of enzymes, for example. 4.2

Symmetry Breaking and Chirality

The fundamental symmetries of physics are at the origin of conserved quantities or constants of molecular processes, permanent and immutable for eternity. But, small violations of these symmetries lead to slight disturbances in this static world and introduce some possibilities for the emergence of new chemical phenomena. In traditional quantum molecular and particle physics it was assumed that the Hamiltonian of a molecular or particle quantum system is invariant under the following fundamental three operations: (1) P -operation of parity, i.e. the inversion of all particle coordinates in the center of mass with x → −x, y → −y, z → −z; (2) T -operation of time, i.e. time reversal t → −t ; (3) C-operation of charge, i.e. the exchange

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of all particles by their antiparticles. In elementary particle physics, these three symmetries are all more or less inexact or violated, while the combined operation CP T is still assumed to be an exact symmetry. Parity-violation has immediate consequences in molecular chemistry with perhaps even effects on the origin and evolution of life. In general, the symmetry of reflection (inversion) means that right-handed and left-handed structures of chiral molecules can be distinguished in space. But energetically they seem to be completely equivalent. It was van’t Hoff who found the geometrical explanation of chiral and optically active molecules. Mathematically, we can use coordinate systems with right and left orientation to distinguish both forms of chirality. In quantum chemistry the symmetry of chirality is represented by a quantum number (“parity”) with two possible values +1 for positive parity and −1 for negative parity. But the symmetry of chirality is violated by observations and measurements in the laboratories of biochemists. Macromolecules like, for instance, L-amino acids or D-sugars which are building blocks of living systems possess a characteristic homochirality or dissymmetry. Sometimes the enantiomers (i.e. the reflections of isomers) can be distinguished by simple tests of taste: S-asparagin has a bitter taste, while R-asparagin has a sweet taste. We can perceive this kind of symmetry breaking, because our body is a handed (chiral) biochemical system. In the 19th century Pasteur already presumed that living systems are characterized by typical dissymmetries of their molecular building blocks having emerged during biological evolution. Then the handed receptor molecules of our taste organs fit the chiral forms of the tasted molecules such as the right or left hand fits the right or left glove. But it cannot be explained why the actual molecular form of symmetry breaking was realized during the evolution and why the other form was unable to survive. As usual in classical physics, the two stable enantiomers can be illustrated by two minima of a symmetric potential curve V (q) where q is the reaction coordinate for the chemical transformation of the molecular substituents. Mathematically the potential curve of the reaction equation is assumed to be completely symmetric with re-

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spect to inversion. There are three solutions as equilibrium points with the two stable minima of the left- and right-handed forms and an unstable solution of a symmetric achiral form. The symmetry is broken by the actually realized stable form with respect to peculiar supplementary conditions. In quantum chemistry the framework of classical physics must be replaced by the principles of quantum mechanics. Molecular states are described by wave functions that can be superposed as pure entangled states according to the superposition principle. Thus for every temperature and energy there is not only the possibility of chiral molecules with either a left-handed or right-handed form, but also a third possible form which is both lefthanded and right-handed. Spontaneous symmetry breaking in quantum chemistry can be introduced by superselection rules forbidding the symmetric achiral superposition states that can be realized by a special physico-chemical environment (e.g. certain radiation fields). The classical and quantum mechanical concept of spontaneous symmetry breaking can only explain that a chiral molecule must emerge under some supplementary conditions. But it cannot explain why the actual form was realized instead of the other possibility. Therefore, the question arises if a selection happens by chance or by the necessity of a natural law. An explanation has been suggested with respect to the parity violating weak interaction that can be evaluated at least numerically in chiral molecules. In case of parity (P )-symmetry the right- and left-handed forms would be energetically exactly equivalent, transformed into each other by inversion. But parity was violated by the symmetry breaking of weak interaction during the cosmic evolution. Thus, if the parity violation can be measured by a small energy difference ∆Epv , we get the non-equivalence of the two isomers or enantiomers which are no longer simple mirror images of each other. The corresponding potential curve is no longer symmetric, but the two minima differ with the energy difference ∆Epv . Obviously the chemical law itself is no longer symmetric [4.18]. Then the actually realized forms of chiral biomolecules can be explained by their greater stability with respect to the parity violating energy. The emergence of a chemical phenomenon is reduced to a physical symmetry breaking.

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Fig. 55.

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Parity violation by small energy difference

As these energy differences are extremely small, they have so far not been measured in laboratories although schemes for such experiments have been proposed [4.19]. Nevertheless, such experiments in combination with accurate calculations provide insight in a fundamental link between cosmology, particle physics, molecular chemistry, and evolution. Even if these tiny energy differences increase proportionally during polymerization they still remain very small under laboratory conditions. But in evolution, nature itself was the laboratory [4.20]. For amino acids, for example, we can accurately calculate the prebiotic evolutionary conditions in which homochirality can be selected, e.g. in a lake with a certain volume of water and over a certain period of time. These calculations are based on an ab initio method (Hartree method) of numerical quantum chemistry, which currently has the best claim to accuracy. Therefore homochiral biochemistry can be interpreted as a direct result of the parity violation of weak interaction. Pasteur’s suspicion of a universal dissymmetrical force in nature is therefore reasonable, at least in terms of quantum chemistry. We could go even further and classify the chirality of biomolecules in a sequence of symmetry breakings that took place in the cosmological growth of the universe. Elementary particle physics intends to unify all the known physical interactions by deriving them from one interaction scheme based on a single symmetry group. Physicists expect to arrive at the actually observed and measured symmetries

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of fundamental forces and their elementary particles of interaction by spontaneous symmetry breaking processes. Electromagnetic and weak forces could already be unified by very high energies in the laboratories of high energy physics (for instance the accelerator ring of CERN). That means that at a state of very high energy the particles of weak interaction (electrons, neutrinos, etc.) and electromagnetic interaction cannot be distinguished any longer. They can be described by the same symmetry group U(1) × SU(2). At a particular critical value of lower energy the symmetry breaks down in two partial symmetries U(l) and SU(2) corresponding to the electromagnetic and weak force. The emergence of weak interaction with its particular violation of parity would be a result of cosmic symmetry breaking during the expansion of the universe. “C’est la dissym´etrie qui cr´ee le ph´enom`ene,” said P. Curie in 1894 [4.21]. Molecular dissymmetry, asymmetry and time irreversibility seem to be consequences of cosmic evolution with decreasing symmetry and simplicity and increasing complexity and variety [4.22]. Obviously, biopolymers of life are made of L-amino acids and D-sugars and not L-sugars and D-amino acids. We live in a world of matter and not of antimatter, which occurs only in small amounts as an exception such as with positrons from β-decay. Time seems to exclusively run forward, never exactly backwards in the molecular and macroscopic world. Parity, charge and time violation seem to result from early phase transitions of the universe. Symmetry breaking of time starts with the cosmic expansion of the tiny early quantum universe. Cviolation of matter and antimatter happens after symmetry breaking of the unified strong-weak-electromagnetic force. A tiny surplus of matter had become the stuff of future galaxies, our earth and life. Symmetry breaking of the unified weak-electromagnetic force caused parity-violation with the selection of biomolecules. This apparent symmetry breaking in the molecular world should be related to the fundamental CP T -symmetry of quantum chemistry. It turns out that chiral molecules provide an ideal test of CP T -symmetry. Tests of CP T -symmetry have been concerned with comparing the mass of proton and antiproton, proving equiva-

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lence and thus CP T -symmetry at a level of precision measured by ∆m/m ≈ 10−9 of mass m and mass difference ∆m. Further on, it has been suggested to compare the optical spectra of, for example, hydrogen and antihydrogen atoms with an accuracy in the test of about ∆m/m ≈ 10−18 . Spectroscopic experiments on chiral clusters and their antimatter equivalents are planned for testing the validity of CP T -symmetry at about 10−30 relative precision. They include the synthesis of chiral antimatter molecules. Tiny differences in these experiments between spectra of left-handed and right-handed molecules and clusters could lead to a complete revision of the traditional belief in fundamental symmetries. These observations of matter versus antimatter, left and right handedness of space and time directions forward and backward would not be granted with CP T -symmetry being valid. Besides spatial symmetries chemists are involved in the fundamental problems of time symmetry. While the laws of classical physics and quantum chemistry assume symmetry with respect of time inversion, the factual chemical reactions in the laboratories proceed only in one direction to the chemical equilibrium. Chemical processes are irreversible. Their reversion seems to be unnatural. Since Boltzmann’s statistical interpretation of the second law of thermodynamics irreversible processes have been discussed for complex molecular systems like gases, fluids, etc. The second law states that closed systems irreversibly approach the thermal equilibrium of maximal entropy. It is remarkable that Prigogine explains the irreversibility of dissipative processes far from thermal equilibrium by a universal symmetry breaking of time. Time has now the status of a mathematical operator only allowing physically asymmetric states. While the spontaneous symmetry breaking of elementary particles in high energy physics assumes the symmetry of its laws with respect to unitary (gauge) groups, Prigogine’s time operator delivers (non-unitary) semi-groups representing both directions of time [4.23]. The second law of thermodynamics is a kind of selection principle for the realized symmetry breaking process. In short: the law itself has become asymmetric.

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4.3

Symmetry and Complexity

Complexity, Dissipation and Nanosystems

Complex structures in nature are generated by conservative and dissipative self-organization. Dissipative systems are not closed like conservative systems, but open with interacting exchange of energy and matter with their environment. Thus, living organisms are typical examples of dissipative systems. But even in physics and chemistry, we can study dissipative self-organization. The emergence of dissipative structures far from thermal equilibrium is an irreversible process of symmetry breaking which can be geometrically illustrated by a bifurcation scheme. In other words: the bifurcation tree of a dissipative system represents the growth of forms in an irreversible time direction. In physics, fluid dynamics provide an example of pattern formation with increasing complexity if the system is driven away from equilibria by increasing velocities of flow. In chemistry, a dissipative system in which chaotic motion has been studied experimentally is the BZ-reaction. In this chemical process an organic molecule is oxidized by bromate ions, the oxidation being catalyzed by a redox system. The rates of change for the concentrations of the reactants in a system of chemical reactions are again described by a system of nonlinear differential equations with a nonlinear function. The variable signaling chaotic behavior in the BZ-reaction is the concentration of the ions in the redox system. Experimentally, irregular oscillations of these concentrations are observed with a suitable combination of the reactants. The oscillations are indicated by separated colored rings (Fig. 56). This separation is a fine visualization of nonlinearity. Linear evolutions would satisfy the superposition principle. In

Fig. 56.

Oscillation of Belousov–Zhabotinsky reaction

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this case the oscillating rings would penetrate each other in superposition. Analogously to fluid dynamics or laser systems, the pattern of oscillation is destroyed and transformed to chaos if the input rate of energetic substances surpasses certain thresholds of a control parameter [4.24]. The class of oscillatory, metal ion catalyzed oxidation of organic compounds by ionic bromate is an example of autocatalytic diffusion-reaction processes. This class of reactions involves the nonlinear diffusion of both excitatory and inhibitory molecules supporting self-exciting ring waves on the macroscopic level. In the language of chemistry, the autocatalytic process in a simplified case is modeled by reaction schemes with variable substances X = HBrO2 , Y = Br− , Z = Ce4+ , P = HOBr and constant substances A = BrO− 3 and B = BrMS: A+X → X +P A + Y → 2P A + X → 2X + Z 2X → A + P B + Z → fY A chemical autocatalytic term, e.g., in the third reaction rule, corresponds to a mathematical nonlinearity. Thus, in the language of mathematics, we get three differential equations of chemical concentrations cX , cY and cZ with constants ki of velocity of the i-th chemical reaction equation: dcX /dt = k1 cA cY − k2 cX cY + k3 cA cX − 2k4 cX 2 dcY /dt = −k1 cA cY − k2 cX cY + f k5 cB cZ dcZ /dt = k3 cA cX − k5 cB cZ The phase transition of the BZ-reaction is represented by typical patterns of time series analysis from periodicity to chaos. Chemical oscillations can also be represented by trajectories running into a limit cycle or chaos attractor of the corresponding phase space.

The phase transitions of the BZ-reaction are associated with symmetry breaking of spatial patterns. While the homogeneous surface of the chemical liquid in the equilibrium state has full symmetry, the emergence of ring waves and finally fractal chaos break the spatial symmetry, phase by phase. Again, decreasing symmetry is accompanied with increasing variety of complex structures. In the framework

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(a)

(b)

Fig. 57. [4.25]

Molecular self-organization of complex crystals with symmetric atoms

of dynamical systems theory, this kind of space and time symmetry breaking refers to phase transitions of complex open (dissipative) systems far from thermal equilibrium. Macroscopic patterns (“attractors”) arise from the nonlinear interactions of microscopic elements (i.e. atoms, molecules) when the energetic and material interaction of the dissipative (open) system with its environment reaches some critical value (“dissipative self-organization”). Phase transitions of closed systems near to thermal equilibrium are called conservative self-organization creating ordered structures with low energy at low temperature. An example at the boarderline of physics and chemistry is the growth of crystals by annealing the system to a critical value of temperature. Fig. 57a shows the molecular growth of complex crystals with translation symmetry from highly symmetrical metallic atoms. In Fig. 57b, the growth of crystals starts with molecular building blocks having the shape of Platonic bodies. Under more extreme conditions after the input of acid, they arrange themselves in complex crystals of metallic oxide. Their periodicity seems to be typical for the inan-

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imate molecular world. In the biosphere, we will find non-periodic structures with symmetry breaking which inspired E. Schr¨ odinger to introduce his famous concept of “aperiodic crystals” for living organisms. In supramolecular chemistry, conservative self-organization plays a tremendous role. In this case molecular self-organization means the spontaneous association of molecules under equilibrium conditions into stable and structurally well-defined aggregates with dimensions of 1–100 nanometers (1 nm = 10−9 m = 10 ˚ A). Nanostructures may be considered as small, familiar or large, depending on the view point of the disciplines concerned. To chemists, nanostructures are molecular assemblies of atoms numbering from 103 to 109 and of molecular weights of 104 to 1010 daltons. Thus, they are chemically large supramolecules. To molecular biologists, nanostructures have the size of familiar objects from proteins to viruses and cellular organelles. But to material scientists and electrical engineers, nanostructures are the current limit of microfabrication and thus are rather small [4.26]. Nanostructures are complex systems that evidently lie at the interface between solid-state physics, supramolecular chemistry and molecular biology. It follows that the exploration of nanostructures may provide hints about both the emergence of life and the fabrication of new materials. But engineering of nanostructures cannot be mastered in the traditional way of mechanical construction. There are no man-made tools or machines for putting together their building blocks like the elements of a clock, motor or computer chip. Thus, we must understand the principles of self-organization used by nanostructures in nature. Then, we only need to arrange the appropriate constraints under which the atomic elements of nanostructures associate themselves in a spontaneous self-construction: the elements adjust their own positions to reach a thermodynamic minimum without any manipulation by a human engineer. In the beginning of nanoscience there was the vision of an ingenious physicist. In an article entitled “There’s Plenty of Room at the Bottom”, Feynman declared that the principles of physics do not speak against the possibility of maneuvering things atom by atom

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[4.27]. Feynman proclaimed his physical ideas of the nanoworld in the late 1950s. The belief in a new world needs new instruments of observation and measurement for confirmation. Since the start of the 1980s, the nanoworld could actually be explored using the scanning tunnel microscope. At the end of the 1980s, E. Drechsler described a revolutionary vision of technological applications [4.28]. With nanotechnology, atoms will be specifically placed and connected in a fashion similar to processes found in living organisms. Historically, the idea of a supramolecular interaction dates back to the famous metaphor of E. Fischer (1894), who described a selective interaction of molecules as the lock and key principle. Today, supramolecular chemistry has by far surpassed its original focus. Molecular self-assemblies combine several features of covalent and non-covalent synthesis to make large and structurally well-defined assemblies of atoms. Single van der Waals interactions and hydrogen bonds are weak relative to typical covalent bonds and comparable to thermal energies. Therefore, many of these weak non-covalent interactions are necessary in order to achieve molecular stability in selfassembled aggregates. In biology, there are many complex systems of nanoscale structures such as proteins and viruses formed by selfassembly. Living systems sum up many weak interactions between chemical entities to make large ones. How can one make structures of the size and complexity of biological structures, but without using biological catalysts or the informational devices coded in genes? Many non-biological systems also display self-organizing behavior and furthermore provide examples of useful interactions. Molecular crystals are self-organizing structures. Liquid crystals are selforganized phases intermediates between crystals and liquids with regard to order. Micelles, emulsions and lipids display a broad variety of self-organizing behavior. An example is the generation of cascade polymers yielding molecular bifurcacional superstructures of fractal order [4.29]. Their synthesis is based on the architectural design of trees. Thus, these supramolecules are called dendrimers (from the Greek word “dendron” for tree and “polymer”). The construction of dendrimers follows two basic procedures of monomer addition. Divergent construction begins at the core and builds outward via an

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increasing number of repeating bifurcations. Convergent construction begins at the periphery and builds inward via a constant number of transformations. The divergent construction displaces the chemical reaction centers from the center to the periphery, generating a network of bifurcating branches around the center. The bifurcations increase exponentially up to a critical state of maximal size. They yield fractal structures such as molecular sponges which can absorb smaller molecules, which can then be dispersed in a controlled way, e.g., for medical applications. Examples of cave-like supramolecules are the Buckminsterfullerenes, forming great balls of carbon atoms [4.30]. The stability of these complex clusters is supported by their high geometric symmetry. The Buckminsterfullerenes are named after the geodesic networks of ball-like halls constructed by the American architect R. Buckminster Fuller (1895–1983). The cluster C60 of 60 carbon atoms has a highly Platonic symmetry of atomic pentagons forming a completely closed spheroid. Cave-like supramolecules can be arranged using chemical templates and matrices to produce complex molecular structures. Several giant clusters comparable in size to small proteins have been obtained by self-assembly. Giant clusters may have exceptional novel structural and electronic properties: there are planes of different magnetization being typical for special solid-state structures and of great significance for material sciences. A remarkable structural property is the nanometer-sized cavity inside the giant cluster. The use of templates and the selection of appropriate molecular arrangements may well remind us of Fischer’s lock and key principle [4.31]. Molecular cavities can be used as containers for other chemicals or even for medicaments needing to be transported within the human organism. An iron-storage protein that occurs in many higher organisms is ferritin. It is an unusual host-guest system consisting of an organic host (an aprotein) and a variable inorganic guest (an iron core). Depending on the external demand, iron can either be removed from this system or incorporated into it. Complex chemical aggregates like polyoxometalates are frequently discovered being based upon regular convex polyhedra, such as Platonic solids. But

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their collective electronic and/or magnetic properties cannot be deduced from the known properties of these building blocks. According to the catch-phrase “from molecules to materials” supramolecular chemistry applies the “blue-prints” of conservative self-organization to build up complex materials on the nanometer scale with novel catalytic, electronic, electrochemical, optical, magnetic and photochemical properties. Multi-property materials are extremely interesting. The exploration of the nanoworld and applications in nanotechnology depend on better instruments of observation and measurement. The scanning force microscope is a further development of the scanning tunnel microscope and can be used like a fountain pen to write down molecular structures of nano size. A thin film of thiolmolecules is used as “nano ink”. In a tiny drop of water the thiolmolecules organize themselves as monolayer. Nanocrystals of a few hundred atoms can organize themselves with cadmium ions, selen ions and organic molecules into a ball-like structure (Fig. 58). In ultraviolet light they fluoresce with a certain color. Thus, they could be used as markers (“quantum dots”) of molecules, cells, and substances in medicine, for example. Complex systems of carbon molecules can organize themselves as tiny tubes of 1 nm diameter according to certain catalysts and templates. Their symmetric order of bonding results in great hardness and toughness. Carbon nanotubes might be used as conductors for miniaturized chips beyond the limits of silicon technology. Supramolecular transistors are an example that may stimulate a revolutionary new step in the development of chemical comput-

Fig. 58.

Self-organization of complex nanocrystals [4.32]

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ers. Actually, there is a strong trend towards nanostructures in electronic systems which may realize small, fast devices and highdensity information storage. But one can also imagine nonelectronic applications of nanostructures. They could be used as components in microsensors or as catalysts and recognition elements in analogy to enzymes and receptors in living systems. In natural evolution very large complex molecular systems are also produced by stepwise gene-directed processes. The conservative self-organization processes of nanomolecular chemistry are non-gene-controlled reactions. Only a clever combination of conservative and non-conservative selforganization could have initiated prebiotic evolution before genes emerged. But even during the evolution of complex organisms, conservative self-organization must have occurred. Open (“dissipative”) physical and chemical systems lose their structure when the input of energy and matter is stopped or changed (e.g. laser, BZ-reaction). Organismic systems (like cells) are able to conserve much of their structure at least for a relatively long time. On the other hand, they need energy and matter within a certain interval of time to keep their structure more or less far from thermal equilibrium. In the technical evolution of mankind, the principles of conservative and dissipative self-organization have once more been discovered and open new avenues of technical applications.

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Chapter 5

Symmetry and Complexity in Life Sciences

Biochemistry and molecular biology are the bases of modern biology and thus serve as the bases for explanations of life processes of organisms such as bacteria, plants and animals. But organisms are not merely complex aggregates of atoms and molecules. Some characteristics of symmetry are determined by the building blocks, of course (e.g. the dissymmetry of proteins). On the higher level of organization of organisms, however, new characteristics of symmetry, dissymmetry and asymmetry arise, which become necessary as a result of functional requirements (e.g. adaptation to the environment, preservation of the species, metabolism). The whole, i.e. the organism, is therefore more than the sum of its parts. It is more correct to say that the parts of the organism and the environment are connected to one another by a number of functions which, when separated, lead to the death of the organism. Mathematically, the wholeness of an organism corresponds to the nonlinearity of its functional structure. Thus, the emergence of new forms of life with increasing complexity has been made possible by the nonlinear dynamics of self-organization. In the first section of this chapter, we analyze the functional symmetries of organisms. Then, in the following two sections, their evolution is explained by biological phase transitions and symmetry breaking, generating the complexity and diversity of life. 5.1

Symmetry in Biology

There are apparent symmetries in the living world. They all seem to be distinguished by special functions of an organism. But in 199

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contrast to elementary subatomic particles, atoms and molecules, the organism as a new and complex living unit cannot be so easily and clearly defined. All that can be indicated are a few necessary and by no means sufficient characteristics such as metabolism, selfreproduction, selection and mutation [5.1]. Viruses show how difficult it is to define life. They are organisms to the extent that they consist of complicated organic molecules such as nucleic acids and proteins, and possess genetic information for selfreproduction. On the other hand, they are constructed too simply to live and reproduce independently. A virus particle can only reproduce in the context of a living cell. Its short nucleic acid molecules can only carry the information of a few types of proteins. Therefore, the outer shell of the virus particle must be constructed from these few protein molecules. Fig. 59 shows the molecule of an adeno-virus [5.2]. Externally, the 252 capsomeres and 1500 identical protomers form a regular Platonic body (icosahedron) with 20 equilateral triangles, but where the capsomeres are not completely identical to one another. The extensions projecting from the 12 corners infect the host cells. On account of this extremely symmetrical structure, virus particles can coagulate to form crystals and thus assume a form of organization like that found in non-living matter. If we are aware of the effect of the virus particles in mammals (infection and cancer), the model illustrated in Fig. 59 has a certain ghostly beauty and recalls an ominous space ship from an alien microworld. The virus particle is a clear example of the fact that the dynamics of life processes require symmetry breakings. As a Platonic body in Fig. 59, for example, it is quite lifeless. To participate in the life of the host cell and reproduce, the virus particle must trigger an infection and thus give up its symmetry. In the sense of structural symmetry, an organism such as the amoeba is altogether asymmetrical. But it is an almost perfect survival system, in which the functions such as metabolism, motion, nutrient transport etc. are optimally harmonized with one another. But mathematically, the concept of symmetry is not bound to the geometric shape. A structure whose elements are functionally har-

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Fig. 59.

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Symmetry of a virus

monized with one another and which remains conserved (invariant) during self-reproduction in the sense of a mathematical mapping in itself also forms a symmetry. In contrast to a geometric, structural symmetry, we therefore speak of a functional symmetry [5.3]. If we establish a scale of aggregates between maximum symmetry and chaos, for example, from crystals through paracrystals, liquid crystals, gels and real fluids up to ideal fluids and gases, then living organisms are obviously classified somewhere in the middle. Their symmetries are of an altogether statistical nature, since in selfreproduction invariant patterns of characteristics of course recur, but more or less random changes also occur, and thus make possible the variety and individuality of living nature. The sudden changes in the genome (mutations) are typically local symmetry breaking which is part of evolution.

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No two leaves are exactly alike. All that exists is the type of a certain leaf, around which the real leaves are statistically scattered. Statistical symmetries are therefore scattered around an average value with a standard deviation. The patterns of complex cell wall pores or muscle fibers are microscopic examples while, for example, the pattern of veins in a leaf or the arrangement of blades of grass in a meadow is directly visible to the eye. But perfect symmetry in these cases is merely a human fiction that is projected on nature by abstraction. These are rather the structures of biological growth programs which are realized in the context of certain statistical deviations.

Fig. 60.

Cellular symmetries of self-reproduction

Among these restrictions, we shall first examine the geometric symmetries of more or less complex organisms, from single-cell organisms through the higher plants up to the animals. Spherical bacteria and algae can be broken down through many planes into symmetrically equal halves. Single-cell organisms such as the single-cell green algae frequently have a main axis. Vegetative reproduction then customarily takes place by longitudinal and diagonal division of the cell (Fig. 60). In other bacteria and algae, depending on the variety, there is a regular division in one, two or all three 3-dimensional directions, which can lead to different associations of cells, for example, chains of streptococci which look like strings of pearls, threads like the thread-forming conjugatae or colonies which feature a division of labor among the individual cells [5.4]. In the higher plants, it becomes possible to distinguish the elements of symmetry such as translation, rotation and reflection. Seldom is there a pure translation symmetry, in which the leaves grow

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isometrically along with axis of the branch with the same translation length (e.g. the leaflets of the ailanthus tree (Ailanthus altissima) illustrated in Fig. 61a). More frequently, growth is related to a regular increase of the translation units, such as in the leaflets of the mountain ash in Fig. 61b. In addition to longitudinal symmetry, these two examples also exhibit a lateral symmetry, since the left and right halves of the leaf are symmetrical to one another. On the other hand, the elm branch in Fig. 61c is generated by a glide reflection, in which a translation and a reflection form a combined symmetry operation. In the positions of leaves, helical symmetries also occur, in which the translation of the growth motion is connected with a unit of rotation. One example is the helical pattern of the leaves of the branch of a variety of stonecrop, on which each leaf differs from the preceding leaf by a certain angle. In a helical motion, in general, every point of the space not laid on the helical axis describes a helix. The positions that the moving

Fig. 61.

Symmetries of leaves

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point assumes at uniform moments of an interval of time are uniformly distributed over this helix, analogous to the steps of a spiral staircase. It is apparent that the ratios µ/v with which the helixlike arrangements of leaves are represented are frequently elements of the Fibonacci sequence (1/1, 1/2, 2/3, 3/5, 5/8, 8/13,. . . ). This consequence results from √ the expansion into a continued fraction of the irrational number ( 5 − 1)/2, i.e. the proportional ratio of the Golden Section. If we replace the cylinder on which the helix moves with a sphere, we get a symmetry operation in which in addition to translation and rotation, a dilatation also occurs. One example is the arrangement of the scales on a pine cone. If we check the Fibonacci numbers on the scales of the pine cone, we find frequent deviations, so that in the best case, we can only speak of a statistical symmetry of the phyllotaxy. When the giant sunflower (Helianthus maximus) is in bloom, the

Fig. 62a–d.

Symmetries of influorescenes

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Fig. 62e.

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Spiral tendency of nature

small blossoms are arranged in logarithmic spirals, whereby two sets of spirals occur with opposite directions of rotation (Fig. 62e). Both Bonnet and in particular Goethe noted the “spiral tendency of nature” in the latter’s theory of the metamorphosis of plants. For Goethe, however, the decisive factor was not the arithmetic law of the Fibonacci sequence, but the regular pattern in which a temporal rhythm of growth is revealed in three dimensions, and which contains clues to the shape of an ancestral plant. Spirals of plants occur as right-hand and left-hand rotations. There is no parity-violation. A high degree of symmetries of rotation and reflection can be demonstrated in many inflorescenes [5.5]. In Fig. 62a, the symmetry of a geranium is completely defined by the dihedral group D5 , which in addition to the rotations by the angle n·360◦ /5 with 1 ≤ n ≤ 5 also contains the possible reflections. On the other hand, Vinca herbacea

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is a realization of pure symmetry of rotation C5 . The narrow-leafed wild rose in Fig. 62c therefore has only the symmetry D4 , since its growth was influenced by artificial manipulations of gravity. In nature, there is the bilaterally symmetrical shape (Fig. 62d), which was prevented by gravity from becoming symmetry of rotation. As a rule, functional requirements are the reason for the structural symmetries of animals, the purpose of which is to guarantee viability and which cannot be explained by molecular reasons alone. These are of course characteristics that only occur at a certain stage of evolution on account of new life circumstances (“emergent characteristics”). But we need not assume any “vitalistic” forces of zoology to explain them. One of the most frequent symmetries in animal bodies is reflectional symmetry. More than 95% of all types of animals are included among the Bilateralia with simply symmetry of reflection [5.6]. Forward motion in a direction at right angles to gravity is certainly an important objective for crawling, walking, swimming and flying Bilateralia such as snakes, lizards, fish, insects and birds. Since Antiquity, philosophers have speculated whether the bodies of human beings and animals are designed according to a certain law of proportion (e.g., the Golden Section). For example, Fig. 63 shows the proportions of a butterfly (b) and a fish (c). The ratio of the Golden Section, which occurs in the context of statistical symmetries, is determined by the flow conditions of the respective medium (air or water) in which the motions of flight or the flight-like motions of swimming must be executed. In place of the Platonic assumption of an ideal form defined by nature, the theory of evolution suggests a selection advantage by means of which a certain symmetry is preferred over another. Rotational symmetry has advantages in terms of natural selection for free-floating or stationary animals. Examples of stationary animals (Fig. 64), which must spread their snares in all directions, are sponges (a), rotifers (b), the pterobranchia (c) and echinoderms (d). The anthozoae, a group which includes the water lilies and coral polyps, also have almost exclusively rotational symmetry. Jellyfish also have rotational symmetry, as recorded in minute detail by E. Haeckel in his book “Artistic Forms of Nature” (1899)

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a

b

c

Fig. 63.

Bilateral symmetry of animals [5.7]

[5.8]. Fig. 64f shows an octagonal symmetry D8 . With a pronounced Platonic slant, D’Arcy Thompson, in “On Growth and Form” describes the geometric symmetry of the jellyfish [5.9]. There is also the approximately spherical shape of organisms that float freely in the water. Physically, the sphere always presents the same resistance to the water. Organisms frequently inflate themselves with water to achieve a spherical external shape, such as plankton organisms or marine larvae. The interior of these organisms is in no way constructed centrally symmetrically, but only the

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Fig. 64.

Rotational symmetry of Radiolaria

outer shell, as a result of the above-mentioned physical environmental conditions. The Radiolaria have fascinating sphere-like shapes, and Haeckel and D’Arcy Thompson have already described their siliceous skeletons and cell body with extreme precision. The sphere-like symmetry

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of the Radiolaria also corresponds to the centric symmetry of the cell. Fig. 64g shows the skeleton of a Periphylina Radiolaria, whose spherical skeleton is equipped with spines in three planes of reflection/lines of intersection at right angles to one another. The Radiolaria Acantharia in Fig. 64h acts exactly like a space satellite. Haeckel drew skeletons of various Radiolaria, the symmetries of which are intended to recall Platonic bodies such as octahedron, icosahedron and dodecahedron. But there is some question about the extent to which his pen was guided by his desire to stylize the “artistic forms of nature”. In the animal kingdom, spiral symmetry is found among the snails. In the shell of the “Turitelle duplicata” (Fig. 64i), the symmetry is defined by a continuous group whose symmetry elements consist of a combination of translation, rotation and dilatation of the radius of rotation. The dilatation of the size of the successive helical rotations satisfies with approximate accuracy the mathematical law of a geometric sequence. The snail’s shell holds the animal’s intestinal cavity which, with increasing age, is wound in a spiral fashion. The symmetry of this shell can be disrupted by growth and environmental conditions. Examples are a thickening of the intestine as the snail ages or the growth of the sexual organs when the animal reaches sexual maturity, but also changes by deposits of coralline limestone. From the standpoint of structural symmetry, therefore, simpler forms of life such as jellyfish, Radiolaria, snails etc. exhibit a significantly higher degree of symmetry than the Bilateralia of fish, birds, mammals etc. Apparently, gravity is a central requirement for the bilaterality of organisms. Gravity makes it possible for the respective body to be designed as a vector with a pronounced direction of motion at right angles to gravity. The work of motion on two reflectively symmetrical sides of the animal can itself be bilateral, as with the simultaneous flapping of wings, in the form of sliding reflections as during the walking of two-legged or four-legged animals (e.g. horses and human beings), helical as with serpents, crawling animals, etc. In any case, the tendency of organisms to bilaterality, from the crawling of amphibians to the upright stance of the human being, is a remarkable symmetry characteristic in evolution.

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The increasing complexity of types of motion is also characteristic. A human being does not move in short jerks like a robot or jumping jack. Only at first glance do such mechanical motions have a greater symmetry than the natural motions of primates. Human motor skills are characterized by a high degree of flexibility and a complex coordination of many sequences of motion, which express high proportionality and thus also symmetry. The bilaterality of the human body seems to have enforced the reflection symmetry of the two halves of the brain. In general, the bilaterally symmetrical structure of many animals extends to a corresponding structure of the nervous system (Fig. 72). Actually, there are some neural functions distributed symmetrically on both halves of the brain. But higher animals also exhibit a symmetry breaking in that the two halves of the brain do not always retain the same function: that is, copying the left or right domain of the external world via organs of touch, sight and hearing. The two halves of the brain are locally specialized, not only in perception, but also in higher functions of human intelligence. The cerebral asymmetry associated with this specialization is now considered to be a result of evolution. As mirror-image duplication was increasingly dispensed with, the freed-up capacity could be used by the right and left halves of the brain for new specializations. Brains are distinguished by high flexibility. If a local part of one half is damaged, the failing function can often be compensated by some neural part on the other half of the brain. Rigid symmetric domains are surpassed in order to guarantee the highly complex coordination of the organism. The question of what role symmetry plays in perception and knowledge involves cognitive science and philosophy, and is examined in Sec. 8.1. 5.2

Symmetry Breaking and Evolution

In the framework of nonlinear complex systems, the emergence of new phenomena is made possible by phase transitions in critical states of a system. The symmetry breaking of phase transitions can be illustrated by a tree of bifurcation points where old equilibria become unstable and new branches are generated with new local

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points of equilibria and the emerging phenomena as leaves (Fig. 48). In nonequilibrium dynamics, a bifurcation tree spread out with increasing complexity when the dynamical system is driven further and further away from its initial equilibrium to local equilibria, becoming unstable under changing conditions. Mathematically, the emerging structures at the branches of the bifurcation tree are represented by order parameters of differential equations, modeling the time-depending evolution of the dynamical system. Order parameters of complex systems are selected at critical points of instability when unstable modes of microscopic elements dominate stable ones and determine the emergence of new macroscopic phenomena (Fig. 49). In short: decreasing symmetry (“symmetry breaking”) and increasing complexity are the ingredients of nonlinear dynamics. That was true in physics and chemistry for reaction-diffusion processes and self-exciting media. It is also true for biological evolution if the mathematical terms of the formalism are interpreted in an appropriate manner. In the framework of nonlinear complex systems, many models have already been suggested to simulate the molecular origin of life. Complexity on the molecular scale is characterized by a large potential number of states which could be populated given realistic limits of time and space. Certain microstates may strongly influence macroscopic behavior. Such fluctuations may amplify and cause a breakdown of formerly stable states. Nonlinearity comes in through processes far from the thermal equilibrium. Classical and only necessary conditions for life demand: (1) self-reproduction (in order to preserve a species, despite steady destruction), (2) variability and selection (in order to enlarge and perfect the possibility of a species, biased by certain value criteria), and (3) metabolism (in order to compensate for the steady production of entropy) [5.10]. These criteria can be realized by a mathematical optimization process. In this model, the nucleation of a self-reproducing and further evolving system occurs with a finite expectation value among any distribution of random sequences of macromolecules such as proteins and nucleic acids. The initial copy choice for self-reproduction is accidental, but the subsequent evolutionary optimization to a level

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of unique efficiency is guided by physical principles. In this model, life should be found wherever the physical and chemical conditions are favorable, although some molecular structures should show only slight similarity with the systems known to us. The final outcome will be a unique structure, for example, an optimized molecular sequence. Darwin’s principle of the survival of the fittest is mathematized by an optimization principle for possible microstates of molecular sequences. It is assumed that in simple cases biomolecules multiply by autocatalysis. For instance, two kinds of biomolecules A and B from ground substances GS are multiplied by autocatalysis, but in addition the multiplication of one kind is assisted by that of the other kind and vice versa (Fig. 65a). In more complicated cases with more kinds of biomolecules, the latter are assumed to multiply by cyclic catalysis (Eigen’s “hypercycles”) (Fig. 65b). This mechanism combined with mutations is able to realize an evolutionary process.

Fig. 65a–b.

Cyclic catalysis

According to M. Eigen and P. Schuster (1979), a hypercycle is a cycle of cycles of cycles [5.11]. The phase transitions of prebioic evolution start at the microlevel with molecular microdynamics driven by a basic catalytic cycle. An example is the citric acid cycle, which uses a molecule of oxaloacetate over and over again to extract the energy from acetic acid, driving a variety of higher level cycles that

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Fig. 66.

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Emergence of prebiotic life by self-organization

require consumption of energy. At a higher macrolevel of organization (Fig. 66), several such basic cycles can comprise an autocatalytic cycle that is able to instruct its own reproduction. The new emerging macroscopic phenomena are, for example, DNA-protein structures that can reproduce itself. Several of these autocatalytic cycles can be organized at yet a higher macrolevel into a catalytic hypercycle (Fig. 66). An example of a new emerging entity at this second macrolevel is a virus. In Sec. 5.1, a virus was explained as a system at the threshold of life from a “dead” crystal to a “living organism.” It is a first example of a complex adaptive system [5.12]. Obviously, we get a hierarchy of macrolevels with new emerging entities of increasing complexity. According to the general scheme of Fig. 49, each level can be modeled by appropriate differential equations. The order parameters of the macrodynamics characterize the new emerging entities. In the biological context, order parameters are mainly interpreted as fitness degrees or selection values of molecular species. A simplified model of an appropriate complex molecular system is a so-called evolution reactor [5.13]. In the reactor there are macromolecules such as nucleic acids, which are continuously constructed and broken down. Nucleic acids consist of four different building blocks. The population consists of i alternative sequences or concentrations of uniform chain length. The chain length of the macromolecules is ν, their coefficient of occupation ni , the total num-

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bers of all sequences z =



ni . It is assumed that the number 4ν of

i

all combinatorially possible alternative sequences is very small in relation to the total population z. In that case, the expectation value for the occurrence of a certain sequence of molecules is also very small and is equivalent to the condition which can be assumed to have existed at the origin of life. High-energy molecules are continuously fed to the reactor from outside to construct the nucleic acids. For that purpose, autocatalytic processes of self-reproduction are assumed, which Eigen and Schuster described in their models of hypercycles. The low-energy byproducts are continuously removed. A constant total population can be established and maintained by appropriate regulation from outside. The construction and destruction of macromolecules takes place as in independent organisms. Let the construction parameter of the molecule type ni be γi , and the destruction parameter δi . Both can be a function of the concentrations nj of other types. As a result of mutations, only a certain fraction of copies of a sequence will be error-free. The proportion of correct copies is designated by a quality parameter λi , which is a fraction 0 < λi < 1. The rates of evolution of the changes of ni over time are then dni /dt = (γi λi − δi )ni − e(t)ni (“evolution equations”),  in which e(t) = (γi − δi )ni /z is then the average generation rate of all types of i

molecules [5.14]. Then e(t)ni in the evolution equations is a destruction quantity which can be used to designate the proportion of ni in the conservation of the constancy of z. From the evolution equations, the following necessary characteristics of living systems can be derived: (1) The metabolism of the open system is measured by   yi ni and δi ni of high-energy and low-energy molecules. the reaction terms i

i

(2) The self-reproduction is expressed in the evolution equations by the fact that the evolution rate of a molecule type ni is proportional to its concentration. Thus the potential dependence of the construction and destruction parameter γi and δi on the other concentrations is not affected. (3) The capacity for mutations is taken into consideration by the quality parameter λi . The evolution equations thus indeed satisfy the characteristics of molecular Darwinism. The quantity wi = γi λi − δi is interpreted as the selection value of the type of molecule. This interpretation is justified because the equations of evolution are written as dni /dt = (wi − e(t))ni .

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An evolution equation can be understood as an extremal principle, according to which the types of molecules are optimized on the basis of selection values. If the selection value wi of one type is less than the average production rate e(t) of all types, then negative growth rates occur, and the type dies out. In the other case, there are positive growth rates. Thus e is constantly pushed upward, while simultaneously an increasing number of types have lower selection values and thus die out. This selection process is stabilized only when e has reached the maximum selection value wmax of the production e → wmax and the system is in selection equilibrium. But this equilibrium state is only preliminary. As soon as a new mutant ni+1 occurs which is more favorable in terms of selection than the dominant type, the equilibrium collapses. Again, a new selection equilibrium is then established, which is defined by the now dominant type ni+1 . A system of molecules optimizing itself (“evolution reactor”) therefore runs through a series of selection evolutions, which correspond to an ascending series of maximum selection values which belong respectively to the currently dominant type wmax 1 < wmax 2 < · · · < wopt . According to S. Wright and Eigen, this optimization path, on which the system climbs to increasingly higher “peaks” of selection values, can also be represented in three dimensions [5.15]. For that purpose, a sequence of molecules is defined with ν positions as a point in the ν-dimensional sequence space. With 4 possible components, the ν-dimensional sequence space has 4ν points. For simplification, we will limit ourselves to the binary case with two symbols 0 and 1, by means of which, theoretically, the 4 molecular letters A, T, G and C can also be codified. The binary system also facilitates the calculations on a computer. In this case, the ν-dimensional sequence space has 2ν points. Each point has ν neighbor points, each of which represents a single-error mutant, i.e. mutants which differ by only one position. Between the two extreme points of a pure 0 and 1 sequence, there are ν! possible connections. Fig. 67 shows examples of ν-dimensional sequential spaces in the binary case. The major advantage of this 3-dimensional representation is the very short distances involved and the dense network of possible connections. For example, the longest distance in the 1000-dimensional

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Fig. 67.

ν-dimensional sequential spaces

space is only 1000 length units (“meters”). In the 23-dimensional space with 1014 points, it is only 23 “meters”. Moreover, the 23-dimensional sequence space is sufficient to represent all points on the surface of the earth at an interval of 1 meter. In this space, optimal strategies can be pursued to find the highest mountains on the earth. To do that, a value function is used to assign an “altitude” to each point. On a hike, the strategy is to go uphill as much as possible, and to lose as little altitude as possible. Therefore we are looking for a local peak, so that we can then climb to an adjacent, higher peak, etc. Mathematically, therefore, the gradient of the hike is “uphill”, and we can make local decisions regarding which of the next peaks should be climbed next.

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On the surface of the earth, therefore, the hiker will try to reach as many peaks as possible along a 1-dimensional grade or col, without losing too much altitude in between. But the hiker is severely restricted by the 1-dimensionality of his path. In the 23-dimensional space, the hiker could go toward each position in 23 different directions, of which k ≤ 23 lead upward and 23-k lead downward. The probability of finding maximum peaks in the immediate vicinity is therefore very great. In the ν-dimensional space of molecule sequences, the points are assigned selection values instead of altitudes. Analogous to the altitudes of the mountainous landscape, the selection values in the molecular sequence space are not distributed randomly, but are assembled in regions, so that “mountains” and “plains” can be localized, for example. In this model, the origin of life is a successive self-optimization of a molecular system which is achieved by means of a series of intermediate selection steps. It is not a one-time, random event, a unique singularity in which, on account of a random fluctuation of the phase state, the inanimate matter becomes unstable and spontaneously changes into a new equilibrium state that we call life. According to nonequilibrium dynamics, life does not arise as the result of a one-time, spontaneous symmetry breaking but in a series of local symmetry breakings, in which selection equilibriums that have become unstable are replaced by new and higher-level equilibria. The emergence of proteins and DNA-structure would have not been possible without self-optimizing phase transitions of nonlinear dynamics [5.16]. Throughout the evolution of life on earth, only a small fraction of all possible proteins have been generated. In general, a protein is a polymer of amino acids, coupled by valence bonds. Because 20 distinct amino acids are available to biological organisms, there are some 20200 possible protein molecules that are composed of 200 amino acids each. There is not enough matter in the entire universe to construct a single molecule of each possible protein. Another calculation provides a similar result: the length of a gene is seldom more than 1000 sequential positions. Thus, for four symbols, there are 41000 alternative genes (“mutations”) of length

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1000, which means about 10600 possibilities. We should recall that the content of matter in the whole universe corresponds to 1074 genes and that the age of the universe is less than 1018 seconds. Selection values have been recorded in the genetic codes of living organisms. The nucleic acids, which are primarily responsible for transmitting characteristics through generations of living systems, show characteristic symmetry breakings. Nucleic acids are macromolecules formed by linear polymerization of certain units (nucleotides). According to the double helix model of J.D. Watson and F.C. Crick, the DNA molecule consists of two strands of DNA intertwined in a regular double helix around a common axis [5.17]. The two strands are parallel, but in opposite directions. The sequence of the bases in the one strand determines the sequence in the other strand, so that an A is always opposite to a T and a G is always opposite to a C. Antisymmetry is of fundamental importance for the transmission of genetic characteristics, which can be explained on the molecular level of DNA helix. The genetic information of an organism is encoded in its set of chromosomes (genome) in the form of DNA (Fig. 68). The mechanism of reproduction, which makes possible a clear duplication (replication or reduplication), can be illustrated very clearly: an enzyme allows the hydrogen bridges to open and the double helices to separate into two strands. Each strand reproduces its exact opposite to which it is reconnected by hydrogen bonds, thereby forming a new double helix. On account of the complementary base pair formation of A-T and C-G, which is expressed in three dimensions in the twisting of the double helix, the accuracy of the copy is guaranteed even after many reproductions. If the two strands were orientated parallel and symmetric to one another like a ladder, then the accuracy of the reproductive process would not be guaranteed, with disastrous consequences for the respective organisms. This case demonstrates that antisymmetry or parity violation can have a decisive biological function. In general, an evolutionary process is expected to produce new kinds of species [5.18]. A species may be considered as a population of biomolecules, bacteria, plants or animals. These populations are

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Fig. 68.

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Antisymmetry of DNA

characterized by genes, which undergo mutations producing new features. Although mutations occur at random, they may be influenced by external factors in the environment, such as changing temperature or chemical agents. At a certain critical mutation pressure new

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kinds of individuals of a population come into existence. Genetic selforganization generates Darwin’s evolutionary tree with local points of instability and bifurcating branches where new species emerge (Fig. 69). Obviously, the biological bifurcation tree can be compared with the bifurcation tree of nonequilibrium dynamics (Fig. 48). In case of Darwin’s evolutionary tree, fluctuations at the critical points of bifurcation are interpreted as mutations and biological random events. Selection is the driving force in the bifurcating branches. (The numbers at the branches in Fig. 69 measure the substitutions of nucleotids in the DNA of a species.) Instead of dissipative pattern formation in, for example, liquids or gas we observe the emergence of new forms of organisms. Of course, there are main differences between pattern formation in physics, chemistry, and biology. While, for example, physical and chemical systems lose their structure when the flux of energy and matter is switched off, much of the structure of biological systems is still preserved for a certain time. The reason is that biological systems do not only consist of dissipative processes, but also conservative structures like, for example, the skeleton of human body, which decay on a longer scale of time than organs or liquids. Further on, biological systems satisfy certain functions and tasks in a living organism. But, in the biological context, this kind of purposes is explained by selection values or fitness degrees which, in nonequilibrium dynamics, can be represented by order parameters of nonlinear differential equations. When at a critical point of mutation pressure new kinds of individuals of a population emerge, then the rate of change of these individuals is described by an evolution equation. As these individuals have new features, their growth and death factors differ. A change (mutation) is only possible when fluctuations occur in the population and the environment. Thus, the evolution equation determines the rate of change as the sum of fluctuations and the difference of growth and death factors. A selection pressure can be modeled when different subspecies compete for the same living conditions (e.g. the same food supply). If the mutation rate for a special mutant is small, only that mutant survives which has the highest gain factor and the smallest loss fac-

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Fig. 69.

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Bifurcation tree of evolution

tor and is thus the fittest. The competition at a critical point of instability can be formalized by a linear-stability analysis and adiabatic elimination: unstable mutants begin to dominate the stable ones. They determine macroscopic features, which become order parameters of the new organism and species.

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Biological organisms function on many levels of dynamic activities that have emerged step by step during cosmic and biological evolution [5.19]. The lowest are that of physics and chemistry. The diagram of Fig. 70 is not definitive or complete in an ontological sense. But, from a methological point of view, it represents a formal scheme of research to explore the hierarchical structure of biological organisms and, in general, biological systems. There may be further hierarchical levels between the indicated ones. The hierarchy may bifurcate for different evolutionary trends. New levels may emerge and disappear under changing circumstances. Thus, the hierarchy is not static and fixed for ever in the sense of Aristotle’s philosophy of nature. On the other side, it is not only a mere ordering scheme which has been invented by human mind for pragmatic concerns, but a dynamical principle of biological organization which is well confirmed by observation. According to the general scheme of nonequilibrium dynamics (Fig. 49), we start with the microdynamics of a complex system. It is a question of granulation how “deep” we like to lay the initial layer of microdynamics. As far as we know at least atomic dynamics influence states of living organisms. From the nonlinear dynamics at each level, there emerge new entities that are characterized by order parameters. The macrodynamics of these order parameters determine the microdynamics of the new entities, providing the basis of macrodynamics on the following level. In principle, the dynamics of each level could be modeled by appropriate nonlinear differential equations. In this case, the succeding hierarchical level could be mathematically derived from the previous one by a linear-stability analysis and adiabatic elimination. Actually, in many cases, we only have simplified models that may inspire future research. In some cases there are already well established procedures encouraging the use of the hierarchical scheme [5.20]. The hierarchical diagram of Fig. 70 is a scheme of self-organization at several levels in a biological system. According to the laws of quantum dynamics, atoms organize themselves into molecules. Out of the nonlinear interactions of molecules emerge the proteins acting as catalysts in biochemical cycles. Biochemical cycles support the repli-

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Fig. 70.

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Emergent structures of life by self-organization

cation of biomolecules. Now, cellular dynamics come in. In the next step, cells arrange themselves in organs, interacting in organisms. Their nonlinear interactions determine the population dynamics of a species, which is embedded in the environmental dynamics of nature. Each level determines the following one. But sometimes, there are not only direct feedbacks to the immediately preceding level, but also to deeper ones. It is a well known fact of medicine that, for example, physiology and cytology of our body are influenced by nutrition, depending on the conditions of environment. The whole of our body is more than the sum of its parts. 5.3

Complexity and Biodiversity of Life

The spontaneous emergence of organic forms has seemed to be a miracle of life. Thus, in the history of science, morphogenesis was a

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prominent counterexample against physical reductionism in biology. Today, morphogenesis is a prominent example for modeling biological growth by complex dynamical systems. In this context, pattern formation is understood as a complex process wherein identical cells become differentiated and give rise to a well-defined spatial structure. The first dynamic models of morphogenesis were suggested by N. Rashevsky, Turing, and others [5.21]. In the complex systems approach, the emergence of cellular patterns is explained by competing interactions of activator and inhibitor molecules. In a mathematical model due to A. Gierer and H. Meinhardt, two evolution equations were suggested, describing the rate of change of activator and inhibitor concentrations, which depend on the space-time coordinates [5.22]. The change of rates is due to a production rate, decay rate and diffusion term. Obviously, inhibitor and activator must be able to diffuse in some regions in order to influence the neighboring cells of some transplant. Furthermore, the effect of hindering autocatalysis by the inhibitor must be modeled. The interplay between activator and inhibitor can even lead to growing periodic and symmetric patterns, satisfying functional tasks (compare Figs. 61–63 in Sec. 5.1). To derive such patterns, it is essential that the inhibitor diffuses more easily than the activator. Long range inhibition and short range activation are required for a non-oscillating pattern. By methods of mathematical analysis, the evolving patterns described by the evolution equations of Gierer and Meinhardt can be determined. A control parameter allows one to distinguish the stable ones in the sense of a linear-stability analysis. According to the mathematical procedure of an adiabatic elimination, the stable modes can be eliminated, and the unstable ones deliver order parameters determining the actual pattern. Thus, actual patterns come into existence by competition and selection of some unstable solution. Selection according to linear-stability analysis and adiabatic elimination means reduction of complexity that stems from the huge number of degrees of freedom in a complex system. Thus, evolution does not only mean increasing complexity as H. Spencer proclaimed. The evolution of order parameters requires reduction of complexity.

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Fig. 71.

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Phase transition of activator and inhibitor concentrations

Biochemically, this kind of modeling of morphogenesis is based on the idea that a morphogenetic field is formed by diffusion and reaction of certain chemicals. This field switches genes on to cause cell differentiations. Independently of the particular biochemical mechanism, morphogenesis seemes to be governed by a general scheme of pattern formation in physics and biology. We start with a population of totipotent cells corresponding to a system with full symmetry. Then, cell differentation is effected by changing a control parameter that corresponds to symmetry breaking. The consequence is an irreversible phase transition far from thermal equilibrium. In Fig. 71, the phase transition of activator and inhibitor concentration is illustrated in a computer simulation [5.23].

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Since Antiquity, living systems were assumed to serve certain purposes and tasks. Organs of animals and humans are typical examples of functional structures that are explored by physiology and anatomy. The complex bifurcations of vessel networks are examples of fractal structures. The form of trees, ferns, corals and other growing systems are well described by fractals. Trees branching into open space have room to expand. But hearts, lungs and other organs occupy a limited space. The networks of nerves or vessels that penetrate them are servants to the principal occupants of the space. The structure of the microvascular network is virtually completely defined by the cells of the organ. In skeletal and cardiac muscles the capillaries are arrayed parallel to the muscle cells, with some cross-branches. The system is guided in its growth by the need for the nerve or the vascular system to follow the lines of least resistance. Fractal illustrations of a bronchial network are an inspiration for physicians to apply these approaches to the lung. Physical systems, from galactic clusters to diffusing molecules, often show fractal behavior. Obviously, living systems might often be well described by fractal algorithms. The vascular network and the processes of diffusion and transmembrane transport might be fractal features of the heart [5.24]. These fractal features provide a basis which enables physicians to understand more global behavior such as atrial or ventricular fibrillation and perfusion heterogeneity. Nonlinear dynamics allows us to describe the emergence of turbulence, which is a great medical problem for blood flow in arteries. Turbulence can be the basis of limit cycling, as can be shown with water flowing through a cylindrical pipe. A variety of control systems produce oscillations. It might also be expected that some oscillating control systems show chaotic behavior. Atrial and ventricular fibrillation are the classic phenomena that appear chaotic. The clinical statement on the heart rate in atrial fibrillation is that it is completely irregular. The observations are that the surface of the atrium is pulsing in an apparently chaotic fashion. However, the studies of reentry phenomena and of ventricular fibrillation show that there are patterns of excitation, again illustrating that this is organized (“mathematical”) chaos [5.25]. Fractal and chaotic

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algorithms for this have been described. Nevertheless, chaotic states cannot generally be identified with illness, while regular states do not always represent health. There are limited chaotic oscillations protecting the organism from a dangerous inflexibility. Organs must be able to react in flexible ways, when circumstances change rapidly and unexpectedly. The rates of heart beat and respiration are by no means fixed like the mechanical model of an idealized pendulum. The coordination of the complex cellular and organic interactions in an organism needs a new kind of self-organizing controlling. That was made possible by the evolution of nervous systems that also enabled organisms to adapt to changing living conditions and to learn from experiences with its environment. In Fig. 72 the levels of organization in the nervous system of the human body are illustrated [5.26]. The hierarchy of anatomical organizations varies over different scales of magnitude, from molecular dimensions to that of the entire central nervous system (CNS). The scales consider molecules, membranes, synapses, neurons, nuclei, circuits, networks, layers, maps, systems and the entire nervous system. On the right side of the figure, a chemical synapse is shown at the bottom, in the middle a network model of how ganglion cells could be connected to simple cells in visual cortex, at the top a subset of visual areas in visual cortex, and on the left side the entire CNS. The research perspectives on these hierarchical levels may concern questions, for example, of how signals are integrated in dendrites, how neurons interact in a network, how networks interact in a system like vision, how systems interact in the CNS, or how the CNS interact with its environment. Each stratum may be characterized by some order parameters determining its particular structure, which is caused by complex interactions of subelements with respect to the particular level of hierarchy. Beginning at the bottom we may, for instance, distinguish the orders of ion movement, channel configurations, action potentials, potential waves, locomotion, perception, behavior, feeling and reasoning. The neurons of animal organisms have been generated by evolution to carry impulses of electric voltage from one to another along

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Fig. 72.

Emergent structures of the nervous system

interconnecting fibers or axons. Axons can be considered as tubes of membrane which are semipermeable to ions of sodium (Na+ ) and potassium (K+ ). In the early 1950s, A. Hodgkin and A. Huxley measured the sodium and potassium components of membrane ionic current for several squid axons [5.27]. From these data they formulated phenomenological expressions for the ion current flowing out the cell per unit area of the membrane. Furthermore, they introduced variables to represent the opening and closing of ionic channels across the membrane. These empirically measured activities were modeled by a nonlinear diffusion–reaction equation with an exact solution of a traveling wave (Fig. 73), giving a precise prediction of the speed and shape of the impulse of electric voltage. On a fiber with x-direction, V (x, t) is the transverse voltage across the cell membrane and i(x, t) the electric current, flowing through the tube. Ap-

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plying Kirchoff’s voltage and current laws delivers the differential equations ∂V /∂x = −ri and ∂i/∂x = −c∂V /∂t − ji with the longitudinal resistance r per unit length of the fiber, the membrane capacitance c per unit length, and the total ionic current ji flowing across the membrane per unit length. The term c∂V /∂t represents a displacement current flowing through the membrane capacitance. The ionic current is carried across the membrane through protein pores or channels. The combination of these two equations delivers the Hodgkin–Huxley equation ∂ 2 V /∂x2 − rc∂V /∂t = rji . A nerve impulse as solution of this equation is transmitted in the x-direction without attenuation because of the nonlinearities in the representation of the ionic current [5.30].

Although the large amplitude impulse in Fig. 73 is a stable solitary wave, moving with fixed shape and speed, it differs essentially from solitons. A collision between two nerve impulses destroys both of them. The Hodgkin–Huxley equation is no Hamiltonian with conservation of energy, but a nonlinear diffusion equation. A linear-stability analysis confirms empirical evidence that impulses of greater speed and amplitude are stable, whereas the slower and smaller travellingwave solution is unstable. The unstable wave in Fig. 73 determines

Fig. 73.

Emergence of a nerve impulse [5.28]

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threshold conditions for igniting the stable impulse. In general, nerve impulses emerge as exact solutions of nonlinear differential equations. Thus, they are new dynamical entities like ring waves in BZ-reactions or fluid patterns in nonequilibrium dynamics. They are the units of neural interactions which provide the dynamics of neural pattern formation, representing all kinds of motor, sensory or cognitive activities of the brain. In short: they are the “atoms” of the complex neural dynamics. In order to model the brain and its complex abilities, it is quite adequate to distinguish the following categories. In neuronal-level models, studies are concentrated on the dynamic and adaptive properties of each nerve cell or neuron, in order to describe the neuron as a unit. In network-level models, identical neurons are interconnected to exhibit emergent system functions. In nervous-system-level models, several networks are combined to demonstrate more complex functions of sensory perception, motor functions, stability control, etc. In mental-operation-level models, the basic processes of cognition, thinking, problem-solving, etc. are described. According to the complex systems approach, we have to define the state variables and their dynamical equations generating the patterns of interaction for each hierarchical level. On the neural basis, we distinguish neurons and synapses between them. Neurons have two possible microstates, firing and non-firing a nerve impulse. In the firing state, the axon of a neuron opens small caves (vesicles) in its end, filled with chemical substances (neurotransmitters) which are transmitted into the synapses in order to activate a further neuron. A neuron receives several input signals from a dendritic tree of input channels. It fires a nerve impulse if the weighted sum of its inputs exceeds some threshold. In 1949, D. Hebb already emphasized that to comprehend the activity of the brain one should not only focus attention on isolated neurons, but on cell assemblies [5.30]. He suggested that the brain functions not merely by firing and non-firing neurons, but by activating assemblies of neurons. As in the evolution of living organisms, the belief in organizing “demons” is dropped and replaced by selforganizing procedures of the complex system approach. Roughly

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speaking, Hebb’s synaptic rule demands that the synaptic connection between two neurons should be strengthened if both neurons fired at the same time. Thus, the complex system approach offers a view of self-organizing networks changing their synaptic connections, which are induced by synaptic activation and depend on the degree of activation. In the framework of neural complex systems, the microscopic level of interacting neurons is distinguished from the macroscopic level of global patterns produced as cell assemblies by self-organization. Neural self-organization means that an assembly is formed by synchronous activation according to a Hebb-like rule. C. von der Malsburg modified Hebb’s hypothesis by demanding that assembly formation is produced by rapid synaptic changes [5.31]. Thus, there is no “mother neuron” that can feel, think, or, at least, coordinate the appropriate neurons. The binding problem of pixels and features in perception is explained by cell assemblies of synchronously firing neurons dominated by learnt attractors of brain dynamics. The binding problem asked: How can the perception of entire objects be conceived without decaying into millions of unconnected pixels and signals of firing neurons? H.B. Barlow’s theory assumed single neurons for each property of a perceived object, other neurons for clusters of properties, and, finally, a neuron for the entire object (“grandmother neuron”) [5.32]. Thus, the brain needs an exploding number of specialized neurons which must be postulated in ad hoc hypotheses for every new perception of changing situations. W. Singer and others confirmed von der Malsburg’s concept of synchronously firing neurons through observations and measurements [5.33]. In the complex systems approach, the microscopic level of interacting neurons should be modeled by coupled differential equations modeling the transmission of nerve impulses by each neuron. On the macroscopic level, they generate a cell assembly whose macrodynamics is characterized by some dominating order parameters. For example, a synchronously firing cell-assembly represents some visual perception of a plant which is not only the sum of its perceived pixles, but characterized by some typical macroscopic features like form, background or foreground. On the next level, cell assemblies

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of several perceptions could interact in a complex scenario. In this case, each cell-assembly is a firing unit, generating a cell assembly of cell assemblies whose macrodynamics is characterized by some order parameters. The order parameters could represent similar properties between the perceived objects. By that, we get a classification of similar objects. For example, several similar plants may belong to the same botanic category. In a next step, we may reflect on botanic categories in general, and on categories of categories, and so on. In Fig. 74, we distinguish a hierarchy of emerging levels of cognition, starting with the microdynamics of firing neurons. Analogous to Fig. 70, the diagram is not definitive or complete in an ontological sense. But, from a methological point of view, it represents a formal scheme of research to explore the hierarchical structure of neural and cognitive dynamics of the brain. The dynamics of each level is assumed to be characterized by differential equations with order parameters. For example, on the first level of macrodynamics, order parameters characterize a visual perception. On the following level, the observer becomes conscious of the perception. Then the cell assembly of perception is connected with the neural area that is responsible for states of consciousness. In a next step, a conscious perception can be the goal of planning activities. In this case, cell assemblies of cell assemblies are connected with neural areas in the planning cortex, and so on. Even high-level concepts like self-consciousness or historical consciousness can be explained by self-reflections of self-reflections, connected with a personal or public memory which is represented in corresponding cell assemblies of the brain [5.34]. Neural states and the macrostates of cell assemblies that interact to give brain function, generate a variety of pattern formation that is well known from nonequilibrium dynamics. Brains undergo repeated phase transitions. For example, locomotion is a state, within which walking is a rhythmic pattern of activity that involves large parts of the brain, spinal cord, muscles and bones. The entire neuromuscular system changes instantly with the transition to a pattern of jogging or running. Brains undergo repeated transitions from waking to sleeping and back again, but still, giving the same persons as the night before. Personal identity is an example of stability, equilib-

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Emergent cognitive structures of the brain

rium and symmetry of brain states: The state of identity is invariant with respect to time translation, although there may be local changes with increasing aging. Brain states emerge, persist for a small fraction of a second, then disappear and are replaced by other states. It is the flexibility and creativeness of this process that makes a brain so successful in animals for their adaptation to rapidly changing and unpredictable environments. According to nonequilibrium dynamics, we can distinguish three kinds of attractors. The simplest is the fixed point attractor. In this case, the system is at rest unless perturbed, and it returns to rest when allowed to do so. Examples of neural point attractors are silent neurons or cell assemblies that have been isolated from the brain. A system that gives periodic behavior is said to have a limit cycle attractor. When periodic activity of cell assemblies does occur, it is either intentional, as in rhythmic drumming, clapping and dancing, or it is pathological as in the periodic oscillations of the eyes in

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nystagmus, or of the limbs during Parkinsonian tremor. The third type of attractor gives aperiodic and chaotic irregularities of the kind that is observed in recordings of EEGs. Chaotic dynamics sensitively depends on initial states. Therefore, the system behavior is unpredictable in the long run, although a deterministic evolution equation may be defined explicitly. In the case of deterministic chaos, when the systems only have a small number of components and a few degrees of freedom, the attractors are mathematically well known. They can be derived from the corresponding nonlinear differential equations or by time-series analysis. They are mainly low-dimensional, stationary and noisy-free. Large and complex real-world systems, which include neurons and neural populations, are noisy, nearly infinite-dimensional, nonstationary and non-autonomous. These brain states are called “stochastic chaos” by W. Freeman [5.35]. The source is postulated to be the synaptic interaction of millions of neurons, which create local fields of microscopic noise in the cortex. These activities are revealed by spatial patterns of amplitude modulation (AM patterns) of a spatially coherent aperiodic carrier wave in the gamma range of the EGG. AM patterns play an essential role for perceptions, which has been discovered in the olfactory brain by Freeman [5.36]. Perceptions are not only passive sensory mappings of the external world like the pictures of a camera. Perceptions begin with selections according to goals, motivations and preferences. They are goal-directed actions through the participation of the limbic system with the neurochemical nuclei in the brain stem that express and directly control the state of the organism, body and brain. The limbic system is the neural part that provide emotional and motivational states of the brain. AM patterns represent the intentionality of an individual, guiding the brain to selected perceptions. In this sense, they can be considered as order parameters constraining and determining neurons and cell assemblies for the selected features and context of a perception. The discovery that brain dynamics operates in chaotic domains has profound implications for the study of higher brain function. A chaotic system has the capacity to create novel and unexpected patterns of activity. Phase transitions between chaotic states consti-

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tute the dynamics that we need to understand how brains perform such remarkable feats as abstraction of the essentials of figures from complex, unknown and unpredictable backgrounds, and constant updating by learning. Stochastic chaos attractors are aperiodic with limitations of long-term predictions and local islands of noise, but still highly complex correlated mathematical structures. Complete global noise is a stochastic state without any correlation between its components. If the brain of an individual is at rest with no evidence of overt behavior, then it is in a state of noise. The neurons fire continually, but not in concert with each other. Any correlation between neurons and assemblies has decayed. Therefore, the state of noise has continual activity with no history of how it started and without chance of prediction of when the next pulse and which next state will occur. Nevertheless, noise is the substrate from which chaos and all the other attractors emerge. Noise is essential for maintaining the health of neurons, because they must stay active in order to survive. Brains of animals are no isolated monads, but embedded in the social dynamics of their species. Animal populations can be characterized on a scale of greater or lesser complexity of social behavior. There are populations of insects with a complex social structure, which is rather interesting for sociobiology. The interactions between individuals are physically realized by sound, vision, touch and the transmission of chemical signals. The complex order of the system is determined by functional structures like the regulation of the castes, nest construction, formation of paths, the transport of materials or prey, etc. Ants synthesize chemical substances, which regulate their behavior. They have a tendency to follow the same direction at the place where the density of the chemical molecules reaches a maximum. Collective and macroscopic movements of the animals are regulated by these chemical concentrations. In order to model the collective movements, two equations are suggested, considering the rate of change for the concentrations of insects and chemical substances. There is a critical value of an order parameter (“chemicotactic coefficient”) for which a stationary homogeneous solution becomes unstable. The system then evolves to an inhomogeneous stationary state. Accordingly, different branching

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Fig. 75.

Bifurcation tree and attractors of ant population

structures will appear, as observed in different ant societies. Fig. 75 shows the collective movement of ants with two types of structure characteristic of two different species [5.37]. The social complexity of insects can also be characterized by such coordinated behavior as nest construction. This activity has been well observed and explored by experimental studies. A typical observation is that the existence of a deposit of building material at a specific point stimulates the insects to accumulate more building material there. This is an autocatalytic reaction which, together with the random displacement of insects, can be modeled by three differential equations. These equations refer to the observation that the termites, in manipulating the construction material, give it the scent of particular chemical substance, which diffuses in the atmo-

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sphere and attracts the insects to the points of highest density, where deposits of building material have already been made. Thus, the first equation describes the rate of change of the concentration of building material, which is proportional to the concentration of insects. A second evolution equation refers to the rate of change of the scent with a certain diffusion coefficient. A third evolution equation describes the rate of change of the concentration of insects including the flow of insects, diffusion and motion directed toward the sources of the scent. The complex social activity of nest construction corresponds to the solutions of these equations. Thus, an uncoordinated phase of activity in the beginning corresponds to the homogeneous solution of these equations. If a sufficiently large fluctuation with a larger deposit of building material is realized somewhere, then a pillar or wall can appear. The emergence of macroscopic order, visualized in the insect’s architecture of nests, has been caused by fluctuations of microscopic interactions. There is no insect as commander-inchief with a master plan in its brain, but a chemical diffusion field, guiding the local activities of individuals. The guiding field can be represented by an order parameter, which is sometimes called “swarm intelligence”: the swarm knows more than its individuals. Again, the whole is more than the sum of its parts. Insect colonies are only examples of populations in complex ecosystems that have been generated during evolution. Ecosystems are the results of physical, chemical and biotic components of nature acting together in a structurally and functionally organized system. Ecology is the science of how these living and nonliving components function together in nature. Obviously, in the framework of the complex system approach, ecology has to deal with dissipative and conservative structures of very high complexity depending on the complexity of the individual physical, chemical and biotic systems involved in them and the complexity of their interactions [5.38]. Since the co-evolution of human civilization, the ecological equilibria on earth have become highly critical. It is a challenge of complexity research to support a sustainable future of ecological dynamics on earth.

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Chapter 6

Symmetry and Complexity in Economic and Social Sciences

Sociodynamics of human societies have emerged from population dynamics during the evolution of life on earth. After thermodynamic, genetic, and neural self-organization, we have to explore a new kind of complex systems dynamics. The self-organization of social and economic structures is explored in social and economic sciences. In this context, symmetry is a functional principle that is associated with basic ideas of social organization. Justice in law, equality of people in constitutions, human welfare, social and economic stability have been founded by principles of symmetry. In early cultures, social symmetry was ontologically identified with the symmetry of the universe. In modern societies, it is a dynamical principle that can be modeled in the framework of nonequilibrium dynamics. Symmetry breaking and socio-economic transitions are related to critical instabilities and shifts of historical, economic, and political developments. In the age of globalization, local regions, nations and industries compete in global markets and political conflicts of high instability. The increasing complexity of social, economic, political and cultural problems is a challenge of global governance, which means the decision on the order parameters of human future.

6.1

Symmetry, Social Balance, and Economic Equilibrium

In all early cultures of mankind, the stability of a society has been associated with ideas of symmetry. For example, justice means a state of completion if the society is arranged in its harmonic propor239

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tions in equilibrium, like the static balance of a scale. In a picture of ancient Egypt (2nd century B.C.), Gods arrange the “last judgment of Osiris” with a divine scale (Fig. 76). According to Confucian, as well as Greek philosophers, the rules of justice are founded on the cosmic laws of harmony (compare Sec. 1.1).

Fig. 76.

Divine balance in the last judgment of Osiris

In modern times, symmetry and equilibrium of societies have been conceived in the framework of classical physics. Since the religious and political unity had been destroyed in Europe in long civil wars, people longed for security and peace. T. Hobbes (1588–1679), for example, projected a mechanistic model of society and state at the end of the Middle Ages and the beginning of modern times [6.1]. In order to guarantee the “law of peace”, all citizens have to transfer their natural right of power to an absolute sovereign (“Leviathan”) who is alone legitimated to apply political power and to rule the state. Thus, Hobbes’ social contract legitimates the state’s monopoly of power in order to keep society in an absolute equilibrium. The “phase transition” from the natural state of chaos to political order and equilibrium is realized by a social contract of all citizens. In

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the same manner, the economic system was modeled by the physiocratic economists as a mechanistic clockwork of cog wheels, springs, and weights [6.2]. A clock is a sequentially operating system with programmed functions. Analogously, the physiocratic economy cannot regulate itself. Advances in agriculture, which are the driving forces of the physiocratic economy, are compared with the weights and springs of a clock. Economic production was referred to the composed movements in a clock. Consequently, economic prosperity is only guaranteed by a regular economic circulation like a clockwork. Its causal determinism without any kind of self-regulation or individual freedom corresponds completely to the political system of absolutism during the 17th and 18th century. The citizens are reduced to functioning elements in a political and economic machine. While the physiocrats designed their economic model against the Cartesian background of a mechanistic clockwork, Smith referred to Newton’s classical physics. In Newton’s celestial mechanics, material bodies move in a system of dynamical equilibrium that is determined by the invisible forces of gravitation. The physical concept of freely moving individuals in dynamical equilibrium corresponds to the liberal ideas of a free economy and society with the division of independent political powers. Contrary to the liberal ideas, the Cartesian clockwork of nature seems to accord with the state machinery of absolutism with its citizens as cog wheels. Locke’s liberal ideas of democracy with division and balance of power mainly influenced the American and French constitutions. In his famous “Inquiry into the Nature and Causes of the Wealth of Nations” (1776) [6.3], Smith emphasized that human self-interest is not a theoretical construction of economists, but an empirical fact of experience. Self-interest is a strong and natural impulse of single human beings, and therefore a human right. But the interactions of several single micro-interests achieve the common macro-effect of welfare by the mechanism of the market. The mechanism of the market is regulated by supply and demand, driving the competing micro-interests to the macro-effect of welfare and the “wealth of the nation” in the equilibrium of the market. In the mechanistic view, the

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micro-interests seem to be drawn to the common macrostate of equilibrium by an “economic demon” or mechanical spring. According to the Newtonian method, Smith prefers the picture of an “invisible hand” directing the micro-interests like the “invisible” long-distance force of gravitation in astronomy. Obviously, Smith describes an economy as a complex system of many competing micro-interests [6.4]. The dynamics of their interactions is a self-organizing process of competition with a final state of equilibrium between supply and demand. Smith provided the first model of an economic equilibrium theory that has become the hard core of classical and neo-classical economics. In the 19th century, predecessors of modern mathematical economics propagated the use of the mathematical methods of physics in economics [6.5]. They spoke of a more or less rough correspondence between the play of economic forces and mechanical equilibrium. Actually, much of their vocabulary was borrowed from mechanics and thermodynamics, for instance, equilibrium, balance, stability, elasticity, expansion, inflation, contraction, flow, force, pressure, resistance, reaction, movement, friction and so on. A market exists for a commodity if the commodity can be transferred from any individual to any other without cost. A market clears when, at a certain price, the demand and supply of its commodity are in balance. The price that clears the market is called the equilibrium price of the commodity. Idealized equilibrium models assume the existence and clearance of markets for all commodities like physical equilibrium models without friction. In order to understand general ideas of economic equilibrium theory, some more economic terminology must be introduced: an economy is perfectly competitive if all individuals are passive price takers that cannot influence prices. Thus, monopolists and oligopolists, which can influence prices, are failures of perfectly competitive markets. The commodities are the resource of an economy. The resource is allocated through the transformation of commodities in production and the transfer of ownership in exchange. The microstates of these transformations and transfers are called resource allocation. A competitive equilibrium is a resource allocation in which all markets

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clear. Classical and neo-classical theorists have tried to find the conditions under which competitive equilibria exist for perfectly competitive markets. If all engaged individuals of a market are completely self-interested without any cooperation, all competitive equilibria are asserted to be Pareto-optimal. An allocation is Pareto-optimal if any alteration will make at least one individual worse off. In the 20th century, economists gave up physical analogies and tried to found their own mathematical instruments. A milestone for mathematical equilibrium theory has been von Neumann’s and Morgenstern’s game theory [6.6]. Game theory aims to understand situations in which decision-makers interact. Examples are chess as well as firms competing for business, politicians competing for votes, nations competing for dominance or armies in wars [6.7]. A game is a mathematical description of a strategic interaction. Any strategic interaction involves two or more decision makers (players), each with two or more ways of acting (strategies). The outcome depends on the strategy choices of all the players. Each player has well defined preferences among all the possible outcomes. A formal representation of a game makes explicit the rules of interaction, the players’ strategies, and their preferences over outcomes. A possible representation of a game is in normal form. A normalform game is completely defined by a list of players Pi (i = 1, . . . , n), a finite set of pure strategies si for each player Pi , and a utility function ui that gives player Pi ’s payoff ui (s1 , . . . , sn ) for each ntuple of strategies. Simple applications are the so-called zero-sum games of two persons. In this case, the gain of a player P1 is the loss of the other player P2 and vice versa, i.e. u1 = −u2 . If each player has two possible strategies, the possible strategic interactions with their corresponding payoffs are represented in a 2× 2 matrix (Fig. 77a). If, for example, player P1 chooses strategy s11 and player P2 strategy s21 , then player P1 gets payoff 4 and player P2 the corresponding loss −4. Whereas normal-form games are represented by matrices, extensive-form games are represented by trees. A matrix description shows the outcomes, represented in terms of players’ payoffs for every possible combination of strategies the players might choose. A

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(a)

(b) Fig. 77.

(c)

Matrices of strategies in normal-form games

tree representation is sequential, because it shows the order in which actions are taken by the players. In Fig. 78, the nodes of a game tree represent events of decisions. The branches are the alternative decisions. The name of each player P1 and P2 is indicated besides the nodes. After their choices of strategies, the players arrive at the final nodes and get their payoffs, which corresponds to the matrix in Fig. 77a. A game tree looks like a bifurcation tree of events in nonequilibrium dynamics. But, there is a main difference. What is important, is not the temporal order of events, but whether players know about other players’ actions when they have to choose their own. Thus, the information a player has when he/she is choosing an action is explicitly represented in a game tree by an information set (closed curve in Fig. 78). If an information set contains more than one node, the player who has to make a choice at that information set will be uncertain as to which node he/she is at. Not knowing at which node one is means that the player does not know which action was chosen by the preceding player. If a game tree has information sets with more than one node, the corresponding game is one of imperfect knowledge. Because of its uncertainty, each player tries to minimize the maximal disadvantage of its actions. Therefore, player 1 chooses the maximin-strategy si , for which the minimum minj ui (si , sj ) among all alternatives is maximal. If the corresponding maximum is minimal, the strategy is called minimax-strategy. In Fig. 77a player P1 will prefer strategy s12 and player P2 strategy s22 . For all

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Bifurcation tree of strategies in extensive-form games

strategies s1 of player P1 , it follows u1 (s1 , s22 ) ≤ u1 (s12 , s22 ) and u2 (s12 , s2 ) ≤ u2 (s12 , s22 ) for all strategies s2 of player P2 . Thus, point (s12 , s22 ) is called the equilibrium point of this game. No player can improve his/her situation by changing its strategy. But these inequalities of utilities are not true in general. In Fig. 77b, again, player P1 will prefer strategy s12 and player P2 strategy s22 . But contrary to the game in Fig. 77a, the first inequality is violated: player P1 can improve his/her situation with respect to point (s12 , s22 ) by choosing s11 . Nevertheless, a rational decision is possible if the players change from pure strategies to mixed strategies. A mixed strategy s∗i of player i is a probability distribution over its pure strategies si . For mixed strategies the utility functions of payoffs must be extended to the concept of expected utility. Let us assume that the mixed strategy s∗1 of player P1 means to use his/her pure strategies s1k with probabilities pk . The mixed strategy s∗2 of player P2 may be a randomization over his/her pure strategies s2m with probabilities pm . In this case, the function u∗i of expected utility is defined by u∗i (s∗1 , s∗2 ) = Σk,m pk pm ui (s1k , s2m ) with i, k, m = 1, 2. Von Neumann proved a fundamental theorem for zero-sum games of two persons: for each game of this kind, the mixed extension has maximin-solutions s∗1 and s∗2 with (s∗1 , s∗2 ) as equilibrium point. Under this condition, it is useful to choose a pure strategy by a procedure of randomization. Appropriate mixed strategies are found by procedures of linear optimization.

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If players are allowed to enter into binding agreements before the game is played, we say that the game is cooperative. Noncooperative games instead make no allowance for the existence of an enforcement mechanism that would make the terms of the agreement binding on the players. The concept of equilibrium points was extended to noncooperative games by J.F. Nash [6.8]. According to the central dogma of game theory, rational players will always jointly maximize their expected utilities. In this sense a Nash equilibrium specifies players’ actions and beliefs such that (1) each player’s action is optimal given his/her beliefs about other players’ choices, and (2) players’ beliefs are correct. Thus, an outcome that is not a Nash equilibrium requires either that a player chooses a suboptimal strategy, or that some players misperceive the situation. More formally, a Nash equilibrium is a vector of strategies (s1 , . . . , sn ), one for each of the n players Pi in the game, such that si (i = 1, . . . , n) is a best reply to s−i . (All players other than some player Pi are customarily denoted as −i). Note that optimality is only conditional on a fixed s−i , not on all possible s−i . A strategy that is a best reply to a given combination of the opponent’s strategies may fare poorly vis-` a-vis another strategy combination. A pure-strategy Nash equilibrium is a specification of a strategy for each player, such that no single player can increase his/her payoff by changing his/her strategy if the rest of the players stick to their strategies [6.9]. Fig. 77c illustrates a noncooperative game with the payoffs for player P1 as first number and the payoffs for player P2 as second number. The strategy interaction (s12 , s22 ) is a Nash equilibrium point. For player P1 the situation is like this: if player P2 adopts the strategy s21 , then s12 is the better reply. If player P2 adopts s22 , the strategy s12 is better. Therefore s12 is the sure-win choice. Player P2 reasons similarly. So they end up with the payoff (s12 , s22 ). But the obviously better strategy interaction (s11 , s21 ) with payoff (3,3) is not a Nash equilibrium, for if either player sticks to it, the other can profit by defecting. The game is an example of the prisoner’s dilemma. In these situations, the isolated rational behavior of a player does not provide the socially desired result that can only be realized by the cooperation of the players.

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A pure-strategy Nash equilibrium does not always exist. But Nash proved that at least one mixed-strategy Nash equilibrium exists for any finite noncooperative game. An example is the so-called battle of the sexes. A man and a woman engage in a game of deciding whether to attend a football game or a concert. The man prefers the game, the woman the concert, but each prefers having the company of the other to being alone in the preferred activity. They settle in a Nash equilibrium by flipping a coin to decide where both will go. Nash’s result generalizes von Neumann’s theorem that every game with finitely many strategies has an equilibrium in mixed strategies. Often a game has a variety of Nash equilibria. Therefore more restrictive solution concepts with a refinement of the Nash equilibrium has been explored [6.10]. A Nash equilibrium needs not be interpreted as a unique event. If we think of it as an observed regularity, we want to know by what process such equilibrium is reached and what accounts for its stability. When multiple equilibria are possible, we want to know why players converged to one in particular and then stayed there. An alternative way of dealing with multiple equilibria is to suppose that the selection process is made by nature. An example is the evolutionary stable strategy in biology, in order to find the stable equilibrium composition of a population. In social sciences, the formalism can also be applied to human populations or ethnic groups. In evolutionary game theory, a kind of organism is represented by a player, biological actions or reactions by strategies, the interactions of organisms displaying various behaviors by matching various strategies [6.11]. The fitness of a behavior may be the relative growth rate of the fraction of organisms in the population. With the knowledge of the fitness, the composition of the population changes in successive generations can be determined. The game aims to find a stable composition of the population. J. Maynard Smith introduced the solution of an evolutionary stable strategy (ESS). An ESS is a Nash equilibrium that is robust or stable in the sense that slight deviations damp out and the population returns to its equilibrium state [6.12]. In the framework of complex systems, ESS are stable fixed point attractors of evolutionary dynamics. Thus, the game-theoretical ap-

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proach can be embedded into the theory of equilibrium dynamics. But modeling social and evolutionary dynamics needs more than the analysis of these special attractors of equilibrium: it needs the whole framework of nonequilibrium dynamics. 6.2

Symmetry Breaking and Socio-economic Transitions

In cultural and social history, symmetry has been associated with balance and stability of a society. But, actually, as we all know, phases of stability change with phases of instability and sometimes chaos. New political and social order emerges and old ones decay and disappear. Some philosophers (e.g. A.J. Toynbee) suggested a life cycle theory of civilizations and compared their quasi-cyclic development with the stages of childhood, adolescence, maturity, and old stage of an organism. But metaphoric analogies are no explanations. Critical situations lead to phase transitions of global events like wars and revolutions, fashions and life styles. Symmetry breaking is obviously the driving force of political, economic, and cultural development [6.13]. How can it be explained in the framework of complex dynamical systems? In the complex system approach we distinguish the microscopic and the macroscopic point of view. On the microscopic level, individuals interact und generate patterns of behavior, social structures, order or disorder on the macroscopic level of groups, institutions, and societies. Collective social order emerges by contributions of individuals to the macrostate of a society. In a causal feedback loop, the individuals are also influenced by the collective order achieved by themselves. Consider, for example, the voting behavior of people before a political election. Emerging clusters of individual opinions are represented by political parties, which are the “order parameters” of the voting dynamics influencing the individual behavior [6.14]. This is the general scheme of emergence and self-organization of order in complex dynamical systems, which was confirmed in physical, chemical, and biological examples. Although these systems differ in their microdynamics of atoms, molecules, cells, or neurons, they have surprising similarities in their

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macrodynamics of attractors. Therefore, we distinguished, for example, thermodynamic, genetic, and neural self-organization in the previous chapters. In these cases, the individual interactions of system elements can be modeled by nonlinear equations. The emerging macroscopic structures correspond to solutions of these equations. Thus, at least in principle, the macrodynamics can be explained by the microdynamics of the system. But, in contrast to physical, chemical, and biological systems, no equations of motion on the microlevel are available for social systems. People are not atoms or molecules, but human beings with intentions, motivations, and emotions. In principle, their individual behavior and decision-making could be explained by their brain dynamics. Cognitive and emotional dynamics are determined by order parameters characterizing individual thoughts, decisions, and motivations (Fig. 74). But this is only a theoretical option, because the corresponding neural equations are not yet known. Furthermore, they would be too complex to solve them and predict the future behavior of people. In economics, new classical theorists assume the microeconomic model of perfectly competitive markets [6.15]. Perfect competition or perfect market coordination means, for example, perfect wage and price flexibility. People should behave with economic rationality (“homo economicus”) optimizing their utilities under constraints. Then, wages and prices should always adjust themselves to clear all markets. If they are displaced from their equilibrium path by an exogenous shock (e.g. unemployment), wage and price adjustment rapidly returns it to equilibrium with a cleared labor market. In this microeconomic model, people seem to behave like the constituents of a spin-glass which separately attain their individual optimal states of lowest energy. Obviously, the new classical economists believe in equilibrium dynamics. But, the question arises whether wages and prices always adjust themselves in a sufficiently short time and not only “in the long run when we are all dead”, as J.M. Keynes emphasized. Actually, human agents act neither completely rational nor completely irrational. They deviate from game-theoretically predicted equilibria. Complexity, chaos, randomness, and incompleteness of information enforce them to decide and act under conditions

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of bounded rationality. Bounded rationality results from limitations on our knowledge, cognitive capabilities, and time [6.16]. Our perceptions are selective, our knowledge of the real world is incomplete, our mental models are simplified, our powers of logical deduction are weak and fallible. Emotional and subconscious factors affect our behavior. Deliberation takes time and we must often make decisions before we are ready. Therefore an alternative approach is suggested which gets along without microscopic equations, but nevertheless takes into account the decisions and actions of individuals with probabilistic methods in order to derive the macrodynamics of social systems. The modeling design consists of three steps: In the first step appropriate variables of social systems must be introduced to describe the states and attitudes of individuals. The second step defines the change of behavior by probabilistic phase transitions of individual states. The third step derives equations for the global dynamics of the system by stochastic methods [6.17]. In a society we can distinguish several sectors and sub-sectors that are denoted by variables. There are variables for material states, extensive and intensive personal states. The socioconfiguration of a social system is characterized by these material and personal macrovariables. They are measured by usual methods of demoscopy, sociology, or economics. Like in thermodynamics, there are intensive economic variables that are independent of the size of a system. Examples are prices, productivity, and the density of commodities. Extensive variables are proportional to the size of a system and concern, for example, the extent of production and investment, or the size and number of buildings. Collective material variables are measurable. Their values are influenced by the individual activities of agents, which are often not directly measurable. The social and political climate of a firm is connected to socio-psychological processes, which are influenced by the attitudes, opinions, or actions of individuals and their subgroups. Thus, in order to introduce the socioconfiguration of collective personal variables, we must consider the states of individuals, expressed by their attitudes, opinions, or actions. Furthermore, there are subgroups with constant characteristics (e.g. sections or

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departments of a firm or an institution), so that each individual is a member of one subgroup. The number of members of a certain state is a measurable macrovariable. The socioconfiguration of, for example, a company is a set of macrovariables describing the distribution of attitudes, opinions, and actions among its subgroups at a particular time. The total macroconfiguration is given by the multiple of material configuration and socioconfiguration. If all macrovariables of a macroconfiguration remain constant over time, the social system is in a stationary macroscopic equilibrium, which can be compared to thermodynamic equilibrium. If there are dynamics, we must consider the transition rate between macroconfigurations by either increasing or decreasing macrovariables. In the case of material configuration, an elementary change consists of the increase or decrease of one macrovariable (e.g. the price of commodities) by one appropriately chosen unit. The elementary change in the socioconfiguration takes place if one individual of a subgroup changes its state, leading to an increase or decrease in the number of a subgroup by one unit. For example, the variable of employment is diminished or enlarged by one person or a political voter changes his preferred party in a certain period of time. These transitions of individual states should be described by a probabilistic process, because the individual freedom of decision and action should not be restricted. People are not molecules with deterministic microdynamics. Nevertheless, the phase transitions must take into account running trends and motivations in order to get a realistic estimate of the probabilities. These aspects provide the link between the microlevel of individual decisions and the macrolevel of collective behavior. There are well-known statistical procedures of data-mining to measure probabilistic trends, motivations, and attitudes of people belonging to certain sections and subsections of a society. The probabilistic phase transitions can be used for setting up the macroevolution equation of a social system. The probabilistic macrobehavior of a society is described by a probability distribution function over its possible socioconfigurations at a certain time. The distribution function P (m, n; t) can be interpreted as the probability

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of finding a certain macroconfiguration of material configuration m and socioconfiguration n at time t. The evolution of the social system is the time-depending change of its probabilistic macrobehavior, i.e. the time derivative of the probability function dP (m, n; t)/dt. Thus, we get a stochastic nonlinear differential equation which is well-known in thermodynamics as master equation [6.18]. In natural science (e.g. thermodynamics), the evolution of a whole ensemble of millions of equally prepared but probabilistically evolving systems (e.g. gas, fluids) can often be measured. Therefore, the master equation with its probabilistic distribution function of macrostates is appropriate in natural sciences, although only numerical solutions are in general available. This method of modeling macrodynamics is called ensemble approach. But social sciences (e.g. economics, politics, sociology) only dispose of one or at best of a few comparable systems [6.19]. Therefore, the probability distribution of the master equation contains too much information in comparison to the available empirical data. Sociodynamics focuses on the stochastic evolution of a single system in which it traverses probabilistically a sequence of system states. In the corresponding state space of socioconfigurations, we get stochastic trajectories describing the probabilistic dynamics of social systems. The stochastic evolution of a single social system is determined by autonomous nonlinear differential equations for the stochastic trajectories of material, extensive and intensive personal variables. In this sense, the emergence of social structures and patterns of behavior can again be modeled by solutions of nonlinear equations. In the framework of nonlinear dynamics, we can consider phase transitions as examples of social symmetry breaking [6.20]. An economic example is the following model of two competing firms providing a bifurcation into winner and loser at a critical value of competition (Fig. 79). In nonlinear dynamics, order parameters of the macrodynamics of a system are introduced by linear-stability analysis. The idea of our model is that quality is the order parameter of competition dominating all other economic aspects. Thus, parameters of, for example, prices, supply, or purchase activities of customers can be neglected, and we get an macroevolution equation

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of quality. The evolution of the quality variables qi for firm i = 1 and firm i = 2 can be investigated numerically. In Fig. 79, their (stationary) solutions qi (φ) are depicted as a function of the competition parameter φ. It turns out that both firms have the same stationary quality q(φ) of their products and also the same stationary market share, as long as the competition value φ is smaller than a critical value φc . At φc , a bifurcation occurs and for φ > φc there exist two stable quality values q+ (φ) and q− (φ). The winning firm, say i = 1, will have reached the quality q+ (φ), whereas the losing firm, i = 2, arrives at quality q− (φ) with corresponding market shares. Another example of social phase transitions and symmetry breaking is provided by world-wide migration processes. The behavior and the decisions of people to stay or to leave a region are illustrated by spatial distributions of populations and their change [6.22]. The models may concern regional migration within a country, motivated by different economic and urban developments, or even the dramatic worldwide migration between poor and rich countries in the age of globalization. The migration interaction of two human populations

Fig. 79.

Bifurcation tree of economic symmetry breaking [6.21]

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may cause several macro-phenomena, such as the emergence of a stable mixture, the emergence of two separated, but stable ghettos, or the emergence of a restless migration process [6.23]. In numerical simulations and phase portraits of the migration dynamics, the macro-phenomena can be identified with corresponding attractors. The empirical administrative data can be used to test the models. In the case of a stable mixture, the integration of, for example, two ethnic groups, was successful. The phase portrait of the ”melting pot” shows a stable point of equilibrium and the corresponding master equation has a stationary solution with a centralized probabilistic distribution. If two isolated ghettos emerge in the region, the phase portrait shows two stationary fixed points corresponding to solutions with separated probabilistic distributions (Fig. 80a,b). The situation seems to be symmetric, but like scales highly sensible to tiny fluctuations. In reality, there are not only two subsystems, but an environment with nonlinear dynamics. Thus, the balance can break down and end in chaos. Fig. 81 corresponds to a restless migration process with a strong asymmetric interaction between the populations. The phase portrait Fig. 81a shows a limit cycle with unstable origin. The stationary probabilistic distribution of Fig. 81b has four maximum values with connecting ridges along the limit cycle. This case may be interpreted as sequential erosion of regions by asymmetric invasion and emigration of the populations. Mathematically, the limit cycle with unstable origin is similar to the destroying dynamics of a hurricane. If we consider more than two populations, deterministic chaos can emerge in unstable situations. Numerical simulations lead to strange attractors as final states of trajectories. In other cases, a successive bifurcation becomes more and more complex with a final transition to chaos. Another challenge of global phase transitions is urbanisation which means pattern formation of new metropolitan areas [6.26]. Historically, urban evolution started with simple closed patterns of settlements. In Antiquity, early plans of cities and buildings were symmetric according to ancient theories of proportionality. Capitals were considered as mirrors of an eternal cosmic order with

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(a) phase portrait with flux lines Fig. 80a–b.

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(b) probabilistic distribution

Fixed point attractors of migration dynamics [6.24]

(a) phase portrait with flux lines Fig. 81a–b.

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(b) probabilistic distribution

Unstable limit cycle of migration dynamics [6.25]

temples, churches, or palaces of emperors in the symmetric center. Thus, they were planned and built for eternity. In Renaissance and baroque time, the geometric symmetry of a city, settlement or garden (e.g. French garden) was a symbol of “cartesian” rationality that was demanded by Descartes, a leading philosopher and mathematician of the 17th century. Administration and organization of human life should become completely computable like celestial mechanics.

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In the 19th century, symmetries of cities became an expression of the Laplacian spirit of linearity, perfect calculation and centralized administration. Cities with their rectangular or centralized nets of streets and avenues were constructed by architects like closed systems of sub-modules which should be in absolute equilibrium. But, actually, we observe a historical evolution of cities from simplicity and symmetry to complexity, diversity, and fractality [6.27]. The initial symmetric structures had been broken by urban phase transitions. Fig. 82 shows the complex dynamics of metropolitan areas from closed settlements in Antiquity to fractal attractors in the age of globalization, self-organizing during centuries [6.28]. Generations of people had been engaged in the global dynamics with their local activities without a centralized master plan. Even if there are historical settlements with symmetric regularity, they are nowadays embedded and encapsulated as local areas in the complex urban structure of a metropolis like tissue in a growing cellular organism. Cities can be considered as complex urban systems with pattern formation on a square lattice of plots and sociodynamics of socioconfigurations, depending on material variables with, for example, numbers of lodgings, factories, or greens, economic capacity of a plot, extensive personal variables with population of regions, utility functions and distance-depending transition rates of city configuration. The distribution of firms, residential or shopping areas on the macroscopic level interacts with the material, intellectual, political and social life of the inhabitants on the microlevel. With given transition rates between the plots coupled equations for city and population evolution can be set up. They provide a master equation for the probability distribution over city and population configuration, and equations for stochastic trajectories of the systems. According to the complex system approach, solutions of these equations correspond to the emergence of urban substructures. In computer simulations, the emergence of distributation patterns is visualized by phase transitions and symmetry breaking. The computerdrawn pictures show the evolution of the population distribution of a region that starts off initially an area with no interaction between local centers. The urbanization process is revealed in phase transitions

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Fig. 82.

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Phase transition and attractors of urban dynamics

with changing local attractors. For example, the urban structure may start to solidify around some main centers. Finally, a basic structure may be found which is stable for some time. But, suddenly by local fluctuations, a center can undergo a process of decays and new attractors of lodgings, shopping centers, or industrial regions emerge. It is remarkable that urban regions change from small to large ones, and their evolution speed from fast to slow. Actually, we observe industrial regions with rapidly changing architecture according to changing economic markets and technical progress. Lodging regions may depend on changing fashions or the attractivity of neighboring regions. Historical centers change very slowly. In this sense, urban regions may have their own internal time and age. For practical reasons, computer simulations of urban dynamics help to forecast the future and to support decisions for desired developments. In computer experiments, different scenarios can be tested with changing data [6.29]. One can trace back the simulation results to special choices of values and parameters under changing initial conditions, in order to learn and to optimize one’s decisions. Phase transitions and symmetry breaking of urban dynamics is visible in the distribution of real settlements. The dynamics of social

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groups is mainly invisible, but can nevertheless be studied by analytical and computer-assisted methods [6.30]. A social group may be led by ideas and interests of its members with respect to common or controversial purposes and objectives. Interacting forces between individuals with different material, emotional, and mental needs and desires lead to a self-organization of social structures, customs and practices in a society. In the complex system approach, their emergence corresponds to solutions of appropriate equations of sociodynamics. Norms, rules, and laws of social behavior have the function of order parameters determining the microdynamics of individuals. According to the usual methods of sociodynamics, material and personal variables must at first be defined in order to get the socioconfigurations of a social group. In a next step, the transition rates between socioconfigurations are estimated with consideration for motivation potentials and trends in a group. Distribution functions can describe the emergence of, for example, the hierarchical structure of a group. In this case, the points of a plane represent the individuals of the population. But their position in this abstract space has nothing to do with their location in the real space. The space between the groups represents the crowd of the more or less leading members of the group. The opinion leaders are in the center and the other members are distributed in rings of subgroups with decreasing degree of social influence. States of balance and equilibrium may undergo phase transitions with the emergence of new hierarchical structures. Political phase transitions relate to changes of very large-scale events of whole political systems. Political revolutions can be understood as symmetry breaking and phase transition. Social asymmetries between poor majorities without rights and rich minorities in power have often been the driver of political change. For example, with the assault on the Bastille in the end of the 18th century, the system of absolutism in France collapsed. A local event made global history in the sense of the butterfly effect. I. Kant and other philosophers celebrated the age of republican freedom as an attractor of civil history. The Russian revolution in 1917 was also initiated by local events and local groups in St. Petersburg in the sense of a butterfly

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effect. Each of such large-scale phase transitions is a unique event, because it will never recur in exactly the same way. In each event of this kind, however, there appear universal structures of human behavior under special conditions which play a driving role in phase transitions. This is the motivation to study historical events and processes from the point of view of sociodynamics. Obviously, liberal democratic systems have higher degrees of freedom than totalitarian ones. In an open society with a democratic constitution, the freedom of citizens finds its limitation in the freedom of the other individuals. In this framework, the formation of competing political parties with different socio-political concepts is possible. Power is organized in a complex system of checks and balances with the fundamental division in legislative, executive and judicative. The government is approved temporally in elections by a majority of the people. Contrary to totalitarian systems, democracies need more efforts and time for decisions. Further on, the balance of social and political equilibrium is more endangered in a complex open democratic system than in a closed totalitarian system with rigid regulations. On the other hand, democracies are selforganizing systems with higher degrees of tolerance to perturbations not totally depending on the decisions and regulations of a dictorial center. Historical experiences show that phase transitions from totalitarian regimes to democracies are possible as well as the reversel process from democracies to dictatorships. There are also limit cycles with repeating change from one totalitarian system to another. In this case a country is caught in a cyclic attractor with tragic consequences for its political and economic development (e.g. some countries in South America in the 1950s and 1960s). It is a challenge of political science to find the indicators of political crises in the sense of warning systems. Models of complex dynamic systems provide tools for these activities. 6.3

Complexity and Sociodiversity of Globalization

The dynamics of globalization is surely the most important political challenge of complexity in the history of mankind [6.31]. After

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the Second World War, the confrontation of the Soviet system under the leadership of the Soviet Union and western democracies under the leadership of the United States generates a bilateral, but highly dangerous equilibrium of two political, economic, and military systems with nuclear weapons. Therefore, the symmetry of power in the world from 1945 until 1989 was sometimes called the “equilibrium of fright.” After increasing economical and political fluctuations and instabilities at the end of the 1980s, the equilibrium system imploded peacefully, and nonequilibrium dynamics of a worldwide, multi-centered system started. But there was not only freedom of the people, but the emergence of many local ethnical and religious conflicts with dangerous butterfly effects which are obvious in the international networks of terrorism. Thus, in the historical phase transition after 1989, the equilibrium of fright has been replaced by the fright of nonequilibrium [6.32]. In nonequilibrium dynamics after 1989, the order parameters of market systems have dominated the political and economic development of post-communist states. These phase transitions are complex with states of instability and fluctuations. The national trajectories of European nations run into the strong political and economic attractor of the European Union (EU) that has become one of the powerful economic centers of the world [6.33]. China, India and other Asian states with dramatically growing populations and technological-economic power will become dominating centers of the world, while Africa is in a state of socio-economic stagnation. In the age of globalization, mankind is in an unstable phase transition of high complexity, depending on local fluctuations of economic crises, social tensions, and cultural conflicts. Because of the worldwide nonlinear feedbacks, globalization is no zero-sum game that can be won by one nation with the loss of the other ones. Sociodiversity, with many interests and strategies, opens chances of better solutions like biodiversity in biological evolution. But we cannot trust in the self-organization of socio-economic dynamics leading automatically to welfare and the wealth of nations. It is well-known that the self-organization of biological evolution generates failures and sometimes ends in catastrophes with the loss of giant biocapi-

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tal. Human rights demand the realization of a sustainable future for mankind. Thus, the nonlinear dynamics of globalization need order parameters of global governance. What do we know about the laws of globalization? According to Smith, the “Wealth of Nations” is made possible by two essential principles [6.34]. At first, free markets should organize themselves under the conditions of the unrestricted competition of nations. Smith was obviously the first prophet of globalization. In the sense of complex systems, the competition of free markets is a procedure of economic selection, corresponding to the biological process of selection. C. Darwin proclaimed that selection could only lead to optimally adapted species if there was also a great variety of organisms (“biodiversity”). In economy, the wealth and welfare of nations is analogously explained by Smith with the division of labor relating to an increasing variety of highly specialized jobs. Division of labor means reduction of incompetence by increasing specialized know-how and better adaptation to changing conditions. Biodiversity in evolution corresponds to sociodiversity in societies. Smith analyzed the division of labor during the beginning of the Industrial Revolution in the end of the 18th century. Another example is the progress in medicine. Since the 19th century the health of people had been essentially improved by the specialization of competent experts. The modernization of the world is a process of increasing granulation in the division of labor. Efficiency by professionalization is the slogan of the modern world. Obviously, sociodiversity and the division of labor mean the emergence of a new macroscopic order in a society. In the sense of complex dynamics, people of similar intelligence and abilities specialize themselves in different classes of jobs. An originally uniform system is divided into different equivalence classes, symmetry is broken, and order emerges. Besides selection and diversity, fluctuations are needed as causes of change. In biological evolution, mutations are random events leading to new organisms that must survive the biological competition of selection [6.35]. In economy, technical inventions and scientific discoveries are more or less random events leading to innovations that must become successful products in the economic competition

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of markets. It is obvious that diversity, self-organizing competition, and innovation are basic principles of complex dynamical systems in general. Therefore, globalization is not a political ideology we may like or dislike. It is a lawful, nonlinear process that we should analyze in order to handle it and to find solutions according to human interests. In open complex dynamical systems there are dissipative forces as drivers of change and conservative forces stabilizing the system. Only the balance of dissipation and conservation guarantees the existence of complex systems during the phase transitions of nonlinear dynamics. In the process of globalization, economy, technology and science are the driving forces of change, while the national states have the tendency to conserve the status quo in order to protect the interests of their people. With a dense network of regulations, modern democracies try to organize the economic welfare for the majority of their people [6.36]. Social symmetry is no result of competition, but by regulation of the state. Competition needs asymmetry as motivation for economic activities. On the other hand, dissipation of free markets without any social network could lead to social ruin and destabilize the whole system by a dangerous asymmetry between the majority of poor people and a minority of rich ones, which was already prophesied by K. Marx [6.37] in the 19th century. Thus, modern states tried to stabilize the social balance by a welfare system that has been paid by a complex system of direct and indirect taxes. But over-regulation is a handicap for economic competitors who try to maximize their profits. They prefer markets with less regulations and low taxes. Finally national states with highly developed welfare systems lose jobs necessary to earn the money in order to finance their welfare systems. Therefore, overregulated welfare systems produce high costs of complexity by a feedback cycle which can also destabilize the whole system. In short and simplified: one can accept social asymmetry and get employment, or one can oppose social asymmetry and get unemployment. It is a challenge of modern politics to find the right balance of dissipative and conservative forces in socioeconomic systems.

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Under the conditions of nonlinear dynamics, it is not sufficient to have good intentions, which could lead to undesired side-effects in the long run. Thus, for decisions in nonlinear dynamics, one must consider an appropriate window of time. Dynamical systems in nature and society have different scales of time [6.38]. In democracies, politicians prefer to take into account short-term effects only, because they want to be elected again in the next election. Political power in democracies means power for an elected period. This is the reason why political decisions with short-term benefits may lead to disadvantages in the economic system with longer periods of causal effects [6.39]. If, for example, the central bank of a state enlarges the set of money, the first effect is an expansion of the gross national product and employment, which is followed by increasing prices and wages with a contractive effect. On the other hand, the restriction of money at first has a contractive effect on the gross national product and employment, which is later on followed by stabilized prices, wages, and an increasing gross national product. As politicians prefer the short-term benefits, there is permanent inflationary tendency endangering the value of money. Political and economic systems have their own characteristic dynamics and time-scaling. Therefore, they should not be mixed in order to guarantee the welfare of a nation. In democracies, political power is justified by the principle of majority and consensus. In economies, the value of products is decided by the competition of free markets. If an economy is based on democratic majorities, different interests and intentions must be harmonized, competition reduced, and innovation restricted. The effect is stagnation, administration of the status quo and finally pauperization, because no surplus value is produced. On the other hand, political systems must not be dictated by the innovation cycles of products. Free markets are not interested in the social security of people, but in high profits and less burden by taxes [6.40]. Globalization enlarges free markets from nations to the whole world. In international competition, the industry of a country is only competitive as long as the costs of production and services are not higher than the prices that can be gained for its goods on interna-

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tional markets. The most important part of costs are the labor costs. Therefore, the labor costs of a country must not be too high. The wages generated by international competition are called competitive wages. They are a critical control parameter for the international competitiveness of a country. If competitive wages are surpassed by a country for a longer time, then the economy will decrease with high unemployment, dramatically diminished exports and enlarged imports. Finally, the wages must be decreased again. The contrary effect happens if, on the other hand, the actual wages are massively lower than the competitive wages. There is a boom with increasing exports, decreasing imports, growing profits and increasing employment. With the increasing demand for labor the wages also increase and adjust themselves to the competitive wages. Wages consist of the individual wages paid cash and the collective wages paid to public institutions in order to support people in the case of illness, unemployment, etc. It is a decision of national social politics as to how the parts of individual and collective wages should be weighted. If the welfare system of a country is developed too much with high costs of collective wages, the will of its people to work and to produce is diminished. Therefore, the individual wages must not be reduced too much in order to give sufficient stimulus for work. In the age of globalization there is international competition between many countries in trying to attract firms and industries from abroad [6.41]. They compete with their different forms of welfare systems, taxes, and other local conditions. Again, sociodiversity is a feature of complex social and economic dynamics. In complex systems, nobody knows the best solutions. It is a question of learning steps and experience by trial and error. According to F. von Hayek, competition is a procedure of discovering the best solutions [6.42]. In the age of globalization, the discovering process of competition is generalized for countries and nations. They emerge as global attractors in the worldwide dynamics of free markets. On the national level, countries organize themselves with their political and economic systems. National governance is realized by national governments legitimated by democratic elections in the best case. Governance needs power to enforce laws, regulations, and

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norms. How should global governance be made possible in the age of globalization [6.43]? After the Second World War the General Agreement on Tariffs and Trade (GATT) was the first step in this direction. GATT was a global, political framework of nations that wanted to take part in free markets. After 1989, GATT was improved by the World Trade Organization (WTO) with free trade of further products and services [6.44]. With respect to the increasing importance of inventions and their protection by patents, the WTO was supplemented by an agreement on Trade Related Aspects of Intellectual Property Rights (TRIPS). The WTO enforce its national members to satisfy the global rules of their agreement. Otherwise a country is excluded from the privileges of free markets by the other members. But the attraction of free markets is very strong and supports the national interest of a country. Therefore, any member tries to avoid such a situation voluntarily. In this sense the WTO has an effective system of sanctions that are applied by an international court of trade. It is remarkable that for the first time in history a political-economic system is able to enforce sanctions without military power. Military power is even excluded, because otherwise a country loses the economic benefits of the free WTO-market. The dream of Smith that war could be avoided by free trade has now a realistic perspective, at least in the long term. Mankind’s urgent global problems like war, poverty, and ecological pollution cannot be solved, because there are no effective systems of sanctions. Good intentions are not sufficient. The WTO with its effective sanctions could be a model for successful global governance. Less developed countries have obviously less chance to compete successfully with highly developed countries on the monetary market. They need help to improve their financial systems. Nevertheless many of them can already compete with their human capital, i.e. the know-how of their people. Thus the best help is to open the markets of wealthy OECD countries for these people. There is no better way to gain the knowledge of competitors than by competing with them. A sustainable future for mankind depends basically on the solutions for our environmental problems. How can the nonlinear dynamics of globalization be harmonized with the complex system of

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the natural environment on earth? Today, there are 6 billion people living on earth. Human population will increase to 8 billions in 2025 and 9,4 billions in 2050. These giant numbers are a dramatic burden on natural resources and climate. On the other hand, it is an empirical fact that the growth of a population can only be stabilized by an increasing wealth of its people. But increasing wealth needs growing economies which exhaust natural resources. Therefore, thirty years ago, the Club of Rome asked for the “Limits to Growth” and tried to determine a global equilibrium state of economy and environment [6.45]. Obviously population, industries, capital, services, natural resources and environment interact in a complex nonlinear system which needs careful analysis of causal dependencies. Fig. 83 illustrates some feedback loops, although they are not complete. In a further step of modeling the coupled equations of these interacting quantities must be formulated in order to find appropriate solutions of equilibrium. Even if there is a strategy of zero-growth, there is no chance of application without an effective system of sanctions. Wealthy nations will not renounce their welfare systems that need flourishing economies. Poor nations will try to increase their industrial production in order to improve their living standards. Thus, the question arises as to whether there is an ecologically sustainable strategy of economic growth accepted by the majority of countries? According to the WTO, the common agreement must be of urgent national interest for its members combined with an effective system of sanctions. An economic sanction would be a price mechanism evaluating natural resources (e.g. climate) as limited goods [6.46]. In an international agreement on climate protection, emission rights of hothouse gas could get a market price used by the countries with respect to their national interests. Excessive emissions of hothouse gas cause fines on single countries to the favor of a world fund. With that a price is determined for the limited rights of emission. The price can also be used as reward for renouncing the usage of emission rights. The reward is paid by the same fund getting the fines. The agreement demands that the price must be adjusted if the total emission surpasses or remains under some critical value of a control parame-

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population (+) (+) annual births birth rate

(-)

annual deaths

(-)

death rate

education and family planning

public health welfare

industrial production per capita

services per capita

capital of services

non-regeneratable ressources

industrial production efficiency of capital

(+) industrial capital

investments investments rate

Fig. 83.

(-)

wear of capital average using duration of capital

Nonlinearity of global social, economic, and ecological dynamics

ter. In order to get a growing and ecologically sustainable economy, price and sanctions of emission rights must be combined with new innovations and profitable markets. If the price reaches an appropriate value, then it is assumed that new ecologically sustainable technologies will emerge and produce the same or an even larger gross national product with essentially less coal, oil, and gas. Agreements without sanctions (e.g. the Kyoto agreement) and impulses to new profitable markets only have moral importance. Globalization does not only mean ecological and economic problems. After the fall of the Berlin wall, politicians believed in the linear assumption that coupling the dynamics of free markets and democracy would automatically lead to a community of modernized, peace-loving nations with civic-minded citizens and consumers. This was a terrible error in a complex world! From our point of view,

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complexity is driven by multi-component dynamics. Politicians and economists forgot that there is also the sociodiversity of ethnic and religious groups dominating the whole dynamics of a nation at a critical point of instability [6.47]. As we all know from complex dynamical systems, we must not forget the initial and secondary conditions of dynamics. Instability emerges if free markets and elections are implemented under conditions of underdevelopment. Recent studies demonstrate that in many countries of Southeast Asia, South America, Africa, Southeast Europe, and the Middle East the coupling of laissez-faire economics and electoral freedom did not automatically lead to more justice, welfare, and peace, but tipped the balance in these regions toward disintegration and strife [6.48]. One reason is that these countries mainly do not have a broad majority of well-educated people. Thus, minorities of clever ethnic groups, tribes, and clans come to power and dominate the dynamics of markets and politics. In the terminology of complex dynamics, they are the order parameters dominating (“enslaving”) the whole dynamics of a nation. Again, the good intentions of democracy and free markets are not sufficient. We must consider the local conditions of countries and regions. In classical philosophy, the transition from an intended development to a development contrary to the spirit of the philosophy has become famous as a contradiction of dialectics (e.g. G.W.F. Hegel). Good intentions may lead to bad effects. But human agents are sometimes driven by history to good effects without their subjective intentions. Hegel called it a “stratagem of reason” (“List der Vernunft”). Actually, it is a well-known effect of nonlinear dynamics. Therefore, market-dominant minorities are not a priori evil. Minorities are also the driving forces of activity. If they are open-minded and flexible, they prevent narrow-minded “enslaving” which may be successful only for a short time. In their own interest, they must try to stabilize the whole system in the long run. Therefore they should help dampening the social effects of free markets, bridging social cleavages, and transcending class division during a phase transition to democracy and welfare for the majority of the people. But these phase transitions may be different from region to region in the

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world. Responsible decisions require sensitivity to local conditions in the light of the butterfly effect. There are not only local minorities in regions and countries. During the process of globalization, a minority of nations, institutions, and companies can come to power and dominate the whole dynamics of global economics and politics. Recent discussions on globalization show that a lot of people are not happy with the results of globalization. But it is necessary to understand that globalization means nothing more than the global dynamics of political and economic systems in the world. Therefore at first, it is neither good nor evil like the dynamics of weather. But contrary to weather, the dynamics of globalization is generated by the interactions of humans and their institutions. Thus, there will be a chance to influence globalization if we take into account the dynamical laws of complexity and nonlinearity. It is a hard fact that the order parameters of globalization have been defined by a minority of nations. They are the world’s preeminent political, economic, military, and technological powers whether we like it or not. Philosophers, mathematicians, and systems scientists have no power. But, again, we should use Hegel’s “stratagem of reason”: minorities are also the centers of driving power that enable chances for change. Concepts and ideas without political power have no chance. If the dominating minorities of globalization are openminded and flexible, they will prevent narrow-minded “enslaving” which may be successful only for a short time. In their own interest, they must try to stabilize the whole system in the long run. Therefore they should help in dampening the social effects of global free markets, bridging social cleavages, and transcending class division during a phase transition to global democracy and welfare for the majority of the people. Globalization means the critical phase transition to global governance in the world. We need new global structures to manage the political, economic, military, and technological power in the world according to the interests of the majority of people on earth. Global structures emerge from the nonlinear interactions of peoples, nations, and systems. At the end of the 18th century, Kant had already de-

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manded a law of nations leading “To Eternal Peace” (1795) [6.49]. After the First World War, President W. Wilson of the United States strongly influenced the foundation of the League of Nations. After the Second World War, the United Nations (UN) presented a new chance to handle international conflicts, but they often fail because of their lack of power. The dilemma of international law is that law needs power to enforce rights and ethical norms. Therefore nations have to give up parts of their sovereignty, in order to be dominated by commonly accepted “order parameters”. Since September 11, 2001, a global network of terrorism has been threatening the preeminent political and economic nations of the world. This is the reason why, especially the United States, which historically helped found the League of Nations as well as the UN, now hesitates to restrict its national sovereignty and prefers to organize its own national security through global military defense. Clearly it is a long way to global governance among autonomous nations [6.50]. On the other hand, we must not forget the practical progress made by new social and humanitarian institutions of the UN. New economic, technological, and cultural networks of cooperation emerge and let people grow slowly together in spite of reactions and frictions in political reality. On the way to “eternal peace,” Kant described a federal (multi-component) community of autonomous nations self-organizing their political, economic, and cultural affairs without military conflicts. But an eminent working condition of his model is the demand that states organize their internal affairs according to the civil laws of freedom. It is a hard fact of historical experience that civic-mindedness and humanization have sometimes not only been defended, but also enforced by military power. As long as the demand for civil laws of freedom is not internationally fulfilled, the organization of military power is an urgent challenge to globalization. Globalization and international cooperation is accelerated by the growth of global information and computational networks like the internet and wireless mobile communication systems [6.51]. On the other hand, the electronic vision of a global village implies a severe threat to personal freedom. If information about citizens can be

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easily gained and evaluated in large communication networks, then the danger of misuse by interested institutions must to be taken in earnest. As in the traditional economy of goods, there may arise information monopolies, acting as dominating minorities prejudicing other people, classes, and countries. For instance, consider the former “Third World” or the “South” with its less developed systems for information services that would have no fair chance against the “North” in a global communication village. Are there consequences of symmetry, complexity, and nonlinear dynamics for management systems in the age of globalization [6.52]? Linear decision behavior can obviously lead to desired results only under the conditions of complete information in an environment with linear dynamics. In complex situations, agents must consider positive and negative feedback by side effects of their own nonlinear decisions and by goals and actions of other agents. Short-term thinking is dangerous in a world with delayed and long-term side effects. Learning in complex organizations means a change of mental models, strategies and decisions. It requires nonlinear information feedback. Thus, we should aim at improving complexity in management systems with respect to structural complexity (e.g. flat hierarchies, short length of decision processes, appropriate number of controlling units), with respect to information complexity (e.g. effective knowledge management, information symmetry and transparency), and finally with respect to the individual complexity of co-workers and the diversity of their different creative potentials. But diversity and complexity do not automatically lead to the emergence of self-organizing fruitful effects. In firms, administrations, and other kinds of organizations, we can measure costs of complexity influencing their position in economic competition. Direct costs of complexity are distinguished as single direct costs for too complex products with new expensive materials, tools, and features, and permanent direct costs for service and administration of these products. Indirect costs of complexity are hidden in the organizational structure of a firm or administration. An example is the time of a manager or co-worker not being optimally used. Costs of complexity influence all steps of the value chain during an indus-

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trial production. There are, for instance, the costs of suppliers for various units of products. Furthermore, costs of complexity emerge in purchase and logistic, production and construction, marketing, and administration. If dangerous drivers and traps (“attractors”) of complexity arise, the capital of a firm decreases and the energy of co-workers and managers are dissipated. Examples are offers of firms with a too much variety, over-engineering and defects in quality. Thus, complex nonlinear organizational structures can lead to increasing butterfly effects, spreading to the whole organization (e.g. costs of complexity) like an epidemic. In complex organizations, nonlinear processes cannot be forecast in the long run. Early controlling is necessary to prevent chaotic attractors, i.e. the traps of complexity. In the competition of globalization, complex organizations only survive as learning, rapidly adapting, and flexible systems. In complex organizations, varieties of competent co-workers recognize problems better and react faster than central controlling in an over-regulated hierarchy. Therefore, we should deregulate and support self-regulating autonomy. Complex organizations are nonlinear social systems of people with different abilities, attitudes, emotions and interests. The sociodiversity of people is the human capital for a sustainable progress, not only in single organizations, but in the evolutionary process of globalization.

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Chapter 7

Symmetry and Complexity in Computer Science

The evolution from symmetry to complexity has been analyzed in the physical, chemical, biological, cognitive, and social sciences. In all these contexts, the emergence of order and structure is explained by self-organization with symmetry breaking and phase transition of complex dynamical systems. The increasing power of modern computer technology allows new insights into their nonlinear dynamics, which can often not be solved by analytical methods. On the other hand, the principles of physical, chemical, biological, cognitive and social self-organization have become the blueprints of computer and information technologies. Life and computer science are growing together into a new kind of complex engineering, changing the basic conditions of human life and society. There is a fundamental reason for this obvious tendency: according to the principle of computational equivalence, every nonlinear dynamical system corresponds to an appropriate computational system. In this sense, atomic, molecular, cellular, organic, and social systems are computational systems with phase transitions as information processes. Symmetry, symmetry breaking and complexity are explained by the principles of information and computation.

7.1

Symmetry and Complexity in Information Dynamics

According to Shannon’s information theory [7.1], a message from a sender (e.g. phone, PC) is sent to a recipient by coding the sign of the message into binary digits (“bits”), representing binary technical 273

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signals (e.g. electrical pulses), and decoding them when the message arrives. Communication means the exchange of information. The information content of a symbol is the number of binary decisions leading to it. For example, a storage with four symbols a, b, c, d allows 4 = 22 selecting procedures with two binary decisions which can be represented by a bifurcation tree (Fig. 84). In this sense, information is generated by digital symmetry breaking.

Fig. 84.

Digital bifurcation of information

For N symbols, there are N = 2I selecting procedures with I binary decisions, i.e. I = ld N bits. If the symbols si (1 < i < N ) occurs with different probabilities pi , then their information content is I(si ) = ld p−1 = −ld pi bits. A more probable symbol has less i information content than an improbable one. In this sense, the information content of a symbol can be considered a measure of news for the receiver. The mean information content of a sender with symbols si is the expectation value of the information contents I(si ) of its symbols    si , i.e. H = c i pi I(si ) = − i pi ld pi with i pi = 1. The mean information content H can be considered a measure of uncertainty for the probabilistic distribution of the symbols of a source. The reason being that in the case of the uniform distribution of probabilities, the mean information content Hmax of a source is maximal, i.e. the

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uncertainty of a symbol is maximal. For H = 0 is pi = 1, i.e. symbol si is determined by the source. Information storage and information flow in matter, life, the brain, and societies depend on the dynamics of complex systems. The basic concept linking information dynamics with complex systems is entropy. According to Boltzmann, entropy S is a measure of the probable distribution of microstates of elements (e.g. molecules of a gas) in a complex dynamical system, generating a macrostate (e.g. temperature of a gas), i.e. S = kB ln W with kB Boltzmannconstant and W number of probabilistic distributions of microstates, generating a macrostate [7.2]. According to the second law of thermodynamics, entropy is a measure of increasing disorder in isolated systems. The reversible process is extremely improbable. In information theory, entropy can be introduced as measure of uncertainty of random variables. Random variables are not restricted to randomly produced symbols of a sender. A random variable X denotes states x which can be generated by any kind of complex system. The information which is necessary to determine the probabilistic distributions of microstates, generating the macrostate of a complex system, is given by information entropy. Mathematically, the information entropy H(X) of random variable X is defined as the expectation value of the probabilistic distribution of its values x,  i.e. H(X) = − p(x) log p(x). In complex systems, H(X) is the exx

pectation value of the probabilistic distribution of their microstates. For H(X) = 0, the process X is deterministic. For H(X) maximal, there is uniform distribution with maximal uncertainty of x. Information entropy is obviously considered a measure of uncertainty. Actually, information is the diminution of uncertainty concerning the state of a complex system [7.3]. In thermodynamical applications, the random variable X of information entropy is related to the microstates of, for example, the molecules of a gas. In this case, the entropy of a macrostate corresponds to the information which is necessary to determine the microstates, generating the macrostate, i.e. S = kB H [7.4]. According to the second law of thermodynamics, entropy increases in

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closed systems. Increasing entropy means an increasing number of probabilistic distributions of microstates, generating a macrostate. Therefore, the information which is necessary to determine the microstates, generating a macrostate of a system, also increases. In this sense, information entropy is sometimes considered a measure of potential information evolving in dynamical systems [7.5]. How can we determine potential information in the different complex systems of cosmic, biological, and social evolution? In information theory, information is reduced to bits, the smallest units of binary alternative states, which are denoted by the binary digits 0 and 1. In this case, a state is characterized by a bifurcation tree of binary alternatives which must be decided for its determination. The alternatives are decided when the corresponding events happen. The evolving potential information in closed systems can be represented by the spreading bifurcation tree of alternatives for determining a macrostate. The basic physical theory of cosmic evolution is quantum mechanics. In quantum theory, elementary particles (e.g. photons) have binary spin-states ↑ (up) and ↓ (down) that can be superposed in coherent states. Thus, quantum information theory analyzes quantum information with superposed quantum bits. Again, any quantum state could be characterized by the number of quantum alternatives which must be decided for its determination. The quantum alternatives are decided when the corresponding quantum events happen [7.6]. In this sense, each state of matter can be considered a kind of potential quantum information. Quantum bits correspond to the binary alternatives on which the state spaces of quantum theory can be based. According to C.F. von Weizs¨ acker [7.7] the quantum alternatives define a symmetry group that is isomorphic to the transformation group of special relativity. In this first step, quantum information theory leads to both the existence of a 3-dimensional real space and the validity of special relativity theory. Until now, however, the derivation of actual particles and fields remains just a program. Its goal is to deduce their existence directly from quantum information theory, while avoiding the divergences of quantum field theories that must presently be evaded by ad hoc renormalization techniques. The gauge symmetries of the fundamental physical forces and their particles would have foundations of quantum information theory. In the context of the unification of physics there remains, furthermore, the open problem of the quantum theoretical

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reconstruction of the gravitational fields of general relativity theory, in which the linearity and delocalization of quantum theory collide with the nonlinearity and localization of Einstein’s gravitational equations. If the problems of unification are solved, then quantum information is the original potentiality of the world. In the beginning there was quantum information with high symmetry.

In previous chapters, the cosmic evolution was described by symmetry breaking and phase transitions from early states of high symmetry and uniformity to complexity and diversity. According to the second law of thermodynamics, global cosmic expansion is characterized by increasing entropy which means increasing disorder. Local order of complex structures emerges by self-organization. In this case, a system takes a local state of higher order resp. lower symmetry than its environment. Therefore, local order corresponds to local diminution of entropy with respect to the equilibrium state. In this sense, complex structures have less potential information than states of maximal entropy. But, that is not a contradiction to our intuition. We must not forget that potential information relates to the information which is necessary to determine a state. If potential information increases, then the need for actual information increases and that means increasing uncertainty. Local order of complex structures correspond locally to more actual information than the globally increasing entropy. The universe, with expanding cosmic dust and backgroundradiation, is a global sea of increasing disorder with emerging and disappearing local islands of order such as galaxies, stars, and planets. Our earth, for instance, exports entropy by absorption of radiation of a high temperature and re-emission of radiation of a low temperature. The export of entropy is a necessary condition for the emergence of structure. In order to prevent the system from running into the equilibrium state of maximal entropy, the irreversibly generated entropy in the system must steadily export to abroad. An example is illustrated by the B´enard-effect (Fig. 43). A stream of heat of a high temperature is imported from below and a stream of heat of low temperature is exported from above. The gradient of temperature is responsible for the pattern formation of macroscopic convection cells. In the chemical BZ-reaction (Fig. 56) the export of

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entropy is realized by an autocatalytic diffusion-reaction process. In biology, genotypical uniform cells differentiate themselves into phenotypical different tissues during the ontogenesis of organisms. The growth of an organism is made possible by the import of nutrition with high structural order and the export of output with less structure. In sociodynamics, people with similar intelligence and abilities specialize themselves in different classes of jobs. In this case, export of entropy means reduction of incompetence by specialization, professionalism, and division of labor. In all these examples, an originally uniform system is divided into different equivalence classes, symmetry is broken, and order emerges by reduction of entropy and uncertainty. Reduction of entropy and uncertainty means loss of potential information, but gain of actual information. Information systems reduce uncertainty and enlarge actual information. In this sense, even clouds, stars, and galaxies with their locally emerging order and decreasing entropy can be understood as information systems. Gain of information is by no means restricted to information systems in exchange with human beings. A molecule combining itself with other molecules in a new chemical structure reduces entropy and uncertainty. Therefore, chemists do right to call it “molecular pattern recognition,” when a molecular structure is “recognized” or “selected” by a molecule as appropriate compound. These kinds of information systems need neither sensory perception nor consciousness or biological mechanisms of selection. Nevertheless, in thermodynamic systems, gain of information happens in molecular exchange of matter and energy. The laws of thermodynamics are not sufficient for an explanation of living systems. After thermodynamic self-organization, the emergence of life is made possible by genetic self-organization. The genetic information of an organism is coded by the four chemical compounds Dadenine (A), cytosine (C), guanine (G) and uracil (U). With binary coding A = 00, U = 11, G = 01, and C = 10, we get a genetic code in bits. Structure and function of living organisms need giant sets of information. For a virus, one gets 104 bits, 106 bits for a bactarium, 108 bits for a single-celled organism, and 1012 bits for

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a mammal. Genetic information of a cell is transferred to proteins with hereditary dispositions. Information processing is realized by highly complex molecular pattern recognition. During the evolution of life, the capacity of information processing was not restricted to genetic information systems. In higher developed organisms, populations, and social systems, information processing of nerve systems and brains play a dominant role. They enabled organisms to learn and to adapt to changing environments during life time. Learning and adapting during life time increase the survival of the fittest and have a great advantage to genetic procedures which only allow adapting to ecological niches in sequential generations [7.9]. Three or four billion years ago, genetic information systems emerged and generated a giant biodiversity of cellular organisms. The increasing capacity of information storage can be estimated by the growth of genetic information during evolution [7.10]. In general the information capacity of storage is measured by the logarithm of its number of different possible states. For nucleotide sequences of length n which consists of four building blocks, there are 4n different possibilities of ordering. For bit-units in the binary system, the information capacity is Ic = ln 4n / ln 2 = 2n. For polypeptide with twenty different building blocks, storage capacity is Ic = ln 20n / ln 2 = n · 4.3219 bits. For chromosomes with 109 nucleotides the storage capacity has double length with 2 · 109 bits. Information capacity is independent of the material form of storage. Therefore, different information systems can be compared with, for example, human storage systems like books or libraries. For one of the 32 letters in the Latin alphabet ln 32/ ln 2 = 5 bits are needed. Therefore, in a DNA-sequence, 2 · 109 bits/5bits = 4 · 108 letters could be stored. For an average length of words with 6 letters, one gets 6 · 107 words. If a printed page has 300 words, then one will gets 2 · 105 printed pages. A DNA-sequence of 109 nucleotides corresponds to a storage capacity of 400 books with 500 pages.

Biological evolution on earth produced an exponential growth of genetic information which reached at a maximum of 1010 bits with the emergence of human beings after the development of bacteria, algae, reptiles, and mammals (Fig. 85). A new strategy of information systems was initiated by nerve systems and brains. They started with specialization of few cells for signal transfer. Therefore, the information sets which could be stored by early nerve systems

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Fig. 85.

Phase transition of genetic information on earth

were essentially smaller than the information sets in genetic information systems (Fig. 85). With the emergence of complex organisms like reptiles, neural information systems surpassed the capacity of genetic systems. With increasing complexity, all necessary information of survival could no longer be stored in organisms, but it had to be be learnt by experience [7.11]. Sensorial stimuli of the human organism are analogous signals (e.g. mechanical pressure of skin or muscles, acoustic waves in the ear, electromagnetic waves of the retina, chemical stimuli in the nose) which are received by sensorial cells, coded into digital action potentials, and sent as binary codes (firing and non-firing of neurons) in the central nervous system (CNS) to the brain. Specific nervous signals (neural information) are decoded as sensorial perceptions, emotions, imaginations, or thoughts by specific areas of the brain. A mechanical stimulus (e.g. stretch of a muscle) is received by a sensorial cell as an analogous signal and transformed into digital action potentials. The intensity of the stimulus is coded by the number of equal action potentials which correspond to bits of digital information.

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Information processing of the human brain is made possible by at least 1011 neurons. Each neuron has on average 1000 synaptic connections. Thus, there are 1014 synaptic connections in a brain which is a number larger than the set of stars in our galaxy. Information is stored in neural networks of synaptic connections [7.12]. The steadily firing cells produce a global sea of neural fluctuations, noise and entropy which can be compared with the background radiation of the universe. Our thoughts, feelings, and perceptions correspond to neural cell-assemblies which are the emerging and disappearing local islands of order such as galaxies, stars, and planets in the global sea of cosmic entropy. In this sense, thoughts, feelings, and perceptions mean reduction of uncertainty and gain of information. In general, the concept of information is not restricted to human information processing with brains and nerve systems. The DNAcode is understood by proteins on the molecular level, but in general not by human beings on the level of brains (which nevertheless try to reconstruct its meaning by scientific methods). The meaning of our e-mails is understood otherwise by their human recipients, but not by molecular pattern recognition of cells. Understanding the codes of information depends on contexts. That aspect is sometimes called semantic information. Human brains generate complex cognitive hierarchies of meaning (Fig. 74). Information is represented by signs like letters of texts, numbers of quantities, symbols of mathematical relations, notes of music, etc. Visual perception only can recognize formal (“syntactic”) patterns of signs by the well-known procedures of sensory cells and firing cell-assemblies. The meaning (“semantics”) of, for example, a melody represented by notes, needs the neural activity of further cell assemblies of acoustic areas and memory which are linked with the visual areas. If someone is an expert of music, she/he will associate the recognized melody with her/his whole knowledge. Thus, the coding and decoding process of meaning is based on complex interactions of cell-assemblies. Cognitive hierarchies of meaning emerge by neural self-organization. Different codes of knowledge representation are transformed into one another. Besides the syntactic and semantic aspects of information, pragmatic information is a measure of the effect of information on its recipi-

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ent. Our intentions are related to effects and changes in the external world, generating pragmatic information. Information processing of evolution does not end with brains and nerve systems of organisms. Even populations of animals such as, for example, colonies of ants and termites develop collective information and communication systems for transport nets and arch formation. In sociobiology, these populations are called superorganisms with swarm intelligence [7.13]. Animals communicate with chemical diffusion fields in a self-organizing manner. They specialize themselves for different tasks of labor, transport, and nutrition. Sociodiversity is represented by different clusters of social order. They are the islands of order and collective information in a sea of uncorrelated activities, fluctuations and noise. There is no single termite as architect with a master plan for constructing an arch. The collective information of termites for the construction of arch formation is stored in the chemical diffusion field of communication like in the neural cell-assemblies of a brain. Analogously, there is no neuron which can think, feel, or perceive. Cognitive abilities are generated by populations of neurons in a certain area of the brain. Besides chemical hints, traces and signs of nature are understood by higher developed animals. In populations, collective information is not stored in a single organism, but in extrasomatic fields of chemical and visual communication. In human societies, brains are not isolated, but communicate by complex systems of visual and acoustic signs, gestures, and languages. In early cultures, the collective memory was orally transferred by traditions from generation to generation. After the invention of writing, the memory of a society could be stored in libraries as examples of extrasomatic information storage. After Gutenberg’s invention of printing, information storage in libraries began to surpass the capacity of information storage in single brains (Fig. 85). Networks of telephone, broadcasting and television are further examples of social information systems with increasing capacities. In the age of globalization, the internet and other computer networks are giant information storage systems with exponential growth. Mankind has initiated a technical co-evolution of extrasomatic information pro-

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cessing in order to manage their worldwide societies of increasing complexity and sociodiversity. Obviously, the cosmic, biological, and social evolution from symmetry and simplicity to complexity and diversity can be explained by cosmic, biological and social information dynamics [7.14]. In the beginning there was quantum information with high symmetry and less entropy. Cosmic expansion is characterized by symmetry breaking and globally increasing information entropy (potential information). But thermodynamic, genetic, neural and social self-organization allow the emergence of local islands of higher order, less entropy, and more actual information than global entropy. Galaxies, brains and societies are examples of information systems generating local order and actual information in the cosmic sea of increasing entropy. But memories can also be forgotten for ever. Information disappears in brains when people become older and suffer from, for example, Alzheimer’s disease. Holes of information spread over the whole organ which, in the end, forgets how to live. Dead bodies are relics of disappeared information systems. Black holes are examples of extreme loss of information, structure and order by dying stars and imploding matter. Radiating black holes lose energy and mass. In time, they will disintegrate and, with them, the information of their original stars will disappear in the surrounding universe. In their place, memory gaps will appear in the universe. With the collapse of its galactic structures, a featureless universe expanding into a void is heading for a “cosmic Alzheimer’s disease” [7.15]. From a physical point of view, it may be comforting to know that information cannot disappear totally because of symmetry. The arrow of time is only a macroscopic phenomenon according to the cosmic arrow of expansion and the 2nd law of thermodynamics. On the microlevel, the laws of quantum dynamics demand symmetry of time. The physical question arises: is information lost in black hole evaporation or not? If it is, the dynamics is no longer reversible (“symmetric”) or, in terms of quantum mechanics, unitary. If a book is thrown into the fireside, its information seems to be hopeless lost. Is there a method to reconstruct its letter from the radiation and atomic paths of smoke? Analogously, if a body falls into a black

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hole, the information of its material structure seems to be lost in the center of extreme gravitation and high entropy. Quantum effects cause a black hole to radiate at a steady state. If the radiation from the black hole is completely thermal, then it cannot carry any information. What would happen to all the information of died stars and bodies locked inside a black hole, that evaporated away, and dissappeared completely? It seemed the only way the information could come out would be if the radiation was not exactly thermal, but had subtle correlations of information preserving. Hawking uses Feynman’s sum over all histories of a black hole: mathematically, he takes the path integral over metrics of all possible topologies. Thus, he finds a reversible process with correlation functions that do not decay. In general, quantum gravity is confirmed to be unitary and information is preserved in black hole formation and evaporation: the symmetry of quantum laws is saved. If a body falls into a black hole, its mass-energy will be returned to our universe, but in a mangled form, which contains the information about what it was like, but, perhaps, in an unrecognisable state [7.16]. Information comes back in our universe, although we are not always able to recognize it. In principle, information is conserved as long as the quantum laws of symmetry are valid. But the economist Keynes was also right with his famous quotation that in the long run we are all dead. Therefore, from a human point of view, the conservation of information by symmetry seems to be a hope for eternity. 7.2

Symmetry and Complexity in Computational Dynamics

Information processing can be simulated by computers. In this sense, atomic, molecular, cellular, organic and social systems are considered computational systems with information processes as phase transitions. Symmetry, symmetry breaking and complexity are explained by the principles of computation. What do computation and computability mean? Turing’s concept of a computer does not depend on changing standards of technical development, but it is a general, logical-mathematical definition of computation and computability [7.17].

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A Turing machine consists of (1) a control box in which a finite program is placed, (2) a potentially infinite tape, divided lengthwise into squares, (3) a device for scanning, or printing on one square of the tape at a time, and for moving along the tape or stopping, all under the command of the control box. If the symbols used by a Turing machine are restricted to a stroke | and a blank ∗, then every natural number x can be represented by a sequence of x strokes (e.g. 3 by |||), each stroke on a square of the Turing tape. The blank ∗ is used to denote that the square is empty (or the corresponding number is zero). In particular, a blank is necessary to separate sequences of strokes representing numbers. Thus, a Turing machine computing a function f with arguments x1 , . . . , xn starts with tape · · · ∗ x1 ∗ x2 ∗ · · · ∗ xn ∗ · · · and stops with · · · ∗ x1 ∗ x2 ∗ · · · ∗ xn ∗ f (x1 , . . . , xn ) ∗ · · · on the tape. From a logical point of view, a general purpose computer — as constructed by associates of von Neumann in America and independently by K. Zuse in Germany — is a technical realization of a universal Turing machine which can simulate any kind of Turing program. Besides Turing machines, there are many other mathematically equivalent procedures for defining computable functions. All these definitions of computability can be proved to be mathematically equivalent. Each of these concepts defines a procedure which is intuitively effective like a Turing machine. Thus, A. Church postulated his famous thesis that the informal intuitive notion of an effective procedure is identical with one of these equivalent precise concepts, such as that of a Turing machine. Church’s thesis cannot be proved, of course, because mathematically precise concepts are compared with an informal intuitive notion. Nevertheless, the mathematical equivalence of several precise concepts of computability which are intuitively effective confirms Church’s thesis. Consequently, we can speak about computability, effectiveness and computable functions without referring to particular effective procedures (“algorithms”). According to Church’s thesis, we may say in particular that every computational procedure

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(algorithm) can be calculated by a Turing machine. So every computable function, as a kind of machine program, can be calculated by a general purpose computer. If a physical, biological, or social process can be represented by a computable function, then by Church’s thesis it can be represented by a Turing program which can be computed by a universal Turing machine. Thus, these processes (if they are computable) can be simulated by a technically efficient general purpose computer. Turing computability is a theoretical limit of computability according to Church’s thesis. There are processes with a degree of computational complexity both below and beyond this limit. Below this limit there are many practical procedures concerning certain limitations on how much the speed of an algorithm can be increased. Thus, there are degrees of computability for Turing machines which can be made precise in the complexity theory of computer science [7.18]. Complexity classes of functions can be characterized by complexity degrees, which give the order of functions describing the computational time (or number of elementary computational steps) of algorithms (or computational programs) depending on the length of their inputs. The length of inputs may be measured by the number of decimal digits. According to the machine language of a computer it is convenient to codify decimal numbers into their binary codes with only binary numbers 0 and 1 and to define their length by the number of binary digits. For instance, 3 has the binary code 11 with the length 2. A function f has linear computational time if the computational time of f is not greater than c · n for all inputs with length n and a constant c. A function f has quadratic computational time if the computational time of f is not greater than c · n2 for all inputs with length n and a constant c. A function f has polynominal computational time if the computational time of f is not greater than c · nk , which is assumed to be the leading term of a polynomial p(n). A function f has exponential computational time if the computational time of f is not greater than c · 2p(n) . Many practical and theoretical problems belong to the complexity class P of all functions which can be computed by a deterministic Turing machine in polynomial time. Sometimes it is more convenient to use

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a non-deterministic computer which is allowed to choose a computational procedure at random among a finite number of possible cases instead of performing them step by step in a serial way. In general, NP means the complexity class of functions which can be computed by a non-deterministic Turing machine in polynomial time. By definition every P-procedure is a NP-procedure. But it is a crucial question of complexity theory whether P = NP or, in other words, whether procedures which are solved by non-deterministic computers in polynomial time can be solved by a deterministic computer in polynomial time. A NP-complete procedure means that any other NP-procedure can be converted into it in polynomial time. Consequently, if an NP-complete procedure is actually proved to be a P-procedure, then it would follow that all NP-procedures are actually in P. Otherwise if P = NP, then no NPcomplete procedure can be solved with a deterministic algorithm in polynomial time.

How far can we go with Turing’s concept of computability? The dynamics of physical, biological or social systems could, in principle, be formalized by axiomatic systems. In this case, mathematical theorems are represented by formulas. True formulas are proven by formal derivation from the assumed axioms with logical rules. The steps of a formal proof correspond to a computer program. Is it possible to build a computer which can decide for any formula if it is true or not? Is there a computer which can completely derive all the truths of a formalized axiomatic theory? It was Turing who, in 1936, proved that there cannot be such a universal deciding machine. The reason is that it would be able to determine whether an arbitrary computer program stops after finite steps. But Turing proved that the so-called halting problem is in principle unsolvable. Turing started his proof with the question, are real numbers computable? A real number like π = 3.1415926 . . . has an infinite number of digits that seem to be randomly distributed behind the decimal point. Nevertheless, there are simple finite programs for calculating the digits step by step with increasing precision of π. In this sense, p is called a computable real number. In the first step, Turing constructed an uncomputable real number. Remember that a computer program of a Turing machine, for example, consists of a finite list of symbols. Thus, it can be coded by a natural number called the program number. Imagine a list of all possible computer programs that are ordered according to their increasing program numbers p1 , p2 , p3 , . . . . If a program computes a real number with an infinite number of digits behind the decimal point (e.g. π), then they should be

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written down behind the corresponding program number. Otherwise, there is a blank line in the list: p1

− .d11 d12 d13 d14 d15 d16 d17 . . .

p2

− .d21 d22 d23 d24 d25 d26 d27 . . .

p3

− .d31 d32 d33 d34 d35 d36 d37 . . .

p4 p5

− .d51 d52 d53 d54 d55 d56 d57 . . .

.. . Following Cantor’s diagonal procedure, Turing changed the underlined digits on the diagonal of the list and put these changed digits together into a new number with a decimal point in front: −. = d11 = d22 = d33 = d44 = d55 . . . This new number cannot be in the list because it differs from the first digit of the first number behind p1 , the second digit of the second number behind p2 , etc. Therefore, it is an uncomputable real number. With this number Turing got the unsolvability of the halting problem. If we could solve the halting problem, then we could decide if the nth computer program ever puts out an nth digit behind the decimal point. In this case, we could actually carry out Cantor’s diagonal procedure and compute a real number, which, by its definition, has to differ from any computable real.

The unsolvability of the halting problem refutes the idea of a universal deciding computer. But what can be said about a universal machine which should derive all the truth of an axiomatic theory completely? In the case of a unified theory, all physical truths could be derived by such a machine automatically. But if there is a complete formal axiomatic system from which all mathematical truth follows, then it would give us a procedure to decide if a computer program will ever halt. We just run through all the possible proofs until we either find a proof that the program halts, or we find a proof that it never halts. So if a complete formalization is possible, then by running through all possible proofs while checking which ones are correct, we would be able to decide if the computer program halts. But that is impossible using Turing’s result. This argument confirms G¨odel’s famous incompleteness theorem with the unsolvability

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of the halting problem. There is no omnipotent computer to decide all problems and to prove all truths. But below these theoretical limitations many automatical decisions and proofs with more or less degree of computational complexity are possible. A formal axiomatic theory which describes a physical, biological, or social system has the great advantage of compressing a lot of theorems into a set of a few axioms. Thus, it delivers a shorter description of mathematical truth. Even a physical theory can be understood as a shorter description of many empirical data. In general, a formal theory can be considered a computer program that calculates true theorems or data. The smaller the program is, relative to the output, the better the theory. Obviously, besides running time, the size of a computer program is an important measure of computational complexity. As a program is a finite list of symbols, its length can be measured by its number of symbols in binary coding. For example, consider the following sequences of binary digits: s1 = 111111111111111111 s2 = 010101010101010101 s3 = 011010001101110100 For s1 and s2 , there are shorter descriptions or printing programs than the actual output: “14 times 1” for s1 and “8 times 01” for s2 . But for s3 , there seems to be no shorter description than the actual output itself. G.J. Chaitin and Kolmogorov came up with the idea that the algorithmic complexity of a symbolic s sequence should be defined by the length of the shortest computer program for generating s (measured in bits) [7.19]. Algorithmic complexity is sometimes called the algorithmic information content of a symbolic sequence, which is the subject of the algorithmic information theory. As random sequences have no regularities, they cannot be described by shorter programs. They are incompressible with an algorithmic complexity equivalent to their length. But, again, we are confronted with incompleteness and undecidability. The reason is that we can never decide if an individual string of digits satisfies this definition of randomness and incompressibility. We can never calculate the

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program-size complexity, because, in general, it is not decidable if a certain program is the shortest one. If we have a program generating a sequence, its size is only an upper bound on the program-size complexity of the sequence. But we can never prove lower bounds. For practical applications we can at least refer to standard procedures for detecting regularities in a sequence. If we are not successful, a sequence is called random with respect to these algorithms [7.20]. The theory of computational complexity provides tools to analyze the dynamics of physical, biological and social systems. A Turing machine can be interpreted in the framework of classical physics. Such a computing machine is a physical system the dynamical evolution of which takes it from one of a set of input states to one of a set of output states. The states are labeled in some serial way. The machine is prepared in a state with a given input level and then, following some deterministic motion, the output state is measured. For a classical deterministic system the measured output label is a definite function f of the prepared input label. In principle, the value of that label can be measured by an outside observer, and the machine is said to compute the function f . If decimal numbers are coded in binary ones, then the digits 0 and 1 can be considered alternative states of a machine representing bits of information. But stochastic computing machines and quantum computing machines do not compute functions in the above sense. The output state of a stochastic machine is random, with only a probability distribution function for the possible outputs depending on the input state. The output state of a quantum machine, although fully determined by the input state, is not an observable and so the observer cannot in general discover its label. What is the reason for this? We must remember some basic concepts of quantum mechanics which were already introduced in Sec. 3.1. Quantum computing relates to the smallest units of matter depending on Planck’s constant and the velocity of light [7.21]. The classical laws of physics are restricted in these dimensions. Contrary to classical physics, matter is no longer continuous, but divided into elementary particles like photons or electrons. Atoms change into discrete states. Thus, for instance, a hydrogen atom could be con-

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sidered a processor for quantum information in quantum bits. The ground state corresponds to 0 and the excited state to 1. Even photons could be used for quantum information processing, because they have two quantum states of horizontal and vertical polarization, represented by 0 and 1. Contrary to classical physics, there are also intermediate states as superpositions of both states 0 and 1. Such a quantum bit is half 0 and 1. A classical bit is either 0 or 1. According to quantum mechanics, the two possible states 0 and 1 of a superposition remain undetermined until it is measured by an observer. But a superposition can also collapse by a material interaction which is not intended by the human user. This kind of instability is one of the technical problems which must be solved in constructions of practical quantum computers. In physical terms, classical systems described by a Hamiltonian function are replaced by quantum systems, e.g. electrons or photons described by a Hamiltonian operator. States of a quantum system are described by vectors of a Hilbert space spanned by the eigenvectors of its Hamiltonian operator. The causal dynamics f of quantum states is determined by a partial differential equation called the Schr¨ odinger equation. While classical observables (e.g. localization or impulse of a particle) always have definite values, non-classical observables (operators) of quantum systems in general have no common eigenvector and consequently no definite eigenvalue. A superposition with 0 and 1 is causally determined by the Schr¨ odinger equation. But the two possible states 0 and 1 remain undetermined until it is measured by an observer.

The superposition of quantum states opens new avenues of computational parallelism, because myriads of input bits could be superposed in a quantum state. Thus, a quantum computer could deliver the superposition of perhaps billions of parallel computations in a rather short time, overcoming the efficiency of classical computing systems, working in a serial manner step by step. But quantum computers would still work in an algorithmic way, because their dynamics would be deterministic. The non-deterministic aspect comes in via the act of measurement and the collapse of superposition. Thus, it cannot be expected that quantum computers will perform nonalgorithmic operations beyond the power of a Turing machine [7.22]. But quantum computers will be extremely interesting for computa-

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tional complexity theory and for overcoming practical constraints of computation. The superposition of quantum states also opens new perspectives of quantum communication [7.23]. According to the EPR-experiments, pairs of elementary particles (e.g. photons) which are emitted from a central source into opposite directions remain correlated in the superposition of an entangled quantum state. If one of two entangled quantum bits is measured at one location of a particle, then the value of the other quantum bit is instantly determined at the location of the other particle in opposite distance. Quantum teleporting which is sometimes used in science fiction movies is at least possible. In order to transport a body from one location to another one, the quantum information of the body’s structure could be transferred by a kind of EPR-experiment. In order to reconstruct the body at the distant location, it is necessary to have its materials at this place. In any case, quantum information could be transported by a new kind of quantum communication technology.

Quantum computing does not only mean exponential growth of computational capacity and communication technology. Any kind of matter stores quantum information. Therefore, any elementary particle is a processor of quantum information. The computational rules of these processors are symmetric according to the principles of quantum symmetry. Any computational step is especially reversible according to the quantum symmetry of time (microreversibility). Phase transitions of matter are quantum information processing. The universe is an expanding quantum computer producing quantum information of giant complexity. Furthermore, it is an immense database conserving all quantum information because of symmetry. We must not forget that the concept of a computing machine is not restricted to human technology with symbolic dynamics of data. Symbols only represent states of dynamical systems for human purposes. Information processing does not depend on human purposes and interests. Human knowledge only relates to a tiny part of the information in the world. In principle, quantum information does not depend on an observer or measurement process. Observing and measuring quantum systems is only a special example of an interaction of a quantum system with another system [7.24]. Cosmic evolution from symmetry to complexity is characterized by the emergence of new structures. After elementary particles and

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atoms, molecules arrange themselves in more or less complex clusters, solids, fluids and gases. The emergence of new structures is combined with the gain of information. Thus, any molecular structure is a kind of molecular computer with phase transitions as information processing. That is the reason why molecular computing of nature also becomes a standard for modern computer technology besides quantum computing. For a molecular computer billions of molecular building blocks must be switched to one another. According to nanotechnology single atoms molecules can actually be manipulated by scanning tunnel microscopes. But this technology is not always sufficient in order to arrange millions and billions in certain distances. The theory of complex dynamical systems offers new possibilities. Molecular building blocks of complex systems can arrange themselves according to the laws of molecular self-organization and templates of desired patterns. In Chapter 4 many examples of symmetric structures are discussed which could be used for molecular computing. All kinds of smart and intelligent materials could be understood as information processing systems with molecular computer devices. Symbolic representations of molecular computing are only necessary for human understanding. The next evolutionary step after quantum and molecular computing is DNA-computing [7.25]. During evolution genetic information systems have emerged by DNA-ruled genetic self-organization. In electronic computers information is coded by sequences of bits 0 or 1. DNA-systems used DNA-sequences of nucleotides which are symbolically represented by the letters A, C, G, T. Information processing in DNA-systems happens by chemical reactions which generate, divide or recombine DNA-strands. DNA-replication is realized by a molecule which can be understood as a tiny nano-machine. It moves along a DNA-strand, recognizes its bases step by step and generates a complementary DNA-strand according to the DNA-law of asymmetry. In a mathematical model it is a Turing machine which implements a sequence of symbols A, C, G, T and prints its complement by rules of simple operations. As a Turing machine is a universal concept of computability according to Church’s thesis, the DNA-replication corresponds to a special Turing machine with a cer-

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tain degree of computability. The biological evolution on earth has generated billions of nano-machines for particular tasks. But the fascinating perspective of DNA-computing is less motivated by single nano machines, but by biochemical systems with billions of simultaneously interacting DNA-strands. They are a complex system with information processing of massive parallelity. Parallel computers enable solutions of highly complex problems, which fail to be solved by serially working von Neumann computers. Because of their great packing density and high-speed (e.g. 6 gramm DNA for 1 million tera-operation per second) DNA-computers could be applied for special computational tasks of high complexity. Nature only generated special examples of DNA-computers. According to the laws of genetic self-organization we could develop new types for human purposes and interests. After quantum, molecular and DNA-computing, the emergence of cellular organisms provides the next protype of information and computing systems. Von Neumann’s concept of cellular automata gave the first hints of mathematical models of living organisms conceived as self-reproducing networks of cells [7.26]. The state space is a homogeneous lattice which is divided into equal cells like a chess board. An elementary cellular automaton is a cell which can have different states, for instance “occupied” (by a mark), “free”, or “colored.” An aggregation of elementary automata is called a composite automaton or configuration. Each automaton is characterized by its environment, i.e. the neighboring cells which may have a symmetric form like a square or cross:

Fig. 86.

Symmetric environments in cellular automata

The dynamics of cellular automata are determined by synchronous transformation rules. Von Neumann already proved that the typical feature of living systems, their tendency to reproduce

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themselves, can be simulated by particular cellular automata. In Conway’s game of life, cellular automata are complex systems generating patterns of black or colored cells which remind us of growing, changing, and dying populations of living systems. Even all kinds of 2-dimensional symmetries can be generated by 2-dimensional cellular automata. But cellular automata are not only nice computer games. They have turned out to be discrete and quantized models of complex systems with nonlinear differential equations describing their evolution dynamics. Imagine a chessboard-like plane with cells, again. A state of a 1-dimensional cellular automaton consists of a finite string of cells, each of which can take one of two states (“black” (0) or “white” (1)) and is connected only to its two nearest neighbors, with which it exchanges information about its state. The following (later) states of a 1-dimensional automaton are the following strings on the space-time plane, each of which consists of cells taking one of two states, depending on their preceding (earlier) states and the states of their two nearest neighbors. Figs. 87b–e illustrates the time evolution of four automata. Thus, the dynamics of an 1-dimensional cellular automaton is determined by a Boolean function of three variables, each of which can take either the value 0 or 1. For three variables and two values, there are 23 = 8 possibilities for three nearest neighbor sites. In Fig. 87a, they are ordered according to the corresponding three-digit binary number. For each of the three nearest neighbor sites, there must be a rule determining the following state of the middle cell. For eight sequences and two possible states, there are 28 = 256 possible combinations. One of these rule combinations, determining the dynamics of a 1-dimensional cellular automaton, is shown in Fig. 87a. Each rule is characterized by the eight-digit binary number of the states which each cell of the following string can take. These binary numbers can be ordered by their corresponding decimal numbers [7.27]. The time evolution of these simple rules, characterizing the dynamics of a 1-dimensional cellular automaton, produces very different cellular patterns, starting from simple or random initial conditions. According to S. Wolfram, computer experiments give rise to the following classes of attractors the cellular patterns of evolution

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are aiming at. After a few steps, systems of class 1 reach a homogeneous state of equilibrium independently of the initial conditions. This final state of equilibrium is visualized by a totally white plane and corresponds to a fixed point as attractor (Fig. 87b). Systems of class 2, after a few time steps, show a constant or periodic pattern of evolution which is relatively independent of the initial conditions. Specific positions of the pattern may depend on the initial conditions, but not on the global pattern structure itself (Fig. 87c). In a 3rd class, cellular automata produce patterns that seem to spread randomly and irregularly over a grid (Fig. 87d). In a 4th class, evolutionary patterns with occasional quasi-organic and locally complex structures can be observed (Fig. 87e). Contrary to 1st and 2nd class automata, patterns in the 3rd and 4th class sensitively depend on their initial conditions. Obviously, these four classes of cellular automata model attractor behavior of nonlinear complex systems, a

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fact well-known from self-organizing processes [7.28]. They remind us of the familiar classifications of materials into solids, liquids, and gases, or living organisms, such as plants and animals. In general, the cellular automata approach confirms the intuitive idea that complex systems lie somewhere between regular order (like ice crystals and Buckminsterfullerenes) and complete irregularity or noise (like molecules in a heated gas). Organisms and brains are highly complex, but they are neither completely ordered nor completely random and disordered. Obviously, these four classes of cellular automata model the attractor behavior of nonlinear complex systems, which is well known from self-organizing processes. In the preceding chapters, we have seen many examples in physical, biological and social dynamics. In general, self-organization has been understood as a phase transition in a complex system. Macroscopic patterns arise from complex nonlinear interactions of microscopic elements. There are different final patterns of phase transitions, corresponding to mathematically different attractors. Predictions of future development are easy for cellular automata of the first two classes. In the 1st class, cellular automata always evolve after finite steps to a uniform pattern of rest, which is repeated for all further steps in the sense of a fixed point attractor. As they preserve no information about the arrangement of cells on earlier steps, the evolution is irreversible: we have no chance to go backwards and reconstruct the initial conditions from which the automata actually started. In the 2nd class, the development of repeated patterns is obviously reversible and symmetric for all future developments. It preserves sufficient information to allow one to go backwards or forwards from any particular step. In random patterns of the 3rd class, all correlations have decayed, and, therefore, the evolution is irreversible. For localized complex structures of the 4th class, we perhaps have a chance to recognize strange or chaotic attractors, which are highly complex and correlated patterns, contrary to the complete loss of structure in the case of randomness. Of the 256 simplest 1-dimensional cellular automata with nearest neighbors and binary cellular states (or two colors), only six have

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symmetric (reversible) behavior. They only generate simple repetitive changes in the initial conditions. In these cases, it is always possible to reproduce the configurations of all previous steps, starting from any given configuration. In other words, it is possible to interchange the past and future. The computational system has symmetry of time. If we increase the number of cellular states to three, instead of two, colors, we get 33 = 27 possibilities for three nearest neighbor sites and the gigantic number of 327 = 7 625 597 484 987 1-dimensional cellular automata. Among them, there are 1800 reversible automata, so starting from any configuration of cells, it is possible to generate the configurations of all previous steps. But some of these 1800 reversible 1-dimensional automata no longer only deduce simple repetitive transformations of initial conditions, but show complex, scrambled patterns. Thus, microreversibility with symmetric microrules can generate complex macro-behavior. We can construct reversible rules that remain the same even when turned upside-down. Therefore, the rules of a 1-dimensional cellular automaton are affected by the dependence on colors two steps back. In Fig. 88, we take rule 122 of the 256 simplest 1-dimensional automata with nearest neighbors and binary cellular states (or two colors). We add the restriction that the new state (color) of a cell should be inverted if the cell is black (1) two steps back. With knowledge of not one but two successive steps, it is always possible to determine the cellular configurations of future or past steps.

1 1 1 1 1 1 1 1 111 110 101 100 011 010 001 000 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 111 110 101 100 011 010 001 000 0 1 1 1 1 0 1 0 Fig. 88.

Rule of the reversible cellular automaton 122R

The symmetry and asymmetry of time are an important topic of natural science. All fundamental laws of classical, relativistic, and quantum physics are reversible: they are invariant with respect to the two possible directions of time, t or −t. Our everyday experience

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Fig. 89. Computational simulation of the second law of thermodynamics by a reversible cellular automaton [7.29]

seems to support an irreversible development with one direction of time. According to the second law of thermodynamics, increasing disorder and randomness (“entropy”) is generated from simple and ordered initial conditions of closed dynamical systems. Irreversibility is highly probable in spite of the symmetry (microreversibility) of molecular laws. Some cellular automata with reversible rules generate patterns of increasing randomness, starting from simple and ordered initial conditions. In Fig. 89, the reversible cellular automaton of rule 122R can start from an initial condition in which all black cells or particles lie in a completely ordered pattern at the center of a box. Running downwards, the distribution seems to become more and more random and irreversible, in accordance with the second law. In principle, symmetry of time (reversibility) is possible, analogous to Poincar´e’s famous theorem of reversibility in statistical mechanics, but extremely improbable. By starting with a simple state and tracing the actual evolution, one can find initial conditions that will lead toward to decreasing randomness (Fig. 89). But for cellu-

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lar automata, the computational amount to go backwards and find these conditions cannot be reduced to the actual evolution from simple to random patterns: computational irreducibility corresponds to temporal irreducibility and improbability. Thus, in computer experiments with cellular automata, we get a computational equivalence of the second law of thermodynamics. Different increasingly complex and random patterns can be generated by the same simple rules of cellular automata with different initial conditions. In many cases, there is no finite program to forecast the development of complex and random patterns. The algorithmic complexity is incompressible due to its computational irreducibility. In this case, the question of how the system will behave in the future is undecidable, because there can be no finite computation that will decide it. Obviously, computational irreducibility is connected with Turing’s fundamental problem of undecidability. Whether a pattern of a cellular automaton ever dies out, can be considered analogous to the halting problem of Turing machines. Computational irreducibility means that there is no finite method of predicting how a system will behave except by going through nearly all the steps of actual development. In the history of science, one assumes that the precise knowledge of laws allows for precise forecasting of the future. Even in the case of chaos theory, there are methods of time series analysis that determine, at least, future trends and attractors of behavior. But in the case of randomness, there is no short cut to the actual evolution. Wolfram supposes that the sciences of complexity are basically characterized by computational irreducibility [7.30]. Even if we know all the laws of behavior on the microlevel, we cannot predict the development of a random system on the macrolevel. The brain, as a complex system, is determined by simple synaptic rules (e.g. Hebb’s rule) on the microlevel of neurons that are more or less well-known. Nevertheless, there is no chance of computing pattern formation of neural cell assemblies in all its details. In a philosophical sense, computational irreducibility seems to support personal individuality: our personal life is influenced by many unexpected and random events. The pattern of our way of life is highly nonlinear, complex, and random. Thus, there is no short

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cut to predicting life: if we want to experience our life, we have to live it. From a methodological point of view, a 1-dimensional cellular automaton delivers a discrete and quantized model of the phase portrait which describes the dynamical behavior of a complex system with a nonlinear differential equation of evolution, depending on one space variable. There are many reasons for restricting oneself to discrete models. The complexity of nonlinear systems is often too great to calculate numerical approximations within a reasonable computation time. In that case, computer experiments with discrete models give, at least, a rough idea and feeling of what is going on, similar to laboratory experiments. 2-dimensional cellular automata, which have been used in Conway’s game of life, can be interpreted as discrete models of complex systems with nonlinear evolution, depending on two space variables. Obviously, cellular automata are a very flexible and effective modeling instrument when the complexity of nonlinear systems increases and the possibility of determining their behavior by solving differential equations, or even by calculating numerical approximations, becomes more and more hopeless. In short: all complex dynamical systems are computational, but they are not always computable. The principle of computational equivalence requires that there is always a computational model of a complex dynamical system. For example, in the case of cellular automata, we can always program the simple rules of interacting cells on the microlevel. Therefore, every cellular automaton is a computational system. But, on the macrolevel, an automaton may nevertheless generate complex patterns which cannot be decided in the sense of Turing’s halting problem or which cannot be forcast in the sense of chaos theory. Thus, cellular automata may not be computable in principle or by practical reasons, because the degrees of computational complexity increase exponentially with forcasts in the long term. According to the principle of computational equivalence, the universe could be considered a computational system, if the basic rules of interacting elementary particles on the microlevel are well known. Nevertheless, on the macrolevel, the emergence of

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patterns, clusters and other new phenomena may not be computable in all cases. The next evolutionary step after thermodynamic, genetic and cellular self-organization was neural self-organization of nerve systems and brains. Brains have the possibility to learn and to adapt to changing conditions of an environment. Therefore, after quantum, molecular, DNA- and cellular computing the emergence of learning and cognitive systems in nature is the next standard for new information and computing systems. In the first logical model of the brain, which was offered by the McCulloch-Pitt network, the function of an artificial neuron is fixed for all time [7.31]. But in order to make a neural computer capable of complex tasks, it is necessary to find mechanisms of self-organization which allow the network to learn. In 1949, Hebb suggested the first neurophysiological learning rule which has become important for the development of neural computers. Synapses of neurons do not always have the same sensitivity, but modify themselves in order to favor the repetition of firing patterns which have frequently occurred in the past. In 1958, F. Rosenblatt designed the first learning neural computer, which has become famous under the name “Perceptron” [7.32]. Rosenblatt’s neural computer is a feedforward network with binary threshold units and three layers. The first layer is a sensory surface called a “retina” which consists of stimulus cells (S-units). The S-units are connected with the intermediate layer by fixed weights which do not change during the learning process. The elements of the intermediate layer are called associator cells (Aunits). Each A-unit has a fixed weighted input of some S-units. In other words, some S-units project their output onto an A-unit. An S-unit may also project its output onto several A-units. The intermediate layer is completely connected with the output layer, the elements of which are called response cells (R-units). The weights between the intermediate layer and the output layer are variable and thus able to learn.

The Perceptron is viewed as a neural computer which can classify a perceived pattern in one of several possible groups. In the case of two groups, each R-unit learns to distinguish the input patterns by activation and de-activation. The learning procedure of a Perceptron is supervised. Thus, the desired state (active or not) of each R-unit, corresponding to a pattern to be learnt, must be well known. The

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patterns to be learnt are offered to the network, and the weights between the intermediate and output layer are adapted according to the learning rule. The procedure is repeated until all patterns produce the correct output. But some simple problems show that Perceptrons are not universal. For instance, a Perceptron is not able to distinguish between even and odd numbers. Its limitation depends on the particular architecture of Perceptron [7.33]. A network of supervised learning which solves the problems of Perceptron is the Hopfield system [7.34]. It works with feedback and Hebb-type learning which is practised by biological brains, too. In the case of a homogeneous network of boolean neurons, the two states of the neurons can be associated with the two possible values of electron spin in an external magnetic field. A Hopfield model is a dynamical system which, by analogy with annealing processes in metals, admits an energy function. As it is a non-increasing monotonic function, the system relaxes into a local energy minimum, corresponding to a locally stable stationary state (fixed point attractor). Thus, the dynamical evolution of a Hopfield system may correspond to mental recognition. For example, an initial state representing a noisy picture of the letter “A” evolved towards a final state representing the correct picture, which was trained into the system by several examples. The physical explanation is given in terms of phase transition in equilibrium thermodynamics. The correct pattern is connected to the fixed point or final state of equilibrium. A more flexible generation is the Boltzmann machine with a stochastic network architecture of non-deterministic processor elements and with a distributed knowledge representation, mathematically corresponding to an energy function. The general idea of relaxation is that a network converges to a more or less global state of equilibrium on the basis of local interactions. By iterative modification of the local connections (for instance, by a Hebb learning strategy in the case of a Hopfield system) the network as a whole eventually relaxes into a stable and optimal state. We may say that local interactions lead to a cooperative search which is not supervised, but self-organized. There are networks which use the strategy of cooperative search for mental activities like, for in-

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stance, seeking a probable hypothesis. Imagine that a certain range of competing hypotheses are represented by neural units which may activate or inhibit themselves. The system thus moves away from the less probable hypotheses toward more probable hypotheses. In 1986, J.L. McClelland and D. Rumelhart used this cognitive interpretation to simulate the recognition of ambivalent figures with two symmetric views in gestalt-psychology. Fig. 90a shows a network for cooperative search simulating the recognition of one of the two symmetric orientations of a Necker cube. Each unit is a hypothesis concerning a vertex of the Necker cube. Abbreviations are B (back), F (front), Le (left), R (right), U (upper), Lo (lower). The network of hypotheses consists of two interconnected subnetworks, one corresponding to each of the two symmetric interpretations. The recognition of one of the two views happens by symmetry breaking [7.35]. Incompatible hypotheses are negatively connected, and consilient hypotheses are positively connected. Weights are assigned such that two negative inputs balance three positive inputs. Each unit has three neighbors connected positively and two competitors connected negatively. Each unit receives one positive input from the stimulus. The subnet of hypotheses to find is the one which best fits the input. Tiny initial fluctuations (which means a small detail in the special view of an observer) may decide which orientation is seen in the long run. Obviously, the decision happens by symmetry breaking of an ambivalent situation. To visualize the dynamics of symmetry breaking, suppose that all units are off. Then one unit receives an input of positive value at random. The network will evolve toward a state where all units of one subnetwork are activated and all units of the other network are turned off. In the cognitive interpretation we may say that the system has relaxed into one of the two interpretations of the ambivalent figure of either a right-facing or a left-facing Necker cube. Fig. 90b shows three different evolution patterns, depending sensitively on different initial conditions. The size of the circles indicates the activation degree of each unit. In the third run, an undecided final state is reached which is nevertheless in equilibrium. Obviously, the architectural principles of this network are cooperative computation, distributed representation, and relaxation procedure, which are well known in the dynamics of complex systems.

Pattern recognition is interpreted as a kind of phase transition by analogy with the evolution equations which are used for pattern emergence in physics, chemistry, and biology. We get an interdis-

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Symmetry breaking of pattern recognition (Example: Necker cube)

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ciplinary research program that should allow us to explain neurocomputational self-organization as a natural consequence of physical, chemical, and neurobiological evolution by common principles. As in the case of pattern formation, a specific pattern of recognition (for instance a prototype face) is described by order parameters to which a specific set of features belongs. Once some of the features which belonging to the order parameter are given (for instance, a part of a face), the order parameter will complement these with other features so that the whole system acts as an associative memory (for instance, the reconstruction of a stored prototype face from an initially given part of that face). The features of a recognized pattern correspond to the subsystems which are dominated by the order parameter of the whole pattern. A new technical approach to model symmetry and complexity of nature is the concept of cellular neural networks (CNN) [7.36]. The emergence of CNN has been made possible by the sensor revolution of the late 1990s. Cheap sensor and MEMS (micro-electro-mechanical system) arrays are proliferating in all technical infrastructures and human environments. They have become popular as artificial eyes, noses, ears, tastes, and somatosensor devices. An immense number of generic analog signals have been processed. Thus, a new kind of chip technology, similar to signal processing in natural organisms, is needed. Analogic cellular computers are the technical response to the sensor revolution, mimicking the anatomy and physiology of sensory and processing organs. A CNN chip is their hard core, because it is an array of analog dynamic processors or cells. The CNN was invented by L.O. Chua and L. Yang at Berkeley in 1988 [7.37]. The main idea behind the CNN paradigm is Chua’s so-called “local activity principle”, which asserts that no complex phenomena can arise in any homogeneous media without local activity. Obviously, local activity is a fundamental property in microelectronics. For example, vacuum tubes and, later on, transistors became the locally active devices in the electronic circuits of radios, televisions, and computers. The demand for local activity in neural networks was motivated by the practical needs of technology. In 1985, J.J. Hopfield suggested his theoretical neural network, which,

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in principle, could overcome the failures of pattern recognition in Rosenblatt’s “Perceptron”. But its globally connected architecture was highly impractical for technical applications in the VLSI (verylarge-scale-integrated) circuits of micro-electronics: the number of wires in a fully connected Hopfield network grows exponentially with the size of the array. A CNN only needs electrical interconnections in a prescribed sphere of influence [7.38]. An immense increase in computing speed, combined with significantly less electrical power in the first CNN chips, has led to the current intensive research activities on CNN since Chua and Yang’s proposal in 1988. In general, a CNN is a nonlinear analog circuit that processes signals in real time. It is a multi-component system of regularly spaced identical (“cloned”) units, called cells, that communicate directly with each other only through their nearest neighbors. But the locality of direct connections allows for global information processing. Communication between remotely connected units are achieved through other units. The idea that complex and global phenomena can emerge from local activities in a network dates back to von Neumann’s earlier paradigm of cellular automata (CA). In this sense, the CNN paradigm is an advancement of the CA paradigm under the new conditions of information processing and chip technology. Unlike conventional cellular automata, CNN host processors accept and generate analog signals in continuous time with real numbers as interaction values. But, actually, discreteness of CA is no principle difference to CNN. We can introduce continuous cellular automata (CCA) as a generalization of CA in which each cell is not just, for example, black or white, but instead can have any of a continuous range of grays. A possible rule of a CCA may require that the new gray level of each cell be the average of its own gray level, and that of its immediate neighbors. It turns out that in continuous cellular automata simple rules of interaction can generate patterns of increasing complexity, chaos, and randomness, which are not essentially different to the behavior of discrete CA. Thus, they are useful in approximating the dynamics of systems that are determined by partial differential equations (PDE).

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Mathematically, a CNN is defined by (1) a spatially discrete set of continuous nonlinear systems (“cells” or “neuron”) where information is processed in each cell via three independent variables (“input,” “threshold,” and “initial state”) and (2) a coupling law relating relevant variables of each cell to all neighbor cells within a pre-described sphere of influence. Many CNN applications use space-invariant standard CNNs with a cellular neighborhood of 3 × 3 cells and no variation of synaptic weights and cellular thresholds in the cellular space. A 3 × 3 sphere of influence at each node of the grid contains nine cells with eight neighbor cells, and the cell in its center. In this case, the contributions of the output (feedback) and input (feedforward) weights can be reduced to two fixed 3 × 3 matrices, which are called feedback (output) cloning template A and feedforward (input) cloning template B. Thus, each CNN is uniquely defined by the two cloning templates A, B, and a threshold z, which consist of 3 × 3 + 3 × 3 + 1 = 19 real numbers. They can be ordered as a string of 19 scalars with a uniform threshold, nine feedforward and nine feedback synaptic weights. This string is called a “CNN gene”, because it completely determines the dynamics of the CNN. Consequently, the universe of all CNN genes is called the “CNN genome”. With respect to the human genome project, steady progress can be made by isolating and analyzing various classes of CNN genes and their influences on CNN genomes. A successful application is visual computing which generates nice models of symmetries, symmetry breaking and complexity. Concerning visual computing, the triple {A, B, z}, and its 19 real numbers can be considered a CNN macro instruction of how to transform an input image into an output image. Simple examples are subclasses of CNNs with practical relevance, such as the class C(A, B, z) of space-invariant CNNs with excitatory and inhibitory synaptic weights; the zero-feedback (feedforward) class C(0, B, z) of CNNs without cellular feedback; the zero-input (autonomous) class C(A, 0, z) of CNNs without cellular input; and the uncoupled class C(A◦ , B, z) of CNNs without cellular coupling. In A0 all weights are zero except for the weight of the cell in the center of the matrix. Their signal flow and system structure can be illustrated in diagrams that can easily be applied to electronic circuits, as well as to typical living neurons.

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CNN templates are extremely useful for standards in visual computing. An example of symmetry and symmetry breaking is the visual illusion where some images can be perceived in an ambiguous way, depending on the initial thought or attention. One of the examples of this phenomenon is the face-vase illusion (Fig. 91), where the image can be interpreted either as two symmetric faces, or as a vase. Initial attention is implemented by specifying, via a second binary pattern, one of the two ambiguously interpreted regions. Thus, initial attention initiates symmetry breaking between two possible solutions in an equilibrium system. Symmetry breaking of pattern recognition corresponds to symmetry breaking of pattern formation in nature which was illustrated in Fig. 43 by the emergence of two kinds of convection rolls with opposite orientation. In Fig. 91, the pictures consist of 200 × 400 pixels. Feedback-, feedforward-templates and threshold have the following values:

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The emergence of complex structures in nature can be explained by the nonlinear dynamics and attractors of complex systems. They result from the collective behavior of interacting elements in a complex system. The different paradigms of complexity research promise to explain pattern formation and pattern recognition in nature by their specific mechanisms. From the CNN point of view, it is convenient to study the subclass of autonomous CNNs that cells have no inputs. These systems can explain how patterns arise, evolve, and

Fig. 91. Symmetry breaking of pattern recognition by a CNN (Example: facevase-illusion) [7.39]

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sometimes converge to an equilibrium by diffusion-reaction processes. Pattern formation starts with an initial uniform pattern in an unstable equilibrium that is disturbed by small, random displacements. Thus, in the initial state, the symmetry of the unstable equilibrium is broken, leading to rather complex patterns. Obviously, in these applications, cellular networks do not only refer to neural activities in nerve systems, but also to pattern formation in general. Thus, the abbreviation CNN is now understood as “Cellular Nonlinear Network”. A CNN is defined by the state equations of isolated cells and the cell coupling laws. For simulating diffusion–reaction processes, the coupling law describes a discrete version of diffusion (with a discrete Laplacian operator). CNN state equations and CNN coupling laws can be combined in a CNN diffusion-relation equation to determine the dynamics of autonomous CNNs. If we replace their discrete functions and operators by their limiting continuum version, we get the well-known continuous partial differential equations of diffusion–reaction processes, which have been studied in the complexity paradigms of, for example, Prigogine’s non-equilibrium chemistry and Haken’s synergetics. Chua’s version of the CNN diffusion– reaction equation delivers computer simulations of these pattern formations in chemistry and biology (e.g. concentric, auto- and spiral waves). On the other hand, many appropriate CNN equations can be associated with any nonlinear partial differential equation. In many cases, it is sufficient to study the computer simulations of associated CNN equations in order to understand the nonlinear dynamics of these complex systems. Sometimes, the autonomous CNNs (like digital cellular automata) are only considered approximations of nonlinear partial differential equations for the practical purpose of computer simulations. But, Chua claims nonlinear partial differential equations are limiting forms of autonomous CNNs. Thus, only a subclass of CNNs has a limiting representation of partial differential equations. In short, the CNN paradigm of complexity is more than the conventional approach with differential equations. Pattern recognition is understood in relation to pattern formation. Coupled CNNs with linear synaptic weights open avenues to

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much richer visual computing applications than uncoupled CNNs. In coupled CNNs, there are couplings from the outputs of the surrounding cells to a cell in the center. Thus, at least one element of the feedback (output) template A (which is different from the coefficient of the cell in the center) is not zero. Coupled CNNs are, for example, able to detect holes (i.e. a set of adjacent pixels) on a surrounding background. In particular, it turns out that the famous connectivity problem can be solved by a simple coupled CNN of this kind. This problem is not only important for practical reasons, but also has a long tradition in the history of cognitive science. How can we recognize connected patterns (“gestalt”), such as shapes, figures, or faces from a set of pixels? In a famous proof, M. Minsky demonstrated that the connectivity of certain patterns could not be recognized by neural networks like Rosenblatt’s “Perceptron”. In the case of linear synaptic weights, the characteristics of a synapse or template element are linear. But in technical applications (e.g. with voltage- controlled current sources) or living cells with synaptic communication by neurotransmitter, they are never completely linear. If we use nonlinear templates for modeling synaptic dynamics, the analysis becomes more complex. Thus, a compromise of modeling is the application of uncoupled CNNs with nonlinear, space-invariant weights. Besides visual computing, other functions of behavior are also modeled by neural networks. In nature, complex patterns of movements are not computed and controlled by a central processor, but by self-organizing learning algorithms of feedback nets. An example is a grasshopper with six legs and different motor modules of lifting, swinging, and coordinating. External information of an unknown environment is learned and stored implicitly by the distribution of synaptic weights in neural nets. During evolution, decentralized network modules could be used as building blocks for different organisms according to changing conditions. These biological insights into motor information processing are already being applied to robotics and chip technology (embodied cognition). Soft computing uses fuzzy logic, genetic and learning algorithms for flexibility, adaptive and human-like information systems. Affective computing aims at rec-

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ognizing and modeling emotional states of the brain as information processing. Cyborgs (cybernetic organisms) represent the vision of a brain with implanted chips of neural computers. Neural nets could recognize patterns of brain activities (e.g. EEG signals) correlating to states of cognition and consciousness. In a next step, patterns of neural activities could be scanned and downloaded to a supercomputer. Then, of course, a dramatic ethical problem arises: Could human personality (not only the DNA-genotype) be cloned and influenced by computational systems? In Sec. 5.3, the brain was introduced as a complex cellular system. If its motoric, sensory, emotional, and cognitive dynamics are well understood, then, according to the principle of computational equivalence, they can be modeled by computational systems. Thoughts, emotions, and even consciousness correspond to complex patterns of neural cell-assemblies (Fig. 74) emerging from basic synaptic rules of neural interaction (e.g. Hebb’s rule). But even in the simplified case of cellular automata, local rules of cellular interaction can generate undecidable and chaotic patterns. Thus, the patterns of cellassemblies may be too complex to be forecast in all details by finite programs. The brain would be a computional system, because its basic neural rules could be programmed. But, nevertheless, its dynamics would not be computable. Actually, the brain is a stochastic system with global noise of uncorrelated neural firing. Cognitive or emotional states correspond to locally correlated patterns of synchronously firing cell-assemblies emerging like islands in a sea of noise and entropy. To forecast a specific feeling or thought would be as improbable as forecasting the emergence of a little wave on the wide ocean. Therefore, people are characterized by their particular nonlinear dynamics and development. They have their own history, personality, and intimacy. In this sense, robots with artificial minds like humans could develop their own identity and emotions, although their basic rules are programmed by human engineers. In short, to be a computational system is no contradiction to the concept of free will. The reason is the complexity and nonlinearity of computational systems [7.40].

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(a) star

(b) ring

(d) tree

(e) completely meshed net

Fig. 92a–f.

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(c) bus

(f) partially meshed net

Symmetries of network typologies [7.41]

Human brains and artificial minds are only specific models of computational networks with different degrees of complexity. Their topologies and dynamics are a challenge of present and future research. Different principles of symmetry are applied in network topologies determining the physical shape or the layout of computational networks (Fig. 92). In star topologies (Fig. 92a), all cells are connected to a central cell. All traffic emanates from the central cell. The advantage of the star topology is that if one cell fails, then only the failed cell is unable to send or receive data. The star networks are relatively easy to install, but have potential bottlenecks and failure problems at the central cell because all data must pass through this central cell. In ring topologies (Fig. 92b), all cells are connected to one another in the shape of a closed loop, so that each cell is connected directly to two other cells. In most cases, data flows in one direction only, with one cell receiving the signal and relaying it to the next cell on the ring. In bus topologies (Fig. 92c), all cells

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are connected to a central backbone, called the bus. The structure has a translational symmetry. The bus permits all cells to receive every transmission. It is relatively simple to control traffic flow among cells. The main drawback stems from the fact that only one communication channel exists to service all cells of the network. If a channel between two cells fails, then the entire network is lost. In tree topologies (Fig. 92d) characteristics of bus and star topologies are combined. It consists of groups of star-configured cells connected to a bus. Tree topologies allow for the expansion of an existing network. In mesh topologies (Fig. 92e), each cell is connected to every other cell by a separate wire. This configuration provides redundant path through the network, so if one cell blows up, we do not lose the network. Thus, the full symmetry offers high security. But it demands a high amount of technical effort. Fig. 92f shows a ring

Fig. 93a.

Fully-contracted CNN with 25 cells [7.43]

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Fig. 93b.

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Star CNN with 26 cells [7.43]

topology with less symmetry, because it is only partially meshed. Only critical cells are secured by multiple connections. A star topology can also be applied to neural networks like CNNs [7.42]. A star cellular neural network is a new dynamic nonlinear system defined by connecting N identical dynamical systems, called local cells, with a central system in the shape of a star. All local cells communicate with each other through a central system. Thus, a Star CNN has only N connections from the N local cells to a central system. Since a fully-connected CNN has a mesh topology and self loops, it needs N (N + 1)/2 connections. Fig. 93 shows a fullyconnected CNN with N = 25 cells and a Star CNN with N = 26 cells. The Star CNN has the same bottleneck as that of the star topology.

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However, the Star CNN can be easily implemented in hardware using only N connections, except that a central cell has to supply complicated signals. A Star CNN can store and retrieve complex oscillatory patterns in the forms of synchronized chaotic states like associative memories. Furthermore, the Star CNN can function as dynamic memories. In this case, its output pattern can occasionally travel around the stored patterns, their reserved patterns and new emerging patterns. It is motivated by the observation that a human being’s associative memory is not always static, but sometimes wanders from a certain memory to another memory, one after another. New patterns can emerge like a flash of inspiration which is important for known memories. Changes of memories also sometimes seem to be chaotic. An information storage device is called an associative memory if it permits the recall of information on the basis of a partial knowledge of its contents, but without knowing its storage location. A Star CNN for associative memories usually converges to a stored pattern or to a new one which spontaneously emerge in a flash of inspiration with relation to known memories. The emergence of new patterns can be interpreted as a form of creative activity which is well-known from the human brain. New patterns are usually made up of combinations of stored patterns. Figs. 94a–b shows two new symmetric patterns on the left which are made up of combinations of (right) basic patterns of symmetry. These results are related to the fact that we generally solve problems and generate new ideas by combining various notions, ideas, or memories in our mind, and sometimes have a sudden flash of inspiration [7.44]. Thus, these combinations of symmetric patterns are often used in IQ-tests to measure human intelligence. Without spontaneous memories, the brain would not be capable of learning anything new, and furthermore it would become obsessed with its own strongest memories. Spontaneous memories help the brain avoid this problem and learn something new, albeit similar in some respect to what is already stored in the network. According to the principle of computational equivalence, the evolution of complex dynamical systems in nature and society can be considered evolving computational systems. Actually, computational

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(a)

(b)

Fig. 94a–b. (Left) Spurious patterns which are made up of combinations of (right) basic symmetric patterns [7.45]

networks describe a wide range of dynamical systems in nature and society. For example, a cell is best modeled as a complex network of chemicals connected by chemical reactions. Fads and ideas spread on social networks, whose nodes are human beings whose edges represent various social relationships. Their topology and evolution is governed by principles of self-organization. The development of technical networks seem to continue the evolution of natural and social networks in a kind of technical co-evolution. Evolving computational networks are also characterized by a tendency from symmetry and simplicity to complexity and diversity. Self-organization is a strategy to handle an increasing complexity of data and information which can no longer be programmed, monitored and controlled in all details step by step. Natural evolution has not focused on single organisms with increasing intelligence based on neural information processing. In species and populations, we observe increasing degrees of fitness enabled by increasing capacities of swarm, collective and distributed intelligence with extrasomatic information processing. In sociobiology, populations of ants and termites organize complex transport, information and communication systems through swarm intelligence. There is no central supervisor over the construction of complex networks of paths between their bivouacs. The ordering of the system

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is self-organizing according to chemical signals between thousands of animals. In human history, complex transport and information networks have emerged with more or less self-organizing behavior. Telephone and railway networks are supervised by global control stations, while car traffic in networks of streets depends on the local behavior of drivers. Thus, auto traffic can be considered a complex dynamical system with typical phenomena of oscillation (“stop-and-go”), congestion, and chaos. The capacity to manage the complexity of modern societies depends decisively on an effective communication network. Like the neural nets of biological brains, this network determines the learning capability that can help mankind to survive. In the framework of complex systems, we have to model the dynamics of information technologies spreading in their economic and cultural environment. Thus, we speak of informational and computational ecologies. There are actually realized examples, like those used in airline reservation, bank connections, or research laboratories, which include networks containing many different kinds of computers. Traditionally, complex networks have been studied by graph theory. While graph theory focused on regular and symmetric graphs, large-scale networks with no apparent design principles have been described as random graphs [7.46]. According to the Erd¨ os-R´enyi model, a random network starts with N nodes and connect every pair of nodes with probability p, creating a graph with approximately pN (N − 1)/2 edges distributed randomly. But observations of real complex networks clearly indicate that, for example, the Internet and World Wide Web are neither completely regular and symmetric nor completely random. They are complex systems, and the question arises which principles of self-organization are hidden behind their observed dynamics. Observations lead to the three spectacular quantities of average path length, clustering coefficient, and degree distribution which play a key role in the recent development of complex computational networks. The small-world concept in simple terms describes the fact that, despite their often large size, in most networks there is a relatively short path between any two nodes [7.47]. The distance between

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two nodes is defined as the number of edges along the shortest path connecting them. A popular manifestation of small worlds is the “six degrees of separation” concept, assuming that there is a path of acquaintances with a typical length of about six between most pairs of people in the United States. The average path length of a network is defined as the distance between two nodes, averaged over all pairs of nodes. The diameter of a network is the maximal distance between any pair of its nodes. The emergence of clusters in networks was also at first observed in social systems [7.48]. Cliques organize themselves, representing circles of acquaintances in which every member knows every other member. A selected node i in the network have ki edges which connect it to ki other nodes. If the nearest neighbors of the original node were part of a cluster, there would be ki (ki − 1)/2 edges between them. The ratio between the number Ei of edges between these ki nodes and the total number ki (ki − 1)/2 gives the value of the clustering coefficient of node i, which is Ci = 2Ei /ki (ki − 1). The clustering coefficient of the whole network is the average of all individual Ci ’s. In a random network, since the edges are distributed randomly, the clustering coefficient is C = p. In most real complex networks the clustering coefficient is typically much larger than it is in a comparable random network which has the same number of nodes and edges as the real network. The degree distribution is motivated by the observation that not all nodes in a network have the same number of edges (node degree). The spread in the node degrees is characterized by a distribution function P (k), which gives the probability that a randomly selected node has exactly k edges. Since in a random graph the edges are placed randomly, the majority of nodes has approximately the same degree k of the network. The degree distribution of a random graph is a Poisson distribution with a peak at P (k). But for most large networks the degree distribution deviates from a Poisson distribution. Actually, the degree distribution of, for example, the Internet and the World Wide Web has a power law tail P (k) ∼ k−γ . Since power laws are free of a characteristic scale, such networks are called scale free.

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One of the first examples of large networks in modern civilization was phone call networks. A large directed graph was constructed from long-distance telephone call patterns, where nodes are phone numbers and every completed phone call is an edge, directed from the caller to the receiver. In call graph of long-distance telephone calls made during a single day, the degree distributions of the outgoing and incoming egdes followed a power law with exponent γout = γin = 2.1. The sustained, explosive growth of the Internet and the World Wide Web over the past decade has made them a part of globalization. They have changed the way we do business, communication, entertainment, education and culture. In an increasing flood of information, information retrieval is a challenge of information and computer technology. The Internet is a network of physical links between computers and other telecommunication devices (Fig. 95a). The topology of the Internet has been studied at two different levels. At the router level the nodes are the routers, and edges are the physical connections between them. At the interdomain (or autonomous system) level, each domain, composed of hundreds of routers and computers, is represented by a single node, and the edge is drawn between two domains if there is at least one route that connects them. In each case, the degree distribution follows a power law. The interdomain topology of the Internet, captured on three different days between 1997 and the end of 1998, resulted in a degree exponent γId ≈ 2.2. A 1995 survey of Internet topology at the router level, containing 3888 nodes, found γIr ≈ 2.48. In a 2000 investigation with a connectivity of nearly 150 000 router interfaces and nearly 200 000 router adjacencies, the power law scaling was confirmed with γIr ≈ 2.3. Furthermore, the Internet as a computational network displays clustering and small path length as well. Between 1997 and 1999, the clustering coefficient of the Internet ranged between CI = 0.18 and CI = 0.3, to compare with Crand ≈ 0.001 for random networks of similar parameters. The average path length of the Internet at the domain level was found to be about 4, compared to Lrand ≈ 10 for the corresponding random graph. At the router level, it was around 9, indicating its small-world character.

The Internet is designed to operate over different underlying communication technologies, to support multiple and evolving applications and services. Thus, it is a computational network of vast diversity, combining different kinds of networks. In order to communicate across different kinds of networks, an Internet Protocol (IP) was introduced. The IP number codes the networks which an information

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(a) Fig. 95.

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(b)

Typology of the Internet (a) and the World Wide Web (b) [7.49]

packet has to pass through. The Internet is a computational network of routers that navigate information packets from one computer to another. Contrary to phone call networks, there is no fixed connection between a sender and receiver. Communication is divided into information packets of byte size that are transported in the networks of routers by packet switching. The routers are the nodes of the network determining the local path of each packet by using local routing tables with cost metrics for neighboring routers. A router forwards each packet to a neighboring router, at the lowest cost, to the destination [7.50]. In the sense of the CA and CNN paradigms, the local routing tables can be considered “templates” of local nonlinear information processing. As a router can only deal with one packet at a time, other arriving packets must be stored in a buffer. If more packets arrive than a buffer can store, the router discards the overflowing packets. Senders of packets wait for a confirmation message from the destination host. These buffering and resending activities of routers can cause congestion in the Internet. Fluctuations of information packet congestions can be indirectly observed through echo experiments of control messages between neighboring routers. A monitoring host between two routers periodically sends a series of echo packets to both routers. The packets take a round-trip time (RTT) to the destination and back. Congestion is associated with higher RTT values. RTT fluctuations increase with the sequence of routers in the Internet network [7.51]. In automobile traffic systems, a phase transition from nonjamming to jamming depends on the average car density as the con-

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trol parameter. At a critical value, fluctuations with self-similarity and power law distribution can be observed. From this analogy, a control parameter of data density is defined by the propagation of congestion from a router to neighboring routers and the dissolution of the congestion at each router. The cumulative distribution of congestion duration is an order parameter of pattern formation [7.52]. There are phase transitions between spare and congestion phases. The spare phase corresponds to a case in which the mean input of the information system is smaller than the maximum output. The critical point condition is when the mean input rate is equal to the maximum rate. At a critical point, when the congestion propagation rate is equal to congestion dissolution, fractal and chaotic features can be observed in data flow. On different scales of time series analysis, we can analyze the self-similarity of the information packet’s fluctuations, which is a necessary (not sufficient) condition of strange attractors (Fig. 96). Symmetry as self-similarity is hidden behind the diversity and heterogeneity of the Internet. The World Wide Web (WWW) is the largest computational network for which topological information is currently available. The nodes of the network are the documents (web pages) and the edges are the hyperlinks (URLs) that point from one document to another (Fig. 95b). The size of this network was close to one billion nodes at the end of 1999. The degree distribution of the web pages follows a power law over several orders of magnitude. Since the edges of the World Wide Web are directed, the network is characterized by two degree distributions. The distribution of outgoing edges, Pout (k), signifies the probability that a document has k outgoing hyperlinks, and the distribution of incoming edges, Pin (k), is the probability that k hyperlinks point to a certain document. Both the in- and out-degree distributions are found to be in a power law form: Pin (k) ∼ k−γin and Pout (k) ∼ k−γout . Studies with different subsets of the WWW have showed that the in-degree distribution of the WWW is γin ≈ 2.1, while its out-degree distribution is ranged somewhere in between γout ≈ 2.38 and γout ≈ 2.72.

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Fig. 96.

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Self-similarity of data traffic [7.53]

The directed graph of the WWW does not allow to measure the clustering coefficient. One way to avoid the difficulty is to let the network be undirected, making each edge bidirectional. It was found that the clustering coefficient is much higher than that of a random

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graph of the same sizes and edges, although it is still significantly less than 1. Despite the large number of nodes, the World Wide Web also displays the small world property. The network structure plays a crucial role in determining the spread of ideas, innovations, or computer viruses. In this light, spreading and diffusion has been studied on several types of complex networks regular, random, smallworld and scale-free. It was shown that while for random networks a local infection (“butterfly effect”) spreads to the whole network only if the spreading rate is larger than a critical value, for scalefree networks any spreading rate leads to the infection of the whole network. That is to say, for scale-free networks the critical spreading rate reduces to zero. The information flood in the more or less chaotic World Wide Web is a challenge for intelligent information retrieval [7.54]. Information Retrieval (IR) in the WWW can be considered a decisive procedure for evaluating and selecting the most relevant documents and information according to certain constraints. Procedures of IR are also inspired by natural evolution. There are applications of, for example, genetic algorithms, in order to improve information retrieval. Genetic algorithms optimize populations of chromosomes in sequential generations by reproduction, mutation, and selection. In information retrieval, they are used for optimizing the queries of documents. Information retrieval is also realized by neural networks adapting with synaptic plasticity to the information preferences of human users. In sociobiology, we can learn from populations of ants and termites how to organize traffic and information processing through swarm intelligence. From a technical point of view, we need intelligent programs distributed throughout the nets. There are already more or less intelligent virtual organisms (agents), learning, self-organizing, and adapting to our individual preferences of information, to select our e-mails, to prepare economic transactions, or to defend against the attacks of hostile computer viruses, like the human immune system. Virtual agents are designed with different degrees of autonomy, mobility, reactivity, and learning capabilities

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for communicating. They communicate and cooperate with their virtual environment as local spheres of influence. Global networking is becoming one of the exciting challenges of complexity research. Understanding complex systems in nature and society supports the effective management of communication networks. In the 21st century, information, communication, and biotechnology are growing together. Therefore, information processing requires learning from nature. Information can be generated, transmitted, stored, processed, and represented in nature by sense organs, the nervous system, and the brain. Cognitive processes like learning and thinking, language, motorics, perception, and communication, are simulated using technology by physical, chemical, and biological sensors, light-wave conductors, electronic, optical stores, microprocessors, neural nets, robotics, virtual reality, ubiquitous computing, artificial life and intelligence [7.55]. Together they aim at producing learning, adapting, and self-organizing evolutionary complex systems. Therefore, this approach is called organic computing [7.56]. An exciting example of organic computing is the evolutionary architecture of future automobiles, integrating all aspects of complexity and self-organization. The automobile industry is still one of the driving and dominating engines of the global economy. Thus, complexity research finds a realistic application in the production of future cars as learning, adapting, and self-organizing evolutionary complex systems. A challenge of the automobile industry is the increasing complexity of electronic systems. If we consider the electronic cable systems of automobiles from the beginning through to today, there will be a surprising similarity to neural networks of organisms which increase in complexity during evolution. Contrary to biological evolution, electronic systems of today are rigid, compact, and inflexible. So tiny failures can lead to a collapse of the whole system. In an evolutionary architecture (EvoArch) the nervous system of an automobile is divided into autonomous units (carlets) which can configurate themselves in cooperative functions in order to solve intelligent tasks. Examples of this are the complex functions of motor, brake and light, wireless guide systems like GPS, smart de-

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vices for information processing, and the electronic infrastructure of entertainment. According to the complex systems approach, the functions of a car are considered as macro-features which emerge from self-organizing interactions and the cooperation of autonomous units on the microlevel. Examples of autonomous units of a car (carlets) are: switches, lamps, tuners, controllers, regulators, horns. A car function like “air conditioning”, “turn signal” or “hazard warning” needs one or more switches which must be selected from among more than a hundred candidates. Actually, a car function like “turn signal” needs carlets for a turn signal switch, a terminal switch, turn signal flashing, and several turn signal lamps. In an evolutionary architecture, cooperations are realized in the EvoArch-arena (Fig. 97), where active autonomous units ask for cooperation with passive autonomous units which have the appropriate features to execute a car function. Each unit (carlet) has an ID-number for self-identification. It can declare its property (e.g. turn signal) and its intention (e.g. search for a switch). The interaction of units (carlets) is made possible by a communication system (carCom) with information retrieval procedures, protocols, and contracts of cooperation which are wellknown in the Internet like RMI (Remote Method Invocation) and RPC (Remote Procedure Call). As in the Internet, the networkmanagement is based on the middleware of routing-procedures with routing-protocols and routing-tables. According to the principle of computational equivalence, a car is an example of a dynamical system which can be considered an information and computational system. The increasing diversity and complexity of electronics must be managed by self-organization like in organisms. Cars have been typical products of classical industries. In industrial societies, economies have been characterized by the steps of production, logistics, distribution, marketing and sale of material goods. In computer-assisted information societies, there is an offer and demand of virtual information products and services with steps of information collection, information systematization, information retrieval, production and trade of information-based systems. Therefore, economists distinguish material chains of value in

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Fig. 97.

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Self-organization of car functions in an evolutionary architecture [7.57]

industrial societies and virtual chains of value in information societies. According to Shannon, the content of information goods is measured by the degree of news for a receiver. But it is not sufficient to be well informed in order to handle our affairs. In the next step, information of high value must be evaluated and applied to solve problems. Information must be transformed to knowledge in the sense of know-how for problem solving. Besides matter and life, the chief ingredients of the 21st century are in- formation and knowledge. In a knowledge society, science is a productive power of economic and social growth which needs new strategies of cooperation with economy and politics. The “wealth of nations” (Smith) is the knowledge of their people. Therefore in the process of globalization with competing nations and societies, education must secure the sustainable future of the knowledge society. The technical evolution of computational systems for information and knowledge processing is the fundamental challenge of mankind in the 21st century. Humans will no longer be only products of a blind evolution, but will try to influence their development by use of computational tools.

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Chapter 8

Symmetry and Complexity in Philosophy and Arts

The message of this book is easy to understand: cosmic evolution leads from symmetry to complexity by symmetry breaking and phase transitions. The emergence of new order and structure is explained by physical, chemical, biological and social self-organization, according to the laws of nonlinear dynamics. All these dynamical systems are considered computational systems processing information and entropy. Because of symmetry, no information is lost, although it is sometimes hidden from us. From a philosophical point of view, the question arises as to whether symmetry and complexity are only epistemic projections and models of science or whether they can be understood as universals of reality. In the Platonic tradition, symmetry was even unified with truth and beauty. In modern civilizations, the unity is broken and has been transformed into diversity and heterogeneity. Therefore, in the last chapter, the development of arts from symmetry to complexity is considered in the spirit of nonlinear science.

8.1

The Philosophy of Symmetry and Complexity

Philosophy is the mother of science. It deals with the origin, principles, and universals of knowledge. Since the days of Plato and Aristotle knowledge has expanded into complexity and diversity. Science has split off and specialized in a manifold of disciplines and subdisciplines. Obviously science follows the tendency of professionalization which we discussed in the context of globalization. The growth of 329

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knowledge is embedded in the dynamics of information. But the children seem to forget the origin from their mother. Philosophy is sometimes only a historical relic reminding scientists of their common origin, like background radiation as cosmic trace of the big bang and common origin of the universe’s diversity. Specialization is a necessary tendency of modern science. But the splitting of knowledge endangers the common orientation and overview of scientists. Actually, since antique philosophy, mankind has generated a subtle web of knowledge with meshes of increasing granulation. Even today all knowledge is connected, although the connections are often hidden and not easy to find, like on a complex map of streets and paths in a modern city. The map models a landscape of knowledge with valleys, hills and high mountains. In the valleys, there are the fruitful greens of experience, data, and laboratories, near to the bottom of reality. The hills and mountains represent more or less abstract concepts and theories with more or less distance to reality. More or less applied disciplines are linked like the summits of different heights in a mountain-range. On the top of high mountains, with very general principles and universals, we have a wide view over-looking many other disciplines and their network of connections, but in the thin and clear air of abstraction. This is the area where philosophers feel at home. Philosophers are specialists for principles and universals of knowledge. In this sense, they are part of science, on the top of some hills and mountains, sometimes wandering into the valleys to test their general view and to study concrete models. But philosophy should not be in the clouds without connection to experience and science. On the other hand, science should not be encapsulated in some deep valleys without orientation and connection to the rest of the world. Aristotle was one of the first engineers of knowledge who designed classifications of more or less abstract concepts in hierarchies of knowledge. On the top of his tree-like hierarchies, there are the general principles of entities which are divided into less abstract subconcepts which lead to the concrete concepts of observation and applied sciences. The Aristotelean taxonomies are called ontology which is still a term of informatics for knowledge classifications in,

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for example, databases [8.1]. In modern times, knowledge has specialized, but, nevertheless, deals with general principles. Newton called his main book “Philosophiae naturalis principia mathematica” (Mathematical Principles of Natural Philosophy). He analyzed the general principles of force and mass with mathematical methods and founded the physical theory of dynamics. It is noteworthy that he had a chair for natural philosophy. His famous countryman Smith had a chair for moral philosophy and established the economic theory of sociodynamics. Einstein founded the principles of space-time, and physicists like Bohr, Dirac or Schr¨ odinger analyzed the quantum principles of matter and energy. Today, thousands of scientists are engaged in more or less theoretical research and search for the principles of matter, life, the mind, economies and societies. Therefore, they all are connected with philosophical principles in the web of knowledge. Symmetry and complexity are general universals of structures and systems. It is amazing that it is not only our image of nature that is described by means of ever more complex structures, but human culture and society as well. In the course of modern time institutions, industries, markets, social roles, etc., have achieved such complexity and interconnectedness that the resulting profusion of information can scarcely be mastered any more. Just as reductions in complexity are often what make knowledge about nature possible, our social and political behavior require that we make simplifications and reductions so that the complexity of political, economic and social reality will not render us helpless. Goethe said: to take action one must be without a conscience. To know anything, one must leave out a piece of the truth. Our contemporary image of nature is based on abstract structural species that are linked by a complicated net of pre-theories, observations, instruments of observation and work in laboratories. Structures are mathematical concepts of, for example, abstract sets or spaces which are characterized by relations, transformations or operations. Thus, a question arises as to whether they are merely tools for thought based on axioms for ordering measurement data, or whether they in any way provide information about structures

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of nature. In the history of philosophy this question as to the status of structures and symmetries is clearly in the tradition of the quarrel of the universals that was carried out on the eve of modern philosophy and has subliminally determined the discussions on the foundations of logic, mathematics and the natural sciences ever since [8.2]. The example of the concept of structure and symmetry shows the scale of possible positions in the quarrel of the universals — from heavily realistic-ontological presuppositions to nominalism and positivism. Platonic ontology makes the most ambitious claim. With regard to the problem of universals one could summarize it in the expression: “Symmetria est ante res” (symmetry is behind the things). It holds that symmetrical structures are the real realities, and that we perceive breaks of symmetry as appearances and “shadows”. Plato presupposes that perfect and ideal regular bodies are the building blocks of matter, not approximative models as they occur in some crystals, for example. In the Christian-Augustinian tradition the Platonic ideas become the thoughts of God, which give nature its laws. The ontological-Platonic conception of natural laws is also found in the early mathematical physicists such as Galileo and Kepler. The physical world is conceived of as a second scripture (book of nature) side by side with the Holy Scriptures; God reveals himself to human beings through both. The book of nature is written in the language of mathematics, so that in consequence the laws of nature can be grasped only by one who masters this language. By contrast, the point of view of the Aristotelian philosophy of nature can be summarized in the expression: “Symmetria est in rebus” (symmetry is in the things). The multifariousness of being is actualized in the Aristotelian hierarchy of substantial forms. The pure possibility of matter becomes actuality via intermediate stages. According to Aristotle, the distinction between matter and form is only an abstraction we employ to describe the motion of matter. If one conceives of structures as Aristotelian forms, they are “in things,” (figuratively speaking). Thus they do not exist separately from matter. Instead it is in the motions of matter that structures, as potentialities, are actualized.

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In the age of mechanics the Aristotelian doctrine of forms was often misunderstood and was vehemently attacked as an obstacle on the way to mathematical physics. Leibniz is an exception. He interpreted substantial forms as the new mathematical laws of nature. Heisenberg interpreted the operators of quantum mechanics as potentialities and related them to the Aristotelian doctrine of forms. Weyl offers an epistemic interpretation of symmetry: the invariance of natural laws shows that their validity is independent of the different frames of reference of different observers. In this sense invariance shows the intersubjective validity of natural laws (categories): “symmetria est in mente” (symmetry is in mind). According to Kant, the forms of natural laws (categories) are already pregiven through our subjective constitution of cognition. Only in this way is it possible for us to formulate natural laws at all. In speaking of natural laws Kant uses a typically political metaphor of the Enlightenment: we human beings do not recognize ontologically alleged natural laws as thoughts of God. Instead, we ourselves are “lawgivers of nature” in the framework of the constitution of our reason. Besides making our own laws within the framework of our political constitutions, we also achieve autonomy vis-a-vis nature. Thus structures are products of reason, intuition and imagination and are applied according to categorial schemata for the purpose of giving order to the diversity of perceptual phenomena by means of physical “images” (sic Kant!). The nominalistic view appears in a philosophically sharper form in the conventionalistic and instrumentalistic orientations. In these orientations symmetry assumptions, characterized mathematically only by their simple and transparent formula, must prove their worth physically in the explanation of measurement data or for purposes of prognosis. Regarded this way, they are at best appropriate formulas of mathematical formalisms. At first glance the advantage of this position seems to be that mathematical symmetry structures are not associated with symmetrical entities. Therefore symmetry is not bought at a high price of ontological assumptions. To some extent instrumentalism wants to shop for the advantages of symmetry assumptions at an ontological discount: “symmetria est vox”

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(symmetry is sound [of a word, symbol or formula]), one might add a characterization of nominalism (from Latin “nomen” for name) in the Middle Ages. Philosophically, the situation has not changed since the days of the controversy about universals. Yet today the logical-mathematical methods are sharper, the results of measurement more exact. For that reason symmetry can be made mathematically precise as a canonical universal (“invariance property”). This is a matter of automorphism groups, as we have seen from many examples in this book. After that, however, the philosophical discussion begins again. Is this structural species a separate immaterial identity “before [all] things” as is assumed in Platonic tradition? Is it a structure of reality (“in things”), which we must presuppose in order to be able to speak mathematically about symmetry in nature? Should we use Occam’s razor to cut off the superfluous Platonic creation of entities and confine ourselves to introducing mathematical structures only as useful and simple instruments for mastering nature? Now physicists do establish relationships between empirical measured data (for instance time and position coordinates) by means of transformations. Consequently the following objection was soon raised to traditional nominalism which claimed only concrete measurements and observations as statements about reality: individual measured values cannot be thought of without presupposing a “general” one, namely their relationship to other measured values. From the standpoint of mathematics this objection views what is assumed to be “general” as a mathematical function or relation. In quantum mechanics the situation is even more complicated. There the measured quantities (“observables”) are already abstract mathematical objects, namely operators over a Hilbert space (thus a function space and not a number space). What is the origin of mathematical structures? Structures are familiar to us from everyday life. In perception we register a figure as a totality. In geometry we decompose it into a set of points; for example, we distinguish straight lines and curves as subsets of the whole point set and particular sections, angles, parallels, etc., as relations

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between these objects, by establishing their characteristics in axioms and definitions. Such a system of sets, subsets and relations is a simple example of a structure. A population of living organisms can also be grasped as a structure that is determined by a relational system of kinship relationships, functional tasks, etc. Likewise an ecosystem such as a forest consists of a system of organisms and populations that are structured by a complicated network of relations such as food chains. As we have seen, a molecule or a crystal is described by a structure that consists of a set of elements (atoms) among which relations of sequence, spacing, etc. are defined. Different objects can be examples of the same structure, as is demonstrated by the group structure of molecules. Thus structures provide the possibility of classifying the complex variety of appearances into units and wholes and of making them easy to overview. In the logical set-theoretical language of modern mathematics there is, in principle at least, no difficulty in defining and classifying structures. On the basis of an axiomatic set theory (for example, according to Zermelo–Fraenkel = ZF), structures are introduced through sets or systems of sets and relations are defined for their elements [8.3]. Relations are themselves sets of ordered pairs or general n-tuples of the basic elements. Thus the 2-tupel relationship “being married” consists of the set of all couples in the assumed set of persons who are married to each other. Likewise the 3-tupel group relationship consists of a set of ordered triples of elements that fulfill the axiomatically defined group characteristics. As has been shown, we can imagine these elements as being actualized in completely different ways, for example, as two rotations in space that are carried out in succession and that together result in a third rotation, but also, for example, as two numbers that are added and provide the result of addition as the third number. On the basis of an axiomatic set theory (e.g. ZF), mathematics as a whole can be understood as the theory of abstract structures. The basic set-theoretical relation x ∈ X denotes that an element x is element of a set X. Mathematical theories are concerned with the various kinds of structures that are introduced in set theory and can be classified in a coherent manner. That is related to the fact that set

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theory, together with a standard logic, also postulates strong nonlogical axioms about sets, for instance, that for every set X there exists also the power set Pot(X) as the set of all subsets of X and that there are infinitely many sets. For a set X the Cartesian product X 2 = X × X can be defined as the set of all pairs of elements of X (in general the set X n as the set of all n-tuples of elements of X). In general a structure is a finite system of sets whose type and their species are determined axiomatically. Thus a group (G, g) is a structure with a basis set G (e.g. real numbers) and a 3-tupel relation g on G, with the typification g ∈ Pot(X 3 ). (If g is an element of the set of all subsets of X 3 , then it is by definition a 3tupel relation.) The structural species is defined by the group axiom α(G, g) according to which, for example, the operation on G defined by g fulfills the axiom of the inverse element [8.4]. A set fulfilling the conditions of a certain structure (e.g. real numbers of a group) is called a model of the structure. What seems so abstract at a first glance provides us with a decisive advantage for the theory of science. Namely, we obtain a single linguistic framework for formulating with logical precision the enormous multiplicity of all thinkable structures, theories, models, and their dependencies. This makes available a coherent framework of all mathematized theories. If it is also possible to connect these structures by means of appropriate mapping principles with experiments and measurements, then even a general framework of the empirical sciences would be at hand. At first this program in philosophy of science seems to recall logical empirism. In the view of logical empiricists like R. Carnap, a scientific theory is a set of sentences, defined as the class of logical consequences of a smaller set, the axioms, laws or hypotheses of that theory, which are assumed to be true. Thus, a language of formal logic is needed to formulate a theory [8.5]. Actually, mathematicians, natural and social scientists are not interested in their language, but in the specific objects, structures and models of their theories. In order to present a theory of a specific symmetry, we define the mathematical structure of the symmetry and analyze the class of its models in, for example, physics, chemistry, biology, or social sciences with an

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informal natural language [8.6]. In short: a scientific theory is identified with its structure or class of models. Therefore, this approach is called structuralism or semantic view [8.7]. As an example, consider Newton’s theory of gravitation. Its basic sets are a set T of points of Newton’s absolute time, a set M of points of Newton’s absolute space and a set P of bodies. These sets are each individually structured by time metrics and space metrics or functions of masses. In kinematics a connection between points of time, space points, and bodies is produced in such a way that at a specific time a specific body assumes a specific location. Therefore, kinematics can be typified as the structural element kin ∈ Pot(T × M × P ). Its structural species is determined by the dynamic law of Newton’s theory of gravitation, the equations of gravitation. The gravitational equations constitute a system of differential equations for real functions, i.e. the motions of bodies in space and time are mapped onto coordinate systems in real numbers. The place of physical structure is taken by an isomorphic structure of number sets in which the physical relationships are mapped. Therefore the corresponding axiom α(T, M, P ;. . . kin. . . ) about the structural species would express that there is a real coordinate system f in which the basis sets T of time, M of space, P of the set of solids, and the structure elements such as kin are mapped and the corresponding differential equations with secondary conditions hold. Obviously this type of structure differs from group structure only in its greater complexity [8.8].

In order to classify structures by features of symmetry, we subdivide structures (X, s) into basic sets X (abbreviation for X1 , . . . , Xn ) and stuctural elements s (abbreviation for s1 , . . . , sm ). The structural type s ∈ σ(X) is established by a ladder set on X, i.e. a set that comes from X by iteration of the operation power set of a Cartesian product. The structural species of (X, s) is established by an axiom α(X, s), which determines the structure uniquely with respect to an isomorphism: if (X, s) is isomorphic to (X  , s ), then α(X, s) and α(X  , s ) are logically equivalent. This requirement imposed on the structural species says that the axiom α does not change its truth value if one replaces the structure (X, s) with an arbitrary structure (X  , s ) that is isomorphic to it. Isomorphisms define an equivalence relation. Thus, the class of structures is divided into equivalence classes with respect to isomorphisms. For example, the group axioms are valid for the rotations of an equilateral triangle as well as for the real numbers. The axioms of the

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Newtonian theory of gravitation are valid for artificial satellite orbits as well as for planetary orbits of the solar system. Isomorphisms are one-to-one (“bijective”) mappings of the basis sets X onto the basis sets X  , whereby the typified set s is mapped onto the corresponding set s . The typification in that case remains unchanged by this, since the corresponding copy is given by the ladder set σ(X). Obviously, the general definition of a structure with respect to isomorphisms implies an invariance postulate, which we shall characterize in the following as the canonical invariance of a structure. It can be shown in detail that the various symmetry characteristics that we elucidated in previous chapters, using examples from natural and social science, can be generally derived from the canonical invariance of a structure. On that subject, let us remember F. Klein’s characterization of geometry by means of group theory. Let M be the space of the geometry in question and G a transformation group of the real number space Rn . Then (M, F ) is a structure with a typified set F ∈ Pot2 (M × Rn ) of cooordinate systems and the structural species αG (M, F ) wherein the axiom αG formulates that F is a set of global coordinate systems of M over Rn that is complete with respect to G. The canonical invariance can easily be proven. Now one can set up a hierarchy of transformation groups of Rn and investigate the corresponding geometrical structures.

Many physical theories can be introduced as an extension of geometric structures. Thus the Poisson–Newtonian theory of gravitation is an extension of Euclidean geometry, in which a gravitational field and a mass density which satisfy the Poisson–Newtonian gravitational equation are added to space and its Euclidean coordinate systems. Likewise relativistic field theory can be considered as an extension of Minkowski geometry. In each case the extensions are achieved by the specification of supplemental structural types and axioms for structural species. For example, planetary orbits or elementary particles appear as new structural types in the examples mentioned. The canonical invariance of the extended structure must be guaranteed. Analogously, quantum mechanics can also be introduced by means of stepwise structural extensions in which the Schr¨odinger equation

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is only the final step. For that purpose one begins with an Abelian group G with an addition which is expanded into a complex vector space by adding a scalar multiplication. Introducing a metric turns it into a Hilbert space. Next comes a self-adjoint linear operˆ (“Hamiltonian”) over the Hilbert space and the ψ-functions ator H (“states”) with ψ : R → G, which satisfy the Schr¨ odinger equation. The structural species introduced in this way is canonically invariant. The familiar invariance of the Schr¨ odinger equation with respect to ˆ  = U HU ˆ −1 the unitary transformation U with ψ  (t) = U ψ(t) and H is a special case of it. Symmetries are examples of invariant structures with respect to automorphism groups. They are one-to-one functions mapping structures onto their own domain and leaving all relevant structure intact. In the sense of canonical invariance, automorphisms define equivalence relations dividing structures and their models into equivalence classes of partitions with the same symmetric structure. In the previous chapters we analyzed several structures of symmetry with different models in science, which are collected in Fig. 98: we started with the (assumed) structure of a supersymmetry with models of strings and p-branes which until now is only sketched in M-theory and not yet completely known. The theory of relativity is characterized by space-time symmetries with models of manifolds and trajectories of photons and gravitons. Conservation laws are consequences of space-time symmetries. Quantum field theories provide dynamical gauge symmetries (e.g. electroweak SU(2)×U(1)-force) as substructures of the unified supersymmetry with models of different elementary particles. Therefore, in a bottom up resp. top down approach of Fig. 98, we get symmetry reduction resp. symmetry breaking. Chemistry is characterized by structures of symmetry with models of atoms, molecules and crystals. In biochemistry, at the boarderline of life, we found typical structures of asymmetry with models of DNA. In thermodynamics, the emergence of symmetric patterns by phase transitions and symmetry breaking provides models of symmetric structures. Structures of functional symmetries characterize models of organisms in biology. Ecological, economic and social balance deliver models of symmetry in ecology, economics and sociology.

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theories

models

symmetries

string theories theory of relativity quantum field theories

strings, p-branes gravitons, photons hadrons, leptons, etc.

chemistry biochemistry thermodynamics biology ecology economics

atoms, molecules, etc. proteins, DNA, etc. open systems with metabolism organisms populations economies, markets

supersymmetry space-time symmetries dynamical symmetries (e.g., SU(3)xSU(2)xU(1)-forces) nuclear-, orbital-, crystal symmetries chirality dissipative structures of symmetry

sociology

societies

Fig. 98.

functional symmetries ecological balance economic balance, economic equilibrium social balance, social equilibrium

symmetry breaking

theory reduction

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But, symmetry breaking of balance is also a characteristic structural process in these disciplines. In a bottom up approach, Fig. 98 seems to suggest a theory reduction of the social and biological sciences to physics. But, that is only true in the sense that the emergence of social, biological, and physical structures is embedded in cosmic evolution which started with a supersymmetry, according to M-symmetry. Canonical invariance and symmetries of structures do not only help to classify scientific theories, but they also support problem solving in science and situations of the everyday world. If two situations are isomorphic with respect to their essential features and if we were successful in one of these situations with a certain strategy, then we should apply this strategy in the other situation again. Isolating the relevant structures is equivalent to defining the set of transformations that leave the problem essentially the same. These transformations are the symmetries of the problem. Therefore, problems which are essentially the same must have essentially the same solution. In this sense, symmetry is a successful principle of methodology. With specialization of science, theories depend on one another in a network of increasing complexity. The belief of logical empirism that isolated theories are tested by “naked” facts of reality becomes an illusion. There is no absolute empirical basis with sensory and measured data, propositions of protocol in a language of observation which would be linked to the theory by rules of correspon-

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dence. Measurements and observations already depend on theoretical assumptions. The domain of reality and the rules of application of a mathematical theory generally depend on the theory itself. The part of a domain of reality that is independent of the concerning theory and its rules of application is called the basic domain of the theory. However, this basic domain is not independent of all science and experience. For example, current does not belong to the basic domain of electrodynamics, since it was first defined in this theory. But mechanical forces are introduced in mechanics and belong in the basic domain of electrodynamics. For that reason mechanics is called a pre-theory of electrodynamics [8.9]. For the Newtonian theory of gravitation the orbit of a satellite belongs to the basic domain. It can be determined by a pre-theory that includes geometrical optics and terrestrial geometry and is independent of gravitational theory. Therefore, a pre-history is pre-given a priori relative to its theory. In this sense it is the task of philosophy of science to reconstruct the network of relative dependencies of theories. As all theories depend more or less on one another, the whole network of knowledge is always faced with reality. We have to decide on the restrictions and constraints for concrete observations, tests and experiments. If a mathematical theory is applied to a domain of reality, then mathematical models are mapped on data of measurements. An example is the results of measurement of planetary orbits according to which a planet is at a particular location at a particular time. Factual samples of measurement data provide data models which are mapped on the mathematical model of an elliptic curve. The elliptic model of planets belongs to the Newtonian theory of gravitation. A theory is confirmed if those real systems like the planets belong to the class of models of the theory. Even in Newtonian time, the theory of gravitation contained a well-known class of real models like the planetary system, high tide and low tide of the ocean, and Galileo’s free fall of bodies on earth. But, the mathematical structure of a theory obviously contains many more models than possible worlds. Thus, theory construction should be done from two points of view: the construction of sufficiently rich models to allow for the

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possibility of described phenomena, and the narrowing down of the system of models so as to give the theory greater empirical content. In theory constructing, there must be a steady interplay between the theoreticians and experimenters. A structural analysis reveals the conditions under which one theory provides more information than another about a domain of reality, and thereby also more solutions to problems. These criteria do not depend upon whether a research group finds a theoretical development to be “better” or “worse”. Rather it is a matter of exactly defining when a structure and the theory characterizing it, is more information-rich and more comprehensive than another one. In scientific practice a case can definitely occur in which one chooses the theory that is structurally poorer because under certain research constraints it provides adequate and fast problem solutions. A structure is called richer than another if both structures possess the same basic set and the same typification, but the axioms of the richer structural species include those of the poorer structural species [8.10]. A mathematical example is the transition from an ordered set to a lattice structure. Both have the same underlying basic set and order relation as their structural type. The structural species of the lattice structure requires more axioms for this structural type than the usual axioms of order. A structure is more comprehensive than another if, in addition, new basic sets, structural types and structural species are added to it. An example of that is the above-mentioned step-by-step development of quantum mechanics, which finally extends from an Abelian group over Hilbert spaces to the self-adjoint operators of the Schr¨ odinger equation. Correspondingly, one theory is called richer in structure than another if the first is determined by a richer structure than the other, but both have the same principal basic domain and the same mapping instructions for the application of the mathematical theory. Thus, the more structurally rich theory makes it possible to make more statements and thereby to provide more information about the same facts than the other theory does. The structurally richer theory is therefore a special case of a more comprehensive theory.

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Further operations known from mathematical structural analysis can be carried over without modification. Thus, one speaks of one structure being “embedded” in another one by means of corresponding mappings. In the same way one structure can be “restricted” to another one by means of corresponding rules. Two theories are called equivalent if they refer to the same principal basic domain and if both can be called reciprocally more comprehensive. Examples from the history of science are at hand. Thus, the transition from the Newtonian space-time theory to Einstein’s is obviously a transition from a less comprehensive theory to a more comprehensive theory. Sometimes the assumption of absolute simultaneity in Newtonian space-time and its negation in Minkowski geometry is depicted as an unbridgeable contradiction that evokes the impression of erratic theoretical progress. But, from the point of view of a mathematical structural analysis, this is misleading. In fact, Newtonian space-time is not false (from the point of view of Minkowski geometry). Einstein’s theory, namely, can be restricted to a spacetime theory with inertial systems that move slowly, compared with the speed of light, with respect to the Newtonian inertial system of the planetary system. Moreover, these subsets of inertial systems are not, in any case, spread out over too much of the cosmos. Thus restricted, Einstein’s space-time theory can now be embedded in the Newtonian theory. Besides, there is at least an approximate region of absolute simultaneity in which the sun does not move, or moves only very slowly. The fact that one theory structure is richer or even more comprehensive than another one thus proves to be an objective relationship between theories that is precisely defineable in logical-mathematical terms. Such a theoretical transition is therefore just as cumulative in natural science (such as physics) as in mathematics, as far as increase in complexity, information content and capacity for problem solving are concerned. Thus, one can talk about “upheaval” and “revolution” in psychological, sociological and ideological contexts only where such structural expansions have historically taken place. This applies to the Copernican change as well as to the historical, philosophical discussion that has been going on since Einstein’s

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introduction of the relativity theory in the twenties of the last century. Indeed, after the First World War many people felt that Einstein’s relativistic revision of the Newtonian conception of space-time was the collapse of an old world that had had absolute standards: “everything is relative” was a popular slogan in an era of disintegrating values and may have furnished the ideology for a greater acceptance of Einstein’s theory by some people or increased reservations and rejection by others. Even today “postmodern” philosophers demand “relativism” and “destruction” of scientific objectivity [8.11]. But again, that is a psychological expression of critical feelings in an era of sceptical attitudes against scientific and technological progress. “Postmodern relativism” does not contradict the structural analysis of mathematical and mathematized theories [8.12]. However, the examples also show that a more comprehensive theory is not necessarily a better one. In many areas of technology — such as automobiles — where we look at slow speeds compared to the speed of light, we are working successfully using classical mechanics. For other areas — such as high-energy — that is no longer true. A unified theory of all physical forces promises spectacular insights into the ultimate solutions of our problems of energy. The analysis of theoretical structures seems to suggest a static view of science. Therefore, a model of elements developing in time (e.g. atoms, planets, people) is called a dynamical system [8.13]. But from a mathematical point of view, a dynamical system is, of course, a model of a specific structure with parameters or operators of time. A dynamical model consists of a multi-component set of time-depending elements with local states. Their local interactions determine a global state. The structure of the dynamical models is characterized by a common state space. Their dynamics, i.e. the change of system’s states depending on time, are represented by linear or nonlinear differential equations. In the case of nonlinearity, several feedback activities take place between the elements of the system. These many-bodies problems correspond to nonlinear and non-integrable equations with instabilities and sometimes chaos. The emergence of new order and attractors of complex systems correspond to solutions of these equations. Thus, emergence is no mys-

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tery, but can be explained by structural analysis. Nevertheless, the emerging order cannot be reduced to the features of single elements in a nonlinear dynamical model. Separation of a system into its parts means mathematically linearization. The emergence of an attractor of fluid dynamics cannot be explained by the single molecules of the fluid. The life of an organism cannot be reduced to the sum of its cells. The emergence of cognitive abilities cannot be explained by the sum of neural cells in a brain. The famous slogan of philosophers “The whole is more than the sum of its parts” is true with respect to the nonlinearity of a dynamical system. In this book, we introduced the principles of nonlinear dynamical systems. They belong to the most general theoretical structures of human knowledge which can be applied in models of physical, chemical, biological, neural, cognitive and social dynamics. The mathematical structures of these models do not depend on special, e.g., physical laws. Thus, we demand no kind of physicalism, but structural analysis of dynamical models in nature and society. Another fundamental phenomenon of symmetry and symmetry breaking can also be explained by structural analysis. On the microlevel, physical laws are symmetric with respect to time (microreversibility), but not at the macrolevel. According to the second law of thermodynamics distributions of, for example, molecules develop on the average with irreversibility. For example, elementary particles are reversible with respect to time. Organisms, people, and social groups become elder without a chance of reversibility. But from a structural point of view, microreversibility and macro-irreversibility are no contradiction. We have to distinguish between the reversible “exterior” time of a complex dynamical system and its “interior” time, or its “age”. While the “exterior time” is the usual real time parameter t that is registered by a clock, the “interior time” is defined as an operator that takes into account the irreversible changes in the system’s states. As a real parameter, the exterior time appears merely as an index in a set of trajectories (in classical physics) or in a wave equation (in quantum mechanics). As an operator, the interior time permits statements about the temporal development of a complex ensemble of trajectories or distribution function that serve

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mathematically as eigenfunctions of the time operator. The connection with the external time rests on the eigenvalues of the time operators being real lengths of time as registered by a normal clock. The distribution is graphical representations of the different interior “ages” of a complex system [8.14]. For example, the different organs of a complex system like the human organism wear out at different rates. The time operator assigns a “mean age” to each state of the system, which increases at the same rate as the exterior clock time. Thus, “age” is a structural property of complex dynamical systems and does not depend on special models of, for example, biological systems. For example, it is not only a metaphor to speak about “old” and “young” cities, organizations or societies. The phase transition in political systems may differ according to the periods of election. Each dynamical system has its own characteristic interior time. From the viewpoint of the history of philosophy this is reminiscent of Aristotle, who distinguished between time as “movement” (1"  and time as “coming into being, growth, and decay” 02.   This connection can be connected to the concepts of reversible time in mechanics and of irreversible time in thermodynamics. Irreversible processes are explained according to the second law of thermodynamics as internal breakings of symmetry (based on the time operator) that violate time reversal symmetry. The time operator has the remarkable property that the past and future are separated by an interval that is quantifiable in terms of a characteristic time. Traditionally, the present is represented as a point on the time axis in which past and future can come infinitely close. Prigogine therefore speaks of the “duration” of the present, which he compares to the concept of duration introduced by the French philosopher H. Bergson [8.15]. The time operator is, however, a mathematically defined functional operating on distribution functions and not on single elements. It must not be confused with subjectively experienced time. There is a precise relation between dynamical systems and computational systems. The dynamics of nonlinear systems is given by differential equations with continuous variables and a continuous parameter of time. Sometimes, difference equations with discrete time points are sufficient. If even the continuous variables are replaced by

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discrete (e.g. binary) variables, we get functional schemes of computational systems (“automata”) with functional arguments as inputs and functional values as outputs. Operators (e.g. time-operator of interior time) can also be digitalized. The structural features of the systems are represented in programs (“algorithms”). Their degrees of computational complexity can be determined according to their size (“algorithmic complexity”) or time length. In Chapter 7, we explained which degrees of computability can be distinguished and why computational systems are not necessary computable. Like dynamical systems, they can develop all kinds of more or less computable patterns and attractors of nonlinear dynamics. Degrees of computability and decidability can even depend on procedures which are unknown and therefore called “oracle”. A so-called ψ-oracle Turing machine uses the usual rules of a Turing program and an operation ψ (e.g., “replace the content x of a register by ψ(x)”) whose computability is unknown. The operation generates values (“answers”) like an oracle. Functions or functionals which are computable by ψ-oracle Turing machines are called relatively computable (with respect to the oracle ψ) [8.16]. Actually, many processes in the world are modeled by tools which are computable or only assumed without knowing their degree of computability. In this sense, all kinds of mathematical structures of dynamical models correspond to degrees of at least relative computability. But they are not all (Turing-)computable. These arguments lead us to the fundamental principle of computational equivalence: every dynamical system corresponds to a computational system. If the world is considered a complex dynamical system, then it can also be considered a computational system. Quantum-, molecular-, nano-, DNA-, cellular-, neural-, and cognitive computing are only special models of computational structures, generating different kinds of information. There are natural dynamical systems (e.g. quantum-, molecular-, cellular-, neural systems), which have been generated by cosmic and biological evolution. But they are also only special models of general structures. Therefore, a task of science and technology is to find and to construct new models fulfilling the principles of computational systems. In this sense,

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a technical co-evolution was initiated and implemented by humans, leading to systems with growing capabilities of artificial life and artificial intelligence. Scientific knowledge consists of theories, structures and models corresponding to computational systems. Thus, the expansion of science and growth of knowledge can be considered a part of information dynamics in a world with increasing complexity. A challenge of complexity is the knowledge representation of all kinds of theories, structures, models, dynamical and computional systems. Knowledge representation, which is today used in database applications, artificial intelligence, software engineering and many other disciplines of computer science, has deep roots in logic and philosophy. In the beginning, there was Aristotle who developed logic as a precise method for reasoning about knowledge. Syllogisms were introduced as formal patterns for representing special figures of logical deductions. According to Aristotle, the subject of ontology is the study of categories of things that exist or may exist in some domain. Aristotle distinguished basic categories for classifying anything that may be said or predicated about anything. Many of these categories (e.g. substance, quality, quantity, relation, spatiality, and temporality) are today applied in, for example, data bases. In the Middle Ages, knowledge representation was illustrated by graphic diagrams and pictures. In Peter of Spain’s “summulae logicales” (1239), an ontological hierarchy with Aristotelian categories represented knowledge by genus (supertype) and species (subtype) [8.17]. The features that distinguished different species of the same genus were called “differentiae”. R. Lullus (13th century) illustrated an ontological hierarchy by a tree with branches for categories. Leaves corresponded to questions or to answers which should automatically be found by a system of rotating disks for combining features of things. Actually, Lullus applied a kind of British Museum algorithm, the first attempt to develop mechanical aids for problem solving and information retrieval. There is a close correspondence between categories, theories and structures of ontologies and hierarchies of types or classes in objectoriented programming languages. An object-oriented programming language (e.g. C++ , Java) combines a declarative style for specify-

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ing objects with a procedural style for defining the action by and upon those objects [8.18]. Object-oriented declarations define the same kind of information as frames or classes with certain attributes which are instantiated as particular objects for specific data. For example, the class or type of an automobile is characterized by certain attributes. A particular car is a specific object of this class. In this sense, a theory or structure can also be considered a class of an object-oriented language. A model of a theory is a concrete specification which corresponds to an object of an object-oriented language. Theories or structures with their models can be classified in hierarchies like plants and animals in biological taxonomies. By the way, Aristotle designed the first botanic taxonomies in the history of biology. An ontology is organized in a class or type hierarchy that supports inheritance of properties from supertypes to subtypes. Fig. 99 shows the ontology of symmetries as mathematical structures which have been analyzed in this book. Contrary to the traditional belief of antique philosophers, ontology in the sense of informatics only means the organization and representation of knowledge and is not necessarily identical with the actual order of reality [8.19]. Another example is the ontology of complexity in Fig. 100, which was explained in Chapter 7 and previous remarks on computational systems. Again, properties of supertypes are inherited to subtypes in the following branches of the tree-like hierarchy. Object-oriented programming languages have advantages with respect to traditional declarative or procedural languages. Instead of separating the declarations that define an object from the procedures that operate on them, the object-oriented programming languages integrate the declarations and the methods for each type of object in a single information packet. By encapsulating objects in this manner, object-oriented programming languages provide a way of distinguishing the external behavior of objects from their internal structure and dynamics. For example, a particular car is an object instance of a class or type of automobiles with attributes (e.g. color, size, weight) and methods determining the internal processes (e.g. motor dynamics, electronic equipment). Furthermore, there may be rules describing external behavior of cars in interaction with traffic lights,

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symmetry (= invariance with respect to automorphism groups)

continuous symmetry

discrete symmetry

plane symmetry

ornamentic symmetries

spatial symmetry

PCTsymmetry

crystal parity charge time symmetries

gauge symmetries

space-time symmetries

Galilean invariance (= classical physics)

Lorentz invariance (= special relativity)

covariance

supersymmetry (= M-theory)

(= general relativity) U(1)xSU(2)xSU(3)symmetry

U(1)xSU(2)symmetry

U(1)symmetry (electromagnetic force)

Fig. 99.

Poincaré symmetry (gravitation)

SU(3)symmetry (strong force)

SU(2)symmetry (weak force)

Ontology of symmetries

warehouses and dispatchers. Thus, each object instance is an autonomous entity whose behavior is determined by its class methods and the inputs it receives from other objects. The advantages of encapsulation and inheritance in objectoriented languages can also be applied to knowledge representation of theories and structures. In the beginning of this chapter, we illustrated the complex network of theories, structures and their models as a landscape of knowledge. This metaphor can now be made precise in object-oriented programming languages. In order to represent the complex network of knowledge by diagrams, ontologies of theories, structures and models are enlarged by Entity-Relation (ER)-diagrams and semantic webs. A class is conceived as an entity which is represented by a box (Fig. 101). The attributes and methods of the class are written in ovals which are related to its box. In ER-diagrams, the subordination of classes in ontologies is only a special relationship. In semantic webs, there are further relations between entities indicated by edges and rhombuses between the boxes (Fig. 101). Besides hierarchical relations of subordination (s), there

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complexity

dynamical complexity

fixed point attractor

periodic attractor

computational complexity

quasi-periodic chaos attractor attractor

randomness

relative computability

Turing-compatibility

computational time

deterministic computability exponential time

non-deterministic computability

program size (= algorithmic information)

compressible size

incompressible size (=random)

NP-time

polynomial time (P) quadratic time linear time

Fig. 100.

Ontology of complexity

are relations of parts (p), instantiation (i) of objects from classes, inheritance (h), etc. In Fig. 101, a small section of the knowledge landscape is represented by a semantic web. A physical theory is characterized as a class with only a few attributes (e.g. name, parameters, constants) and methods (e.g. operators, measuring methods). Models of theories (e.g. elementary particles with high speed near the velocity of light as a model of the theory of special relativity) are considered object instances of classes [8.20]. For example, conservation laws are parts of the theory of relativity. ER-diagrams and semantic webs are illustrations of structures and objects which are defined in mathematical and natural language. For implementation on computing systems, the axiomaticmathematical representation must be transformed into a computer language. These steps of transformation from diagrams with natural language to mathematical and computational representations are used in software-engineering. They have great advantages for programming complex structures and networks. Axiomatic representations allow us to control complex processes step by step with mathematical proof methods. Correct programs must satisfy the ax-

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physical theory

Fig. 101.

Entity-Relationship (ER)-diagram of theories

iomatic conditions of the corresponding structures. Carnap’s idea to represent scientific theories in formal languages is now realized in informatics for all kinds of structures and objects. Leibniz’ old vision of computing systems for all tasks in everyday life is a challenge of modern software-engineering. In software-engineering, the structure of, for example, a bank, firm, or traffic system is represented by ER-diagrams, transformed to axiomatic definitions of relations and functions, translated into an object-oriented programming language, in order to implement the structure on a computer. The formal implementation of structures on computers needs formal labels and types, in order to identify all symbols and their meaning uniquely. The label of a structure (e.g. a group) consists of a set T of notations for types, a set F of notations for constants and functions, and a mapping relating each notation from F to a type. The type of a constant c is a type M from T . “c has type M ” is denoted by c : M . The type of a function f : X1 × X2 × · · · × Xn → Xn+1 is denoted by the types of the arguments X1 , X2 , . . . , Xn and the values Xn+1 , i.e., f : M1 , M2 , . . . , Mn → Mn+1 , with M1 , M2 , . . . , Mn+1 from T . The labels can be enlarged for functionals with functions as arguments and values, too. In a formal language, terms can be constructed with functions, constants, and

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variables. Variables and constants are terms of certain types. If f is a function from F with type f : M1 , M2 , . . . , Mn → Mn+1 and t1 , t2 , . . . , tn are terms of type M1 , M2 , . . . , Mn , then f (t1 , t2 , . . . , tn ) is a term of type Mn+1 . Terms are used to construct equations. With equations and logical constants (e.g. ∧, ∨, ¬, →, ∀, ∃ for “and,” “or,” “not,” “if-then,” all-, existence-quantifier), we can construct propositions to represent the axiomatic definitions of a structure. In Fig. 101, the axiomatic definition of, for example, general relativity is given by Einstein’s field equation with certain preconditions. Models are specific interpretations of the formal propositions of the axiomatic definitions. In, for example, the theory of general relativity, the formal tensor of curvature is interpreted by a notation representing the curvature of a light ray in the vicinity of the sun.

ER-diagrams and semantic webs seem only to illustrate static aspects of a structure. In complex dynamical systems, time-depending processes and histories are described by phase transitions of states with the emergence of new entities. Discrete processes can be simulated by digital computers, but continuous processes are more naturally simulated by analog computers. Yet if the time step is small enough, the granularity of a digital simulation might not be noticeable. Movies and television, for instance, represent continuous motion by a sequence of discrete frames. In software-engineering, state-transition diagrams are used to illustrate all kinds of processes which, after axiomatic definition of transition rules, must be translated into an object-oriented programming language in order to implement them on a computer. The structure of a dynamical system is defined by a state space S, a set of control parameters α, and the axiomatic equation (e.g. differential equation) of a function f . In a simplified form the axiom is represented by the equation st+1 = f (α, st ) with the input-state at time t, the output-state at time t + 1, a control parameter α and a certain condition of an initial state so . State-transition diagrams enlarge ER-diagrams and semantic webs for illustrating dynamic processes. They represent states by boxes and transitions between them by arrows. Fig. 102 is a statetransition diagram of a phase transition (“symmetry breaking”) in a complex dynamical system with the emergence of new order. The transition rules are indicated at the transition arrows. The phase transition is initialized by an initial state. The control parameter

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Fig. 102. State-transition diagrams of phase transition (“symmetry breaking”) with emergence of new entities (“self-organization”)

α is written in a rhomb with a feedback loop. If the transition is below a critical value of α, the old order is stabilized. If it surpasses the critical value, the old order is destabilized. Unstable and stable modes of the system elements compete with one another in a competitive state, depending on an order parameter. Unstable modes start to dominate (“enslave”) stable ones, until a new macroscopic order emerges according to the order parameter. An order parameter determines the emergence of a new order. Therefore, it is indicated in a rhomb of the state-transition system. The feedback loop describes the causal connection between the microlevel and the macrolevel of a complex system (e.g. Fig. 49). In the case of symmetry breaking, the state-transition system generates a bifurcation tree (e.g. Fig. 48). Again, the state-transition diagram can be transformed into a mathematical representation of a complex dynamical system with equations and logical deductions. The transition rules “initialize,” “stabilize,” “destabilize,” “dominate,” and “emerge” correspond to the steps of a linear-stability analysis (compare Sec. 3.3). The new order is derived from the equations of microstates as solution of an appropriate macroscopic equation depending on the order parameter. The transition rules can also be translated into a programming language to implement the system to a computer.

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State-transition diagrams are a common representation for discrete (and approximately continuous) processes. Finite-state machines are the simplest and most widely used version of statetransition diagrams. Petri nets are a generalization of statetransition diagrams for representing concurrent processes [8.21]. For object-oriented design, Petri nets have been adopted as the basis for activity diagrams in the Unified Modeling Language (UML). Petri nets are especially convenient for representing cause and effect: each transition represents a possible event, the input states of a transition represent the causes, and the output states represent the effects. By executing the Petri net interpretively, a computer can simulate the processes and causal dependencies. From a philosophical point of view, we get a pragmatic methodology of knowledge representation which is more than software engineering. There is a broad variety of formal tools to represent data, information, and knowledge for different purposes and different steps of development. In the first step, we can use diagram languages for illustrations of complex networks with, for example, ER-diagrams, semantic webs, state-transition diagrams or Petri nets. In our methodology of knowledge representation, there is no dogmatic distinction of any method or language. They are tools with advantages and disadvantages in different contexts of application which should be correlated in a kind of patchwork to get a flexible methodology of problem solving. For example, ER-diagrams focus on the entities and their relations in a domain. Ontologies only represent hierarchical levels. State-transition diagrams underline the dynamic point of systems. In the next step, diagrams are axiomatically described as structures and dynamical systems in mathematical language. They can be transformed into programming languages in order to implement the structures and systems on a computer. Translations into formal predicative logic are used to check the correctness of computer programs by formal proofs. Representations of complex knowledge need experts from different disciplines. Therefore, in requirements engineering of informatics, interdisciplinary teams consist of experts of informatics and the domain of application, as well as experts of cognitive science, economics and

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management, in order to get a useful product of knowledge representation. They start with the identification and limitation of their issue. From different points of view, they correlate different tools of analysis, harmonize different interests pragmatically and develop a common plan of modeling their issue. According to a catalogue of criteria, their results of modeling and representation are evaluated and verified. Analogously, in the philosophy of science, we need interdisciplinary teams to analyze the growth of knowledge and to explore the complex landscape of scientific theories, structures, and models. Obviously, the chosen tools of representation influence our view of a problem, structure or system. But, on the other hand, structural properties are invariant with respect to particular representations of knowledge. Especially, symmetry and complexity are invariant universals of knowledge which are confirmed in different contexts and models. Nevertheless, the old question of philosophy arises as to whether these principles are only abstract classifications of knowledge or real structures of the world. In modern epistemology after Kant, the question for reality seems to be beyond the limitations of human experience. Our knowledge of the world depends on the conditions of human experience. Even our sensory data, as immediate signals of the world, are represented in data models of knowledge. Knowledge processing is generated by the dynamics of human brains. Our models of the world are constructions of the human mind as cognitive states of the brain. But we must not forget that the validity of mathematical structures and proofs is independent of brain research and cognitive science, although it is generated by brain dynamics. A neural or cognitive scientist who can explain brain and cognitive dynamics of mathematical thinking is not able to find a mathematical proof. Euclid proved his theorems without being aware of his brain activity. The same is true for music and arts. In short: brain research cannot replace, for example, mathematicians or artists. The theorem of Pythagoras is mathematically true in all models of Euclidean geometry, independent of possible applications. Some Euclidean models were even verified with antique measurements and technical con-

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structions and confirmed Euclidean geometry empirically. In this sense, symmetry and complexity are invariant properties of mathematical structures containing models of reality. In the case of comprehensive and general theories like quantum mechanics, evolutionary dynamics, or sociodynamics, single models may be less convincing. But embedded in general structures with different models of application and confirmation, the degree of confirmation increases. Verified models connect the network of knowledge with reality. It is always the whole network of our knowledge which is confronted with experience. In general, structures of scientific theories are containers of information with actual and potential models of the world. Actual models are confirmed by observations and measurements. Potential models satisfy the structural conditions of a theory, but they are not yet realized in nature or society [8.22]. For example, the flight of a bird as well as the flight of an aeroplane satisfy the laws of aerodynamics, but only the flight of birds was an actual model of the laws which was realized by nature. The model of a jet was potential, as long as it was not invented by human technology. In Chapter 5, we emphasized that the biological evolution of life on earth was only one possible actualization of evolutionary laws. In Chapter 6, we discussed actual and possible developments of sociodynamics. In this sense, structures of theories may contain more information than is actualized in the world. One may argue that the information of structures and models is generated in our brains. That is obviously right. Therefore, structures of our theories are not static and may change with new experiences. But the whole network of our present knowledge tells us that our brain is a product of cosmic and biological evolution. In short: the universe is elder than the human cognition of it. The information production of our brain is embedded in the information dynamics of the universe. Our concepts of symmetry and complexity are traces of its universals. 8.2

The Beauty of Symmetry and Complexity

In Platonic tradition, symmetry was a universal law for truth, justice, and beauty in the world. The last chapter considers the evolution

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from symmetry to complexity from an aesthetical point of view. Our thesis is that art is another kind of representation with relation to scientific views even in modern times. In many early cultures, as with Greek philosophy, Taoism or Hinduism, nature was interpreted as a harmonious organism whose parts and movements are proportionately matched to each other. The human life-world with its familiar living beings and cyclical natural events became the measure of, and the model for interpreting, the strange and unknown world. This interpretation of the world applies, not only to the Antique-Medieval philosophy of nature, but also to early art. Proportional relationships of the human body are found in the architecture of those times and, correspondingly, in the dimensional relationships that were assumed for the cosmos. What came into play here seems to have been an early, intuitive kind of human knowledge, not at all based on a highly developed geometry and astronomy like those of the Egyptians and Greeks. An example is the Golden Section which is even applied in early cultures without mathematical geometry. Thorough psychological investigations have been made into the harmonious effect of the Golden Section. Various factors play a role here: along with intuitively familiar proportional ratios of the human body, probably also the perception that in the Golden Section two parts of different sizes form a unity; the smaller part is related to the larger one as the larger one is to the whole, and thereby the unequal parts are harmonized in the whole. Thus, we find this proportional relation in objects of everyday use, in architectural monuments and on to contemporary standard sizes [8.23]. Various authors discern the Golden Section not only in the Classical Greek and Egyptian vases, but also in Chinese pottery or products of the Cretan and Mycenaean culture from the end of the Bronze Age, thus ca. a thousand years before Greek mathematics. It is conjectured that the Golden Section is present in the pyramid of Cheops and in Aztec examples. But the situation of the historical sources is obscure and is evaluated in thoroughly divergent ways [8.24]. The Greeks were the first to try to provide a mathematical basis, a geometrical doctrine of proportions, for the intuitive ideas of prehistory,

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for familiar proportions of the life-world in art, architecture, technology, and for the interpretation of nature. The Greek architect Hippodemos, a contemporary of Anaximander, drew up plans for Greek metropolises of his time, in which streets were laid out crossing at right angles along the compass directions, in the service of better orientation and hygiene. That did please politicians like Pericles. But, then as now, cold architectural rationalism provoked the scorn and the anger of citizens who were affected by it, and therefore Aristophanes too, in one of his comedies has the famous architect appear armed with a compass and a ruler to lay out cities mathematically [8.25]. A canon of proportions for the visual arts was first mentioned by Polykletus [8.26], who, however, presumably had Egyptian predecessors. He created his famous spear-bearer (Doryphoros) in accordance with this canon of proportions. The historical sources and references have come down only fragmentarily and through interpretations into later centuries. Presumably Polykletus derived his proportions from his studies of nature and then reworked them into an idealized form. Thus he was not trying to project into his statues presupposed cosmological or philosophically interpreted proportions as, for instance, the Golden Section. Likewise, symmetry is understood as “balance,” thus proportional relationships adjusted to each other that evoke the impression of harmony. Greek philosophers and mathematicians shared the belief in a well-ordered world that could be expressed in proportions and harmonies. Later, Vitruvius named nine additional artists who were said to have authored the “praecepta symmetriarum”. In all variations of these early authors, Vitruvius’ core idea provides the standard: he began his ten books about architecture (25 B.C.) by recommending that temples be built analogously to human anatomy, which, in his view, possesses a perfect harmony of its parts [8.27]. As a physician in the 2nd century A.D., Galenus observed about symmetry “that health depends on the symmetry of the elements and beauty depends on the symmetry of the limbs . . . as is written in the canon of Polycleteus”. An understanding of art was delineated here in which man was literally the measure of all things. In the Platonic tradition the

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Logos of nature was based on a geometric doctrine of proportions. Certain proportional relationships were singled out as particularly “compatible to the measure of man.” They were supposed to recur in art objects, everyday objects and buildings in order to make them compatible to the measure of man. From this point of view the frequently discussed question of whether the Golden Section is “really” a natural law of human proportions, is secondary. Modern statistical studies do lead to conclusions about such laws of proportion in the growth patterns of humans, animals, and plants. But it will not have escaped the Greeks either, that aesthetic and erotic charm frequently originates from the small “symmetry breakings”. Of course, standards have to be presupposed if such deviations are to be experienced as attractive tensions. As an example, consider the Aphrodite of Cyrene, which was celebrated for its perfected harmonious proportions in later epochs as well. Fig. 103 displays Golden Sections, for example, for the knee and the breast, the navel and the vulva, or the breast and the neck. Here the historical question of whether the Antique artist was consciously patterning his work on the Golden Section is not relevant. Even if we, today, project these or other proportions onto Antique statues, their charm inheres precisely in the slight deviations from whatever the norms of proportion were, that is, in the symmetry breakings. The posture of the Aphrodite deviates from narrow standards, by contrast with rigid archaic statues. Thus, the right breast exhibits a changed proportion to the navel, and likewise the vulva and the right knee. In an overarching integrated frame of reference, however, such local deviations are incorporated, and are superimposed on one another into a perfect overall impression that was brought forth again and again. The attraction we feel to the beauty of actual living women is not based merely on their being made of flesh and blood. Here too the small deviations and symmetry breakings, in comparison with idealizing concepts of proportion, play no inconsequential role. The boredom induced by many modern “beauty queens” and pin-up girls with their “ideal” measurements, may be evidence of that. Symmetry breakings are needed to create individuality and — if you like

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Golden sections of the Aphrodite of Cyrene

— “personality.” Yet this charm of the particular is felt in reality only because it is viewed in reference to standard proportional relationships. Natural, proportional relationships may have the character of laws of nature, and in this sense they may be immutable. But if such proportions are elevated to norms in a canon of art, then, like all aesthetic categories, they are subject to historic changes and can at any time be felt to be restrictive and obsolete, or be rediscovered. This was the fate of the Greco-Roman conception of art. However, in judging it one should not cling to the entranced idealizations that later centuries linked with this conception of art. What is fundamental is not the “letter” of the Greek canon of art, but its “spirit”. What was decisive was the idea of carrying over into art and architec-

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ture standards of proportion drawn from the measure of man, that is, organic ones, and not doing it subjectively, arbitrarily, but in the framework of a shared understanding of nature. For example, Vitruvius recommended that the length of a temple should correspond to twice the width and that the proportions of the open entrance hall (pronaos) and the closed interior space (cella) should amount to 3:4:5 (the depth of the entrance hall being 3, its breadth 4, and the depth of the interior hall 5). In this way one derives musical harmonies such as the octave, 1:2, fourth, 3:4, and fifth, 3:5, which in the Pythagorean conception are determinative for the cosmos [8.28]. Along with proportional relations drawn from the measure of man, however, standards for size that have that basis are also of interest. We in the 21st century A.D. are especially aware of that, having experienced how megalomaniac dictators and cynical ideologies consciously apply the means of the “enormous” to oppress the individual person by means of a brutal architecture, designed to be imposing. The significance of an architecture based on the measure of man is apparent today in the monotony and vacuity of many tenement houses, justified by technical constraints of practicality and economy. The Greeks were not the only ones to have investigated the mathematical proportions that became the basis for the Antique–Medieval concept of symmetry (although they were the first to propose an exact geometrical basis). The Hindu, Buddhist and Islamic cultures, as well, had at their disposal not only highly developed mathematics, but also a canon of art consisting of geometrical proportions and symmetries. In Hindu thought, as in the Pythagorean tradition, number has cosmic significance and is considered to be a means of establishing a reciprocal relation between the universe and man. A common denotation of the temple “vimana” means that which is “well measured” or “well proportioned”. The central symmetry of the Hindu temple [8.29] is striking, with the image of the divinity at the center of the sanctuary. In Hinduism as well as in the Taoist or Stoic philosophy of nature, ideas of energy waves and fields of force play a great role. Thus, the radiation of energy from the center of the sanctu-

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ary decreases outwards in rings — in stepped walls and intermediate courtyards. The degrees of efficacy that are manifested as a result, determine the hierarchy of variously placed images of the Gods, rising toward the center. The temporal cycles and repetitions of cosmic ages also find expression in the forms that are part of the temple, in which, for example, motifs emerge repeatedly in varying sizes at different locations. In Buddhist architecture there are centrally-symmetrical shrines like the Borobudur-Stuba on Java, which consists of eight balustrades ascending like steps, with bas-reliefs and shrines in niches and the largest Buddha figure in the center, on the spire of a tower. This structure symbolizes the different stages of enlightenment. The five lower terraces are square, the three upper ones are circular, and 3, 5 and 8 are numbers in the Fibonacci sequence. The Golden Section and other harmonious proportions can be pointed out here just as with Chinese and Japanese pagodas [8.30]. In Islamic architecture we encounter a culture with highly developed mathematics (especially algebra and number theory) that stands on the shoulders of Alexandrian–Hellenistic and Indian traditions. Islamic ornamentation is so skillful and complex that we can do justice to it only with respect to the modern algebraic concept of symmetry. Islam, like Judaism, is a religion of law. This may have provided a supplementary motivation for not depicting God in (forbidden) anthropomorphic statues, but instead in the mathematical symmetry laws of nature. In the architecture of the mosques [8.31] a mathematical comprehension of symmetry and the particularities of Islamic religion underwent a symbiosis. Some typical ground plans of the space of an Islamic mosque are represented in Fig. 104a–d. The ground plan of the Arab buttressed hall (Fig. 104a) and the pillared mosque of Asia Minor (Fig. 104b) have square grids without a particular center. This is a clear expression of the pragmatic character of the mosque. It is not a shrine in the Hindu or Buddhist sense, and not a house of God in the Christian sense, in which it is assumed that God (a personal God) is present. It is a gathering-place (dschami) and a place of prayer or prostrating oneself (masdchid).

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(a) Fig. 104a–d.

(b)

(c)

(d)

Symmetric ground plans of Islamic mosques

By contrast, the Persian four-iwan court mosque (Fig. 104c) and the Osmanic central-dome mosque (Fig. 104d) are centrally symmetrical. Cosmological aspects are assimilated into the architecture, the dome arching over the believers like the spherical dome of the sky. From the ground plan on, there is complete equivalence of the axes and compass directions — thus, isotropy. The believers have access from all sides. But, as is known, Islam makes provision for one symmetry breaking, namely honoring the direction toward Mecca. Islamic architecture has always had to strive for a compromise between its orientation toward Mecca and its preference for mathematical symmetry. The pre-eminence of regular geometrical bodies is striking here, especially that of the cube and the (hemi)sphere. Again, the synthesis of the Hellenistic–Neoplatonic and the Islamic tradition is clear: the cube is both the form of the Kaaba and a Platonic body. The Persian mosques, especially, appear on the basis of their symmetrical forms and splendid mosaics as solidified crystalline structures — absolute, immovable, and of unalterable beauty [8.32]. The Christian Middle Ages owe their knowledge of Aristotelian philosophy to their encounter with the Islamic culture. The combination of Neoplatonic and Christian ideas comes to light already in the Romanesque. Gothic can be mentioned here, as an expression of Christian symmetry, since it manifests a new concept of unity against the background of Aristotelian-Thomistic philosophy. The Gothic cathedral is, like the Summa of Thomas Aquinas, a coherent system which comprehends all levels of life and of the cosmic order of this world and the next [8.33]. It is the architectural representation of an eternal ontology from the universals in the top of the gothic

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tower to the basic stones of application. Here the problem of unity in variety was central. Thus, the scholastics concerned themselves, in the framework of this hierarchy of the world, with the foundation of a harmoniously organized society in which, for example, the conflicting interest of church (sacerdotium) and state (imperium) are reconciled. The arts were to mirror the proportional relationships of this coherent universe. According to the Platonic conception, the elements of the universe were transmitted through the five regular bodies, which were based on the equilateral and isosceles triangles and the pentagon, as excellent forms of the elementary particles (compare Sec. 1.1). If the divine architect employed these forms to construct the universe, then they had to be used for the cathedrals as well, as symbols of the universe. It is not surprising that in the Neoplatonic tradition even the stability of buildings, that is, a physical matter, was based on appropriate geometric forms. Thus, H. Parler, a German adviser for the construction of the cathedral of Milan (1392) recommended giving the ground plan the form of a square since a structure ad quadratum is in accord with the laws of the cosmos, which guarantee it an unshakeable cohesion [8.34]. The regular pentagon also yielded the Golden Section and was occasionally used for the rosettes of the cathedrals (e.g. in the cathedral of Amiens). This proportion frequently appears in churches of the 12th and 13th century (e.g. Chartres) as an approximate number ratio, as in 5:8. The number ratio 5:8 is at the same time a musical harmony, namely the minor sixth. Augustine had already characterized music and architecture as “sisters of number,” in the Neoplatonic and Pythagorean tradition. One example of the use of musical harmonies is the ground plan of a Cistercian church which was found in the sketchbook of an architect of the 13th century [8.35]. The fifth (2:3) determines the proportion of the width of the transept to the total length; the octave (1:2), the proportion of the side aisle to the middle aisle and of the length and breadth of the transept; the fourth (3:4), the proportions of the choir; while the intersection of the nave and transepts, as the liturgical midpoint of the church, is based on the most perfect number ratio of unison (1:1). Rose win-

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Fig. 105.

Central symmetry of a rose window (cathedral of Chartres)

dows are typical attributes of gothic cathedrals. Fig. 105 shows a drawing of the west rose window by Villar de Honnecourt from his sketchbook. Its central symmetry with the peripheral wheel symmetries expresses the Gothic Medieval idea of unity as no other example does [8.36]. It was not the Renaissance that first dictated that “ars sine scientia nihil est” (art without science is nothing). The great Medieval architects did not depend on their predecessors’ experimentation or rules of thumb and empirical formulas alone, but instead tried to base their designs on mathematics and, indeed, on the philosophy of nature. However, the theological substantiation of symmetry faded into the background in the Renaissance. As the Antique scientific authors became increasingly known, the classical texts of the Antique art canon were studied again. In short, in the Renaissance, symmetry was based again on the proportions of man and nature. Thus, L.B. Alberti [8.37] argued in “De re aedificatoria” (1485), that beauty is a correspondence and cohesion of parts according to a specific number, proportionality and order. He asserted that an art

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form that is designed in this way (concinnitas) is the absolute and highest law of nature (absoluta primariaque ratio naturae). Alberti also pointed to reflection symmetry as the natural law in the case of humans and animals, and concluded that therefore it must be applied in architecture. But reflection symmetry is not a discovery of the Renaissance. Villar de Honnecourt had used reflection symmetry economically in his sketches. In one such case he carefully executed only one reflection half of a building in his sketch of it. In reflection symmetry, the second half has the same forms and proportions; only the direction is reversed. Even Leonardo da Vinci was to use this economy of drawing reflective symmetry in his architectural sketch books. Many sources show that laws of reflection in architecture had to be oriented to man and nature. Another example is G. Vasari, who adopted his concept of symmetry (disegno) almost literally from Vitruvius and then wrote: “. . . thus it is that it [the sketch] takes in the proportion of the whole to its parts as well as the proportion of the parts to each other and to the whole, not only in human and animal bodies, but also in plants, buildings, paintings and sculptures.” [8.38] The intentions of the Renaissance were crystallized in one person: Leonardo. He embodies the universality of the Renaissance artist, integrating the architect with the painter, engineer, inventor, nature researcher and philosopher. In the Pythagorean tradition symmetry spans all of these realms: “Proportion is to be found not only in numbers and dimensions, but also in tones, weights, time intervals, and positions as well as in every dynamic force that there is.” [8.39] Moreover, his conception of the philosophy of nature was by no means original, but remained inside the confines of Medieval–Aristotelian and Neoplatonist–Pythagorean deliberations. Leonardo was not a second Galileo. His accomplishments in architecture, painting and engineering are works of genius — the finished works of art as well as the sketches and visions [8.40]. One aspect of Leonardo’s architecture should be emphasized because it became significant for the development of the concept of symmetry: Leonardo emphasized central symmetry, and he system-

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atically worked out the possibilities of central structures such as churches, castles and other buildings. His octagonal floor plans are especially noteworthy. They articulated the possibilities of optimally filling a building with octagonal rooms [8.41]. Naturally, the central symmetry, with its emphasis on the center, humored the new selfawareness of the princes and their requirements for display. They are in the center, and they want to be able to radiate in all directions, and to be seen. Leonardo’s studies of architectonic central symmetry stand in mathematically close connection to his interest in regular polygons, regular (Platonic) and semi-regular bodies. He asserted that all the arts, not only architecture, should have a scientific basis and be derived from natural laws. In his “Trattato della pittura” Leonardo also tried to provide painting with a scientific basis so that it would no longer belong to the “artes mechanicae”, but would be added to the “artes liberales.” [8.42] This gave a particular importance to the study of the human body and its proportions in anatomy. In a famous study of proportions [8.43], which has become an icon of the modern age (Fig. 106) Leonardo related the human body to the circle and the square. This brought in Platonic-mythological allusions: the square as the symbol of the earth, the circle as the symbol of the sky, and, in the center, man as the conjunction and proprietor of both. Mathematically, Leonardo emphasized the proportional ratios (especially that of the Golden Section). Then the inscribed Pythagorean triangle emerges of necessity. With all the Renaissance artists, harmonious proportions constituted the basis of art in natural law. It was Leonardo also who illustrated the book “De divina proportione” (1509) by L. Pacioli: “Every part is so constituted that it can form a unity with the whole and can thereby free itself from its incompleteness.” [8.44] The perception of symmetries also presupposes an analysis of the sense of sight. Here the geometrical concepts of perspective and projection are basic. Although important groundwork was laid in the optics and cartography of Antiquity and the Middle Ages, the technical practice of perspective and projection became central in

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Fig. 106.

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Symmetry of man (Leonardo da Vinci)

Renaissance painting for the first time. The philosophical background of this development is clear: reality is perceived through the senses. This raises the question of whether the world is represented by the senses as it is, or whether it is changed in any way by the sense of sight, as might be inferred from the sensory illusions that had been much discussed ever since Antiquity. This raised the question of whether symmetry is changed by the central projection that takes place when an object is seen. This problem contained the germ cell of a new discipline in geometry, namely projective geometry, which was to gain great significance in modern mathematics. It is noteworthy, for instance, that central projections do not change any similarity relationships, yet do change representations of size [8.45].

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Leonardo’s interest in these questions [8.46] was shared especially by A. D¨ urer, who, in his “Underweysung der messung mit dem zirckel un richtscheyt . . . ” (1525/1538) [Instructions for Measuring with the Compasses and the Ruler], wanted to give painting a scientific basis in geometrical and optical laws. D¨ urer prescribed technical transposition for the painter in great detail: the bundle of visual rays is made concrete by means of a firmly anchored cord, the end of which is to be conducted to various characteristic points of the object to be represented. The penetration point of the cord through an imaginary picture plane is recorded by coordinates and transcribed onto a sketch sheet. If a sufficiently large number of picture points of the object to be represented are available, the scientific framework for the artist’s work is laid. Sighting tubes for enlargement of the depth of vision also came into use [8.47]. To a Renaissance artist it was completely self-understood that the proportions recorded in this way were not only valid for painting, optics and symmetry, but could also be transposed into acoustic harmonies as tone proportions. However, this unitary concept of symmetry based on a geometric doctrine of proportions in the Pythagorean tradition was no longer understood by the time of the early modern era. The Antique and Renaissance doctrine of harmony, which was held to be evident in geometric proportions, calculable in numerical proportions, observable in astronomical proportions and audible in musical proportions, was soon felt to be ridiculous. Increasingly it became the object of mythical, cabbalistic, astrological and alchemistic speculations, as, perhaps, in the fantastic drawings and allegories of R. Fludd. The painter W. Hogarth, finally, considered it “strange” and an object of derision that the same laws should apply to optical images and acoustical harmonies. For the philosopher D. Hume, beauty is reduced entirely to the subjective perception and sensibility of the observer [8.48]. So the unity of Antique symmetry appears to be broken. Science, technology and aesthetics have become independent and have gone their own ways. But, in fact, at least the natural sciences have not broken with the early history of the concept of symmetry. On the contrary, modern mathematics has created a new, expanded concept of symmetry in which the

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old symmetry concept of Antiquity is only a special case, and which has finally developed into a new ordering and unifying idea for the natural sciences. The old Pythagorean conception included music in the quadrivium of the exact sciences as part of their doctrine of harmony along with geometry, arithmetic and astronomy. For the Pythagoreans musical harmonies had the character of natural laws, that is, they were expressions of symmetry laws of nature such as the harmony of the spheres. In other early cultures, also, we have found that musical harmony, nature, and life are identified with each other. In the occidental tradition this unity was broken by the end of the Renaissance at the latest. Aesthetic interpretations and research in the natural sciences developed their own unmistakably distinct categories and laws; in fact Antique standards of art were explicitly criticized by later periods, and their ontological grounding, as in the Pythagorean tradition, was called into question. On the other hand, we have noticed that representational art and mathematics have areas that overlap. In music one immediately thinks of the Baroque, of course, and especially of Bach with his elaborate fugues [8.49]. However, as long as the concept of symmetry is limited to Antique ideas of proportion, or even — as occurs frequently in modern theory of the arts — to reflection symmetry — symmetries in music must seem more or less accidental and sporadic. In actual fact it was the new revolutionary breakthroughs in music in the 20th century such as 12-tone music and electronic music that made the connections with the encompassing mathematical concept of symmetry clear. However, it is not a matter of forcing another “canon of proportions” onto music. On the contrary, it is apparent that all the fundamental concepts of music theory can be translated into the group theory language of modern mathematical symmetry theory and that this makes it possible to analyze examples of music from the Medieval modes through J.S. Bach, L. van Beethoven, A. Sch¨ onberg and on to K. Stockhausen. This common language of mathematical, scientific and artistic subjects not only fosters the unity of the “two cultures”, a unity which was thought to be lost. Group theory permits new methods of studying music

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such as computer analysis, since the language of group theory is very easily translatable into computer languages. Yet this new unity of methods in mathematics, art and natural science is sustained by fundamentally different intentions than those that obtained for the Pythagorean quadrivium. It is no longer possible to consider tone scales and harmonies to be the expression of particular natural laws. In philosophical terms what we are looking at now is a unity of methods, not an ontologically based unity like that of the Pythagoreans. Although the modern concept of symmetry at first seems to make less of a claim, it actually opens up new possibilities and applications. History shows us not only different periods in cultural evolution, but also a variety of different cultural traditions and approaches, with different aesthetic standards, that frequently seem “strange” to each other. The new methods facilitate intercultural comparison for working out the structures held in common and the difference, analogously to the study of different native languages in linguistics. Music is an international language of cultural globalization. Music theory demonstrates that pieces of music can be considered models of mathematical structures in the formal representation of a particular symbolic language. Laws of symmetry in musical tone scales and harmonics have been pointed out ever since the Pythagoreans [8.50]. In the modern era J.P. Rameau based melody on laws of harmony in his “Trait´e de l’harmonie” (1722), and they played a rˆ ole on into the late Romantic period [8.51]. D’Alembert, the Encyclopedist, mathematician and philosopher, tried to make aesthetic taste an objective matter, in keeping with Rameau. It was Sch¨ onberg who first broke with the classical canon of harmony totally in order to allow for greater potentialities of form in the 12-tone technique [8.52]. The symmetries of tone scales and harmonics can be described as cyclical groupings and thus traced back to rotation symmetries. Thus, for example, the 12-step semi-tone scale constitutes a group C12 with the half tones C, C#, D, D#, E, . . . etc. as rotations 0 · π/6, 1 · π/6, 2 · π/6, 3 · π/6, . . . (Fig. 107, (1)) [8.53]. The two possible whole-tone scales are defined by two cyclical subgroups of C12 , namely those that result from rotations by an angle n · 2π/6 (2).

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Fig. 107.

Group-theoretical symmetries of tone scales

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Rotations by n · π/4 produce three cyclical subgroups for the three types of diminished seventh chords (3), whole rotations of n · 2π/3 analogously produce four cyclical subgroups for the four different triads (4). It is noteworthy that rotations of n · 5π/6 = n · 150◦ starting clockwise from C convert the half-tone circle into the circle of fourths, and counterclockwise into a circle of fifths. In harmonics, triads can be represented by (non-equilateral) triangles in a half-tone circle. An interesting thing about this is that major and minor triads form congruent, but reflection-symmetrical, triangles (5). Further studies of cyclical groups in harmonics can be carried over to other tone scales and permit intercultural comparisons of different conceptions of harmony. The possible symmetry characteristics of concrete examples of composition are especially interesting [8.54]. For the sake of interpreting symmetry operations compositionally, let us first elucidate the concept of a musical space by analogy to geometrical ornamentation and crystallography. Since the 19th century, the dimensions of n-dimensional spaces are interpreted and applied in musically different ways. For that reason it seems useful, as an example, to interpret the temporal course of a composition (notationally from left to right), using the division into measures as metrics, as a dimension or a coordinate axis. The pitch (frequency) which is measureable by the staff lines, provides a second dimension. The volume (dynamics) provides a third. For more precise analyses one could draw on the timbre (tone color) as a fourth dimension. What is decisive for the concept of space is, simply, to agree on an appropriate metric [8.55]. Now individual symmetry operations such as translation, reflection or rotation can be examined in the musical space that is defined by the dimensions of time, frequency and dynamics. Thus translations in the dimension of time are interpreted as repetition of tones; translations in the dimension of frequency are interpreted as the parallel direction of voices; translations in dynamics are interpreted as crescendo or diminuendo. Rotations such as C2 can be examined on the axis of dynamics, that is on the notational plane of time and frequency. Reflections in the plane of time and frequency correspond

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Fig. 108.

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Symmetry of reflection in Sch¨ onberg’s Waltz Number 5

to an increase or decrease in the volume. Reflections in the plane of dynamics and time are possible, that is, on the horizontal plane of the notation. Reflections along the vertical, that is, the plane of frequency-dynamics, are more familiar. This is the so-called retrogression which appears frequently in the 15th and 16th century and in classical pieces by Bach (“Musical Sacrifice”) and is employed systematically in the serial music of Sch¨onberg’s 12-tone technique. Fig. 108 shows the four prototypes of the Sch¨ onberg series from his Waltz Number 5 (“Five Piano Pieces,” Op. 23), which emerges from the normal form by reflection on the horizontal, the vertical (= retrogression) and the combination of both reflections. Because the intervals of semitones were written in an inhomogeneous manner, the symmetries are not shown uniformly [8.56]. The question as to whether the combination of symmetry operations is applicable to musical “ornaments,” marks a highpoint in this kind of group theory analysis. Examples of notation in the works of Bach can be systematically cited for the seven one-sided stripe ornaments in the notational plane [8.57]. An example of a sequence of ornament symmetries is measures 27–29 of the first movement of the piano sonata, Op. 53 (the “Waldstein Sonata”) of van Beethoven (Fig. 109). What comes into consideration here are the stripe ornaments of the frieze groups Fig. 18 a (5), (2), and (1) in Fig. 109. This sequence of ornament clearly presents a reduction in complexity of symmetry if one stops to consider that group (1) consists only of translation, group (2) of translation and reflection, yet group (5)

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Fig. 109.

Ornament symmetries in Beethoven’s Waldstein sonata [5.58]

consists of a combination of the symmetry operations of the frieze groups (1), (2), (3) and (4). Reductions in the complexity of symmetry can be interpreted psychologically as a relaxation of tension that takes place in the listener while the piece of music is being played. This gives an indication of how to objectify even subjective effects of music by analyses of symmetry. Symmetry breakings in music, which also have great psychological charm, can of course be recognized only when the structures of symmetry from which they deviate are well-known. “The heart and the brain in music” (Sch¨ onberg) [8.59] are therefore not opposites but instead different aspects of a work of art that correspond to different human abilities. Obviously, the old Pythagorean idea of the quadrivium can be directly transformed into modern mathematical considerations. A piece of music is a model of a mathematical structure in the formal language of musical notations. Therefore, it can be translated into formal and programming languages in order to implement it on a computer. As formal logic is a universal language allowing all kinds of translations, it can be used as another representation of musical information. Fig. 110 shows the representation of a sample melody (“Fr`ere Jacques” or “Brother John” in English) in predicative logic. At the beginning of the first line in Fig. 110, the key signature indicates one sharp for the key of G. The time signature 4/4 indicates 4 beats per measure with the quarter note having a duration of one time unit. The vertical bars divide the melody in 8 measures, with a total of 32 notes. The vertical position of each note on the staff indicates a tone, designated by a letter from A through G. (The letter may be qualified by a sharp or flat sign or by a number that indicates the octave.) The shape of the note indicates duration: one time unit or beat for a quarter note,

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∃x1 ∃x2 ∃x3 ... ∃x32 : (tone(x1, G) ∧ dur(x1, 1) ∧ next(x1, x2) ∧ tone(x2, A) ∧ dur(x2, 1) ∧ next(x2, x3) ∧ tone(x3, B) ∧ dur(x3, 1) ∧ next(x3, x4) ∧ ... ∧ tone(x32,G) ∧ dur(x32,2)) Fig. 110. Representation of musical information in musical notation and formal logic [8.60]

two units for a half note, or half a unit for an eight note. The horizontal position of each note indicates that it is sounded after the one on the left and before the one on the right. These features, which represent the elements of an ontology for music, can be translated to logic supplemented with three predicates: tone(x, t) (“note x has tone t”), dur(x, d) (“note x has duration d”), next(x, y) (“the next note after x is y”). To represent all 32 notes in the melody, the corresponding formula in predicative logic would require 32 variables, each with an existential quantifier. For each note, there would be three predicates to indicate its tone, duration, and successor. The complete formula would start with 32 existential quantifiers and continue with 32 lines of predicates. The last line is shorter than the others because the last note does not have a successor.

A piece of music has not only a static structure, but it is a timedepending model of a dynamical system. Thus, it can also be represented by all kinds of state-transition systems. Finally, it is a pragmatic question of intention and purpose, which kind of representation for musical information should be preferred in different contexts. Structural properties like symmetry are invariant with respect to different representations. A fundamental revolution in art took place during the first decade of the 20th century in the rise of abstract art. Its goal was to work out a method of representation that would allow the painter to express his view of the world without having to copy reality in the boundaries of our visual experience. There is an amazing parallel to the problems of

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abstract quantum mechanical formalism, which was to be developed beyond the boundaries of human intuition in classical mechanics. Whether artists like P. Picasso and G. Braque studied M. Planck, Einstein or other physicists, is still an open question. Nevertheless, their abstract art expressed the spirit of their cultural and scientific epoch. Actually, their cubism followed C´ezanne’s pronouncement that objects are made of geometrical forms such as spheres, cones and cylinders. In addition there is the recourse to archaic art. The object represented in the picture is broken up into stereometric atoms and then reassembled in a new way for the purpose of making the basic and archetypal forms of the world vivid. In 1912 a theory of cubism was formulated [8.61]. While modern painting has remained limited to pictorial representation, in the twenties of the 20th century the Bauhaus set about giving artistic form to the technical-industrial life-world in a synthesis of the arts of architecture, painting, sculpture and functional art. Here, as in Antiquity, it was a matter of comprehending the human being and his life-world as a unity, but now it was from the point of view of science, technology and industry. Corresponding to the paradigm shift in science there was thus an artistic upheaval and structural change striving towards a new measure of things. The standard is a purpose-oriented functionalism that aims to comprehend the human being in his new life-world. The words “new” and “modern” became the fashion in the twenties, which saw the political and social collapse of old world orders. The use of the word “new” ranged from the “Neues Bauen” (new constructions) and “Neues Wohnen” (new dwelling) by way of the “Neue Sachlichkeit” (new practicality) to Huxley’s “brave new world,” with its disillusionment and irony. Interrupted by war, many projects of modernism were not actualized or further developed until the fifties and sixties. They started with the cultural globalization of a technical-industrial lifeworld. Artists like P. Klee applied the ideas of symmetry and law to abstract art in order to probe the structure and dynamics of forms [8.62]. Klee particularily pointed out the parallel with mathematical natural science. In his study “Exact Experiments in the Realm of

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Art,” he wrote: Art has also been given enough space for exact research, and the gates have been open to it for some time. What was done for music before, until the end of the 18th century, is at least beginning in the field of sculpture. Mathematics and physics are providing the opportunity, in the form of rules for persistence and for alteration. This compulsion to concern oneself first with the functions instead of beginning with the finished form, is a wholesome one. Algebraic, geometric and mechanical tasks provide training toward the essential, the functional, in contrast to the impressive. One learns to see behind the facades and to grasp a thing by its roots. One learns to recognize what is flowing underneath it — the prehistory of the visible — and to dig into the depths and to expose substantiate and analyze. [8.63]

The essay culminates in the demand “learn logic, learn the organism.” The organism, art and mathematical natural science are no longer regarded as antithetical. Instead they are related to each other. In Kandinsky’s book “Point and Line to Surface” [8.64] the elements of form are elucidated in their tensions and harmonious compositions by examples from mathematics and natural science. Concentric star clusters from astronomy, variational possibilities of physical curves, line formations of lightning, structures of animal tissues, swimming movements of microscopic organisms, etc., alternate with constructions of modern technology. Abstraction is no longer “artificial” and “strange,” but corresponds to the newly discovered and created world of forms in nature and technology. In his “theory of sounds” W. Kandinsky demanded an art of rhythms and mathematically abstract constructions beyond traditional configuration like Sch¨ onberg’s 12-tone music: “Every work emerges technically like the universe emerged — from catastrophes generating a symphony from the chaotic roaring of instruments, which is called music of the spheres. Creation of a work is creation of a world.” [8.65] Kandinsky and Klee taught in the Bauhaus in the twenties. Klee was also a practising musician (violinist) who tried to transform musical structures into compositions of pictures. The main challenge of the structural analogy between music and painting is the representation of time. In his “Artistic Doctrine of Forms”, Klee devel-

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oped a kind of graphic state-transition system to represent the phase transitions of rhythms according to fugues of Bach [8.66]. Actually, it should be his “UML” (Universal Modeling Language) providing graphic structures of musical rhythms which he realized in several paintings. In this sense, his paintings can be considered visual models of invariant structures of dynamics [8.67]. With respect to Bach’s polyphony of fugues, Klee called his approach “polyphonic painting,” representing the parallel development of simultaneous voices with tools of painting. From a mathematical point of view, the parallel voices of a fugue illustrate simultaneity, multi-threading and nonlinear interaction. A beautiful example is Klee’s painting “Pastorale (Rhythms)” from 1927 (Fig. 111). The painting has a microstructure with subtle configurations of lines and a macroscopic structure with colored squares which seem to emerge from the microscopic network of meshed lines. The parallel lines with their rich vocabulary of microscopic formal segments remind us of the interacting tones in a polyphonic fugue. The colored squares are the emerging sounds of parallel voices. They are colored, because it is the whole sound which effects our emotions. The microscopic configurations are generated by a variety of symmetries like translation, reflection or rotation in the sense of ornamentics. Similar microscopic segments of lines dis-

Fig. 111. Klee: Pastorale (Rhythms) 1927/20 (K 10) [The Museum of Modern Art, New York. Abby Alderich Rockefeller Fund and Exchange. Inv. No. 157-45]

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play macroscopic clusters and patterns. They seem to emerge one after another from left to right in phase transitions of time-depending dynamics. The colours of the squares are harmonized with the warm brown of earth generating the mood of “Pastorale.” According to the Bauhaus, painting is only one representation of art. The whole environment is to be shaped architectonically, corresponding to the technical-industrial conditions of life. As Gropius wrote “the challenge is to master organizationally the gigantic tasks of our time — the whole traffic, all human work, material and intellectual” [8.68]. Form, function and economic requirements had to be reduced to a common denominator as a task of optimization. One of the central representatives of modernism was, without question, Le Corbusier, who not only achieved magnificent edifices, but also came onto the scene as a theorist. In his guiding principles, “Toward an Architecture” (1920), he says this about the engineer’s aesthetics and architecture [8.69]: The engineer’s aesthetics, architecture: at the deepest level the same, one deriving from the other, the one full-blown, the other secretly developing. The engineer, guided by the law of economy and led by calculations, transposes us into accord with the laws of the universe. He attains harmony. By means of his handling of forms, the architect brings into reality an order that is the pure creation of his spirit: by means of the forms he stirs our senses and awakens our feeling for form. The interconnections that he produces give rise to a deep echo in us: he shows us the standard for an order that we feel to be in accord with the world order. He brings about manifold motions of our mind and our heart: thus beauty becomes experience for us. [8.69]

The outlook on architectural and social problems is striking. Symmetry in construction is required to correspond to the “balance of the social order.” In this context Le Corbusier developed the concept of “mass-produced houses” for which “spiritual preconditions must be created”. A passage follows which elevates Le Corbusier outright to the Platonist of modern architecture: Geometry is the means we ourselves have created for ourselves so that we can overcome our surroundings and express ourselves. Geometry is the foundation. It is at the same time the material bearer of the symbols that signify perfection and the divine. It bestows on us the sublime satisfactions of mathematics. The

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machine proceeds from geometry; its dreams set out to find the joys of geometry. The modern arts and modern thinking, after a century of analysis, seek their salvation beyond accidental facts, and geometry conducts them to a mathematical order, to a more and more generalized posture . . . Such passion ensouls deeds, brings forth actions, drenches them in its color, gives them direction. The name of this passion today is: exactitude. An exactitude carried to great length and elevated to the ideal: the striving for perfection . . . [8.70]

However, the Platonic emphasis on geometry and harmony in the architecture of modernism is by no means associated with elitistaristocratic tendencies. In a paper by F. Schumacher about “Social City Planning”, (1919), he wrote: A contemporary metropolis can become harmonious only when its structures, at defining points, conform to certain rhythms that run through the whole. Their sizes and their arrangement must be regulated to express a sense of unity. Inside this elastically planned framework the particular and individual can unfold all the more freely then, undisturbed by any contingencies. [8.71]

Social and artistic harmony is seen in a unity in which social and cultural politics and economics are to be coordinated with each other: One can see that bringing about social and artistic harmony requires many kinds of laws and therefore all these questions lead directly into politics. Not that it would be possible to actualize a social or artistic idea just with laws — the creative act is required; laws clear a path for it.

Meanwhile the combined functionalism of modernism has gotten on into years and is worse for wear. These “breakings of symmetry” are most noticeable in architecture. Functionalism became debased, in part, into an “international style” of desolate and unimaginative structures that concreted shut the metropolises of a global world, veered into the opposite of its original intentions and single handedly reduced the cities to uninhabitable exhaust- and noise-plagued knots of traffic, municipal administration, banks and business. The idea of the city as an antiform to nature, as modernism presented it again and again, had taken on a life of its own and had turned against human nature [8.72]. In architecture the critique of modernism has used the catchword postmodern for years, and has expanded into a general cultural criticism using catchwords like “post-industrial society,” “post-

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structuralism,” “postmodern scientific theory,” etc. [8.73]. A thread running through this sometimes brilliant discussion is that it does not complain of the “loss of the center” and “modernism’s symmetry breaking” as a cultural decline, but, rather, as an achievement that offers new opportunities. Now, the ironic treatment of “unity” and “Logos” in modernism from the beginning of this century is not new: in Dadaism, modernism was, to a degree, keeping its own court fool, one that had substituted the principle of “accident” for the “Logos” and that in artists’ happenings constantly reminded the protagonists of modernism of the consequences of a “falling away from the center” — just like once the court fools in the Medieval world of divine order. By contrast, postmodern cultural criticism doubts the possibility of a “unity” and a “center” in principle and criticizes them as excessive demands on human reason. Reason is taking dangerous paths in centralization, rationalization and bureaucratization and can shift into totalitarianism, as proven by the most recent historical examples. Adorno’s critique of positivism and rationalism should be understood against the background of such a “dialectics of Enlightenment.” Nevertheless, Adorno emphasizes that critique does not cause “categories such as unity and self-harmony” to disappear without a trace. Rather, “the principle of harmony remains in play, transformed beyond recognition”. Besides, according to T.W. Adorno, “the logicity of art” consists of “the balance of the coordinated, of that homeostasis in the concept of which aesthetic harmony is sublimated as the ultimate.” Again asymmetry is comprehensible only against the background of a hidden symmetry [8.72]. Here it is important for modernism to emphasize again that its “center” and “symmetry” are not to be confused with external simple symmetry characteristics such as reflection symmetry or axial symmetry. In modern natural science as well, the external geometrical symmetry characteristics of individual bodies emphasized since Antiquity, play a rather subordinate role. What is decisive are the uniform, comprehensive (but abstract) symmetry characteristics that are expressed in the mathematical structures of scientific theories. Analogously, the concept of symmetry that is intended in the archi-

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tecture of modernism should be seen as a characteristic of a uniform structuralism and functionalism. In this sense the architecture of postmodernism comes to “breakings of symmetry.” Uniform functionalism is broken up. Ornament and decoration are permitted again, and we find in one and the same postmodern building diverse historical styles quoted in partly ironic alienation — from the small oriel with its Medieval effect, to the Baroque stucco work above the window to the Ionic pillar by the entrance. For many a contemporary it is here that the paucity of ideas of an epigonal era becomes visible, characterized by eclecticism and historicism. Others refer to the ironic breaking of history that allows for playfully fishing in old boxes of building blocks for the styles of former eras. Throughout postmodernism, reflection symmetry and the Golden Section appear as historical quotations of external symmetry which nevertheless are included as set pieces and do not determine the uniform structure of the building — neither in the sense of Antiquity nor of modernism. Where successful postmodern architecture appears, it is characterized by a loosening up of the strict purism, openness and “pointillism” of the styles that seem to come together by chance, but at a second glance, at the latest, prove to be a successful and original ensemble of styles. The emphasis on the “selective,” the “casual” and the “individual” versus “unity” and “generality” mirrors a postmodern outlook on life that — on the basis of relevant experiences — reacts rather skeptically and at best ironically to the claims of being the only true technical reason and the belief that modern rationalism will result in universal feasibility and solution to problems. With a new ecological awareness of the environment, old ideas of natural philosophy come into play again: the living space that a city offers is really liveable only when it is in ecological balance with nature as its environment. The requirements of a natural environment extend from nearby recreational areas to oxygen providers (“green lungs”). On closer examination, however, that does not constitute a rift with modernism, whose functionalism was directly intended to produce a livable environment. But an industrial monstrosity in the landscape does not exactly fulfill these functional requirements.

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Ecological, social, economic, political, cultural and religious problems are common challenges for mankind. In the age of globalization, mankind is growing together, but, on the other hand, the complex dynamics of development generate diversity and heterogeneity with great tensions and conflicts. In short: globalization also means symmetry breaking. Nevertheless, people, regions, nations, and continents depend on one another in complex networks. Thus, we need intercultural understanding and tolerance, because no one can dominate the other without endangering the balance (“symmetry”) of the whole network. Peace and tolerance are enforced by the common interests of mankind in order to realize welfare and a sustainable future. Peace, tolerance and welfare are common universals of mankind. Cultural diversity is a source of human creativity which is expressed in art, music and science. In this sense, globalization demands unity in diversity. The age of globalization is also the age of computers, because the just-in-time society of a global world is only possible by computational networks. In the age of computers, the old slogan of Renaissance “ars sine scientia nihil est” (art without science is nothing) gets new relevance. How is art with computers possible? The commercial design of products, advertising and publicity are mainly generated by computational tools. Computer-assisted movies generate a virtual world of artifical life and culture. Obviously, computer programs open new views on actual and possible worlds. Attractors of complex systems can only be visualized by computational means. The beauty of fractals results from endless iterations of self-similar computational forms and colors. Fig. 112 shows iterated fractal structures in Julia sets of the Mandelbrot set (Fig. 30). Computers discover symmetry of invariant structures hidden behind the appearance of chaotic attractors. Symmetry is still related to complexity. We may be fascinated by the insights into a perfect Platonic world of infinity, symmetry and complexity. But do computers generate art [8.76]? As far as we know symmetry and complexity of fractals are also generated by nature. People like the symmetry of leaves and flowers. In time-series analysis of complex dynamical systems (e.g. EEG-data of brain, data flow of the Internet (Fig. 86)), we de-

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ÿ

Fig. 112.

The beauty of computational symmetry [8.75]

tect self-similarity of patterns on different scales. Because of stochastic fluctuations and natural or technical boundaries, they are not as perfect and infinite as the Mandelbrot set. Nevertheless, people are fascinated by the regularities in a world of diversity. Regularities in patterns are models of mathematical structures which people may like or dislike. Thus, beauty depends on our sensation and cognition of forms, patterns and structures. There may be local (“subjective”) differences of our estimation, but there are also global invariants of beauty. Man, nature and computers generate structures which we feel to be beautiful. But art needs creativity. Are nature and computers creative? A necessary condition of creativity is the emergence of new structures, patterns and ideas. At the end of this book we all know: emergence is no mystery, but a result of phase transitions in complex systems like the universe and the brain. Even if an artist’s idea seems to be a random flash of inspiration, it needs the background knowledge and know-how of an artist to create an artistic

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work. Thus, in the case of brains, creativity also means intentionality. Contrary to cosmic and biological evolution, human minds try to realize goals and intentions which we may like or dislike. In this sense, cosmic expansion and biological evolution are emergent, but not creative. Creativity is an ability of complex cognitive systems like human beings. According to the principle of computational equivalence, artificial systems with cognition, emotion and intentions are not impossible (compare Sec. 7.2). Therefore, in a technical co-evolution, artificial minds could emerge, creating their own artistic work. But because of their nonlinear dynamics, they would have their own intimacy of feelings and artistic estimation which might be completely strange to us or not. The same may be true for aliens which may develop on foreign planets elsewhere in the universe. Like humans, they are children of the same universe. As they all, natural and

Fig. 113. Symmetry and symmetry breaking in complex cognitive systems [Montage based on a figure (left) from Descartes’ “Treatise of Man”: The mirrored portrait (right) shows a brain map of the EEG during perception of virtual shapes. Both portraits together make up the vase-face ambiguity (Fig. 91)] [8.77]

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artificial minds, evolve according to the laws of cosmic and biological evolution, they also could be fascinated by the invariant structures of symmetry and complexity.

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References

Chapter 1 1.1

Reichard, G.: Navaho Religion. A Study of Symbolism. 2 vols. Pantheon Books: New York (1950). 1.2 Nowotny, K.A.: Beitr¨ age zur Geschichte des Weltbildes. Farben und Weltrichtungen. Berger: Horn/Vienna (1970), p. 195. 1.3 Witherspoon, G.: Language and Art in the Navajo Universe. University of Michigan Press: Ann Arbor (1977), p. 154. 1.4 Brown, W.N.: A Descriptive and Illustrated Catalogue of Miniature Paintings of the Jaina Kalpasutra. Freer Gallery of Art: Washington (1934). 1.5 Gopinatha Rao, T.A.: Elements of Hindu Iconography. Vol. 1. The Law Printing House: Madras (1914), Inanarnavatantra X 39. Cf. also Nowotny [1.2], p. 100. 1.6 I Ching — The Book of Changes. Ed. by R. Wilhelm. Diederichs: Jena (1924), repr. D¨ usseldorf/Cologne (1973). 1.7 The 13 Books of Euclid’s Elements. Transl. from the Text of Heiberg with Introduction and Commentary by T.L. Heath. 3 vols. Dover Publications: New York (1956). 1.8 Mainzer, K.: Geschichte der Geometrie. B.I. Wissenschaftsverlag: Mannheim/Vienna/Z¨ urich (1980). 1.9 Kepler, J.: Collected Works. Vol. 6: Harmonice Mundi. Transl. and introduced by M. Caspar. C.H. Beck: Munich/Berlin (1939). 1.10 On the Pythagoreans’ doctrine of harmony, cf. van der Waer389

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1.13

1.14

1.15

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den, B.L.: Die Harmonielehre der Pythagoreer. In: Hermes 78 (1943), pp. 163–199. Cf. Mainzer [1.8] and Mainzer, K.: Real Numbers. In: Ebbinghaus, H.-D. Hermes, H. Hirzebruch, F. Koecher, M. Mainzer, K. Neukirch, J. Prestel, A. Remmert, R.: Numbers. Springer: Berlin/Heidelberg/New York (1990), pp. 28–30. Leonardo of Pisa: Scritti de Leonardo Pisano. Vol. 1: Il liber abbaci (liber abaci 1202/1228). Ed. B. Boncompagni. Tipografia delle Scienze Matematiche e Fisiche: Rome (1857), chapt. XII; also cf. Archibald, R.C.: Golden Section. In: American Mathematical Monthly 25 (1918), pp. 232– 238; Coxeter, H.S.M.: Unverg¨angliche Geometrie. Birkh¨auser: Basel/Stuttgart (1963), p. 206. Aristotle: Nikomachische Ethik. Translated by F. Dirlmeier, notes by E.A. Schmidt. Reclam: Stuttgart (1969), 1131 b 9–32. Sachs, A.: Babylonian observational astronomy. In: Philosophical Transactions of the Royal Society of London A 276 (1974), pp. 43–50; cf. also van der Waerden, B.L.: Geometry and Algebra in Ancient Civilizations. Springer: Berlin/Heidelberg/New York (1983); Neugebauer, O.: The Exact Sciences in Antiquity. Brown University Press: Providence (2nd ed. 1957). Cf. also Needham, J.: Astronomy in Ancient and Medieval China. In: Philosophical Transactions of the Royal Society of London A 276 (1974), pp. 67–82. J. Mittelstraß: Die Rettung der Ph¨ anomene. Ursprung und Geschichte eines antiken Forschungsprinzips. De Gruyter: Berlin (1962); Mainzer, K.: Grundlagenprobleme in der Geschichte der exakten Wissenschaften. Universit¨ atsverlag: Constance (1981), p. 8. Cf. also Hanson, N.R.: Constellations and Conjectures. Reidel: Dordrecht/Boston (1973), p. 101. Bruins, E.M.: La chimie du Tim´ee. In: Revue de M´etaphysique et de Morale LVI (1951), pp. 269–282. Diels, H.: Die Fragmente der Vorsokratiker. 6th ed., revised by W. Kranz. 3 vols. Weidmannsche Verlagsbuchhandlung:

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1.20 1.21 1.22 1.23 1.24 1.25

1.26 1.27 1.28 1.29 1.30

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Berlin (1960/1961) (abbrev.: Diels-Kranz), 12 A 10 (PseudoPlutarch). Diels-Kranz [1.19], 13 A 5, B 1. Diels-Kranz [1.19], 22 B 64, B 30. Heisenberg, W.: Physik und Philosophie. Ullstein: Frankfurt (1970), p. 44. Diels-Kranz [1.19], 22 B 8. Diels-Kranz [1.19], 31 B 8. Here one should recall the writings of Aristotle in natural science, the Physics, consisting of 8 books, De caelo (4 books), Meteorologica (4 books), De generatione et corruptione (2 books), a series of biological and physiological writings, above all, Historia animalium, a natural history of animals (to be sure, not exclusively written by Aristotle), De partibus animalium, De motu animalium, as well as De respiratione. Aristotle, Physics I.1, 184 a. Aristotle, Physics III, 202 a 10–15. Aristotle, Physics II, 3. Aristotle, Physics VII. Feng Yu-Lan: A History of Chinese Philosophy. Vol. 2: The Period of Classical Learning. Princeton University Press: Princeton (1953), p. 120.

Chapter 2 2.1

2.2 2.3 2.4 2.5 2.6

Leibniz, G.W.: Zur Analysis der Lage. In: Hauptschriften zur Grundlegung der Philosophie. Vol. 1. Ed. by E. Cassirer, transl. by A. Buchenau. Felix Meiner: Leipzig (1904), p. 73. Van der Waerden, B.L.: Algebra. Vol. 1. Springer: Berlin/Heidelberg/New York (1966), §9. Veblen, O./Young, J.W.: Projective Geometry. Vol. 1. Ginn: Boston (1918), pp. 61 ff. Weyl, H.: Symmetry. Princeton University Press: Princeton (1952). Cf. also Speiser, A.: Die Theorie der Gruppen von endlicher Ordnung. Birkh¨ auser: Basel/Stuttgart (4th ed. 1956), §4. Shubnikov, A.V. Koptsik, V.A.: Symmetry in Science and

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2.13 2.14

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2.16 2.17 2.18 2.19

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Art. Plenum Press: New York/London (1974), pp. 79 ff.; cf. also Wolf, K.L. Wolff, R.: Symmetrie. Vol. 2. B¨ ohlau: M¨ unster/Cologne (1956), pp. 132 f. Shubnikov [2.6], p. 157; Wolf [2.6], pp. 142. Cf. also Coxeter [1.12]; Weyl [2.4]. Mainzer [1.8], p. 55. Speiser [2.5], §33. Burckhardt, J.J.: Die Symmetrie der Kristalle. Birkh¨ auser: Basel/Boston/Berlin (1988). Klein, F.: Das Erlanger Programm. In: Gesammelte Mathematische Abhandlungen. Vol. 1. Springer: Berlin (1921), p. 463. Lie, S.: Abhandlungen. Vol. 5. Teubner: Leipzig/Kristiana (1924), p. 586. Gauss, C.F.: Disquisitiones generales circa superficies curvas. In: Werke. Vol. 4. Teubner: Leipzig (1880), pp. 217 ff.; Mainzer [1.8], pp.157 ff. ¨ Riemann, B.: Uber die Hypothesen, welche der Geometrie zugrunde liegen. Springer: Berlin (1919). See also: Gesammelte mathemematische Werke. N. XIII Habilitations-Vortrag (2nd ed. 1893); reprinted (with notes by H. Weyl) in the collection Weyl, H. et al.: Das Kontinuum und andere Monographien. Chelsea: New York (1973). Eisenhart, L.P.: Riemannian Geometry. Princeton University Press: Princeton (1926). Lie, S. Engel, F.: Theorie der Transformationsgruppen. 3 vols. Teubner: Leipzig (1888–1893). Freudenthal, H.: Lie groups in the foundations of geometry. In: Advances in Mathematics 1 (1965), pp. 145–190. Cartan, E.: Les espaces riemanniens sym´etriques. In: Verh. Intern. Mathem.-Kongr. I. Z¨ urich (1932), pp. 152–161; see also Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press: New York (1962). Elliott, J.P. Dawber, P.G.: Symmetry in Physics. Vol. 1: Principles and Simple Applications. Macmillan: London/ Basingstoke (1979), chapt. 4.

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2.21 Kaplan, D. Glass, L.: Understanding Dynamics. Springer: Berlin/Heidelberg/New York (1995), p. 210; Haken, H.: Synergetics. An Introduction. Springer: Berlin/Heidelberg/New York (3rd ed. 1983), pp. 106 ff. 2.22 Cf. also Haken [2.21], pp. 108 ff. 2.23 Cf. also Haken [2.21], pp. 110 ff. 2.24 Arnold, V.I.: Ordinary Differential Equations. M.I.T. Press: Cambridge Mass. (1978); Hirsch, M.W. Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press: New York (1974). 2.25 Nicolis, G. Prigogine, I.: Die Erforschung des Komplexen. Piper: M¨ unchen (1987), Fig. 3.18–3.19. 2.26 Nicolis [2.25], Fig. 3.26–3.27. 2.27 Abraham, R.H. Shaw, C.D.: Dynamics — The Geometry of Behavior. Part 4: Bifurcation Behavior. Aerial Press: Santa Cruz (1984), p. 179. 2.28 Thom, R.: Structural Stability and Morphogenesis. Benjamin/ Cummings: Reading Mass. (1975). 2.29 Abraham [2.27], p. 46. 2.30 Cf. [2.28]. 2.31 Cf. also Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman: San Francisco (1982). 2.32 Fatou, P.: Sur les ´equations fonctionelles. In: Bull. Soc. Math. Fr. 47 (1919/1920), pp. 161–271, 48, pp. 208–314; Julia, G.: Sur l’iteration des fonctions rationelles. In: Journal des Math´ematique Pure et Appliqu´ee 8 (1918), pp. 47–245. 2.33 Peitgen, H.-O. Richter, P.H.: The Beauty of Fractals. Images of Complex Dynamical Systems. Springer: Berlin/ Heidelberg/New York (1986), p. 10. 2.34 Douady, A. Hubbard, J.H. Iteration des polynomes quadratiques complexes. In: CRASH Paris 294 (1982), pp. 123–126. Chapter 3 3.1

Cf. also Mainzer, K.: Thinking in Complexity. The Computational Dynamics of Matter, Mind, and Mankind. Springer: Berlin/Heidelberg/New York (4th ed. 2004), chapt. 2.6; Abar-

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3.2

3.3

3.4

3.5

3.6 3.7

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banel, H.D.I.: Analysis of Observed Data. Springer: New York (1996); Kanz, H. Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press: Cambridge (1997). For proofs, see also Hamermesh, M.: Group Theory and its Application to Physical Problems. Addison-Wesley: Reading Mass. (1962). Schmutzer, E.: Symmetrien und Erhaltungss¨ atze der Physik. Akademie-Verlag: Berlin/Oxford/Braunschweig (1972), p. 56 ff. Noether, E.: Invariante Variationsprobleme. I. Nachrichten der Gesellschaft der Wissenschaften zu G¨ottingen. MathematischPhysikalische Klasse (1918), pp. 235–257. Minkowski, H.: Raum und Zeit (Lecture 1908). In: Lorentz, H.A. Einstein, A. Minkowski, H.: Das Relativit¨ atsprinzip. Eine Sammlung von Abhandlungen. Teubner: Leipzig/Berlin (1913), repr. Wissenschaftliche Buchgesellschaft: Darmstadt (8th ed. 1982), pp. 54–66; cf. also Pension, L.: H. Minkowski and Einstein’s Theory of Relativity. In: Archiv for History of Exact Sciences 17 (1977), pp. 71–95; Einstein, A.: The Meaning of Relativity. Princeton University Press: Princeton (1988). Cf. also Jackson, J.D.: Classical Electrodynamics. John Wiley: New York (1975), pp. 515 ff. Weyl, H.: Raum, Zeit, Materie. Vorlesungen u ¨ ber allgemeine Relativit¨ atstheorie. Springer: Berlin (1923), repr. Wissenschaftliche Buchgesellschaft: Darmstadt (1961); Philosophy of Mathematics and Natural Science. Princeton University Press: Princeton (1949); see also Ehlers, J.: The Nature and Structure of Spacetime. In: Mehra, J. (ed.): The Physicist’s Conception of Nature. Reidel: Dordrecht/Boston (1973), pp. 94 ff.; Mainzer [1.8], chapt. 5.35. Cf. also Wigner, E.P.: Symmetry and Conservation Laws. In: Symmetries and Reflections. Scientific Esays of Eugene P. Wigner, Indiana University Press: Bloomington/London (1967), pp. 22 ff.; Fock, V.: The Theory of Space, Time and Gravitation. Pergamon Press: New York (1959). Weinberg, S.: Gravitation and Cosmology. Principles and Ap-

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3.10

3.11

3.12 3.13

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plications of the General Theory of Relativity. John Wiley: New York (1972), pp. 25 ff.; Hawking, S.W. Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press: New York (1973); Hawking, S.W.: A Brief History of Time. Bantam Books: New York (1988); Hawking, S.W. Penrose, R.: The Nature of Space and Time. Princeton University Press: Princeton (1996). Mie, G. Grundlagen einer Theorie der Materie (I). In: Annalen der Physik 37 (1912), pp. 511–534; (II). 39 (1912), pp. 1–40; (III). 40 (1913), pp. 1–66. Hilbert, D.: Die Grundlagen der Physik. In: Nachrichten der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ottingen (1915), pp. 395–407; (1917), p. 201; Mathematische Annalen 92 (1924), p. 1; cf. also Mehra, J.: Einstein, Hilbert, and the Theory of Gravitation. In: Mehra [3.7], pp. 137 ff. See also Hund, F.: Geschichte der Quantentheorie. B.I. Wissenschaftsverlag: Mannheim/Vienna/Z¨ urich (2nd ed. 1975). Historically, the group theory concept of symmetry in quantum mechanics was first applied in the texbooks of Weyl, H.: Gruppentheorie und Quantenmechanik. Hirzel: Leipzig (1931); Wigner, E.P.: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg: Braunschweig (1931); van der Waerden, B.L.: Die gruppentheoretische Methode in der Quantenmechanik. Springer: Berlin (1932). For modern textbook representations see also Boardman, A.D. O’Connor, D.E. Young, P.A.: Symmetry and its Application in Science. MacGraw-Hill: London/New York (1973), chapt. 9; Elliott, J.P. Dawber, P.G.: Symmetry in Physics. Vol. 5. Macmillan: London (1979), chapt. 5; Greiner, W. M¨ uller, B.: Theoretische Physik. Vol. 5: Quantenmechanik II: Symmetrien. Harri Deutsch: Thun/Frankfurt (1985). For a modern textbook explanation, see also Itzykson, C. Zuber, J.-B.: Quantum Field McGraw-Hill: Theory. New York (1980), chapt. 2; Elliott [3.13], vol. 2, pp. 380 ff. Marshak, R.E. Riazuddin/Ryan, C.P.: Theory of Weak Interaction in Particle Physics. John Wiley: New York (1969).

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3.16 Lee, T.D. Yang, C.N.: Questions of Parity Conservation in Weak Interaction. In: Physical Review 104 (1956), p. 254; Wu, C.S. Amber, E. Heyward, R.W. Hoppes, D.D. Hudson, R.P.: Experimental Test of Parity Conservation in Beta-Decay. In: Physical Review 105 (1957), pp. 1413–1415. 3.17 See also Abers, E.S. Lee, B.W.: Gauge Theories. In: Physics Reports 9 C (1973), p. 1; Rollnik, H.: Ideen und Experimente f¨ ur eine einheitliche Theorie der Materie. In: RheinischWestf¨alische Akademie der Wissenschaften Vortrag N 296. Opladen (1979), pp. 18 ff. 3.18 Salam, A.: Weak and electromagnetic interactions. In: Svartholm, N.: Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8). Almqvist and Wiksells: Stockholm (1968), p. 367. 3.19 For a historical overview on quantum chromodynamics cf. Mainzer, K.: Symmetries of Nature. De Gruyter: Berlin/New York (1996), chapt. 4.33 and, of course, the pioneer of quarks and complex systems Gell-Mann, M.: The Quark and the Jaguar. Freeman: New York (1994). 3.20 Bernstein, J.: Spontaneous Symmetry Breaking, Gauge Theories, the Higgs Mechanism and all that. In: Revise Reports of Modern Physics 46 (1974), pp. 7–48; Davies, P.C.W.: Superforce. Simon & Schuster: New York (1984); Georgi, H.: Why Unify? In: Nature 288 (Dec. 1980), pp. 649–651; Georgi, H. Glashow, S.L.: Unity of all Elementary-Particle Forces. In: Physical Review Letters 32 vol. 8 (1974), pp. 438–441; Glashow, S.L.: Interactions. Time-Warner Books: New York (1988); Weinberg, S.: The search for unity. Notes for a history of quantum field theory. In: Daedalus 106 (4) (1977), pp. 17–35. 3.21 Wheeler, J.A.: A Journey into Gravity and Spacetime. Scientific American Library: New York (1990). 3.22 Greene, B.: The Elegant Universe. Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. W.W. Norton: New York/London (1999), p. 144, Fig. 6.2. 3.23 Davies, P.C.W./Brown, J. (eds.): Superstrings: A Theory of

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3.25 3.26 3.27 3.28 3.29 3.30

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3.32 3.33

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Everything? Cambridge University Press: Cambridge Eng. (1988). Kaluza, T.: Zum Unit¨ atsproblem der Physik. Sitzungsberichte der preußischen Akadademie der Wissenschenschaften zu Berlin (1921), pp. 966–972; Klein, O.: Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie. Zeitschrift f¨ ur Physik 37 (1926), pp. 895–906; Zur f¨ unfdimensionalen Darstellung der Relativit¨atstheorie. Zeitschrift f¨ ur Physik 36 (1926), pp. 188– 208; J. Phys. Rad. 8 (1927), p. 24; for a generalization see also Veblen, O. Hoffmann, D.: Projective Relativity. In: Physical Review 36 (1930), p. 810. Cf. Greene [3.22], Part IV. Audretsch, J. Mainzer, K.: Vom Anfang der Welt. C.H. Beck: M¨ unchen (2nd ed. 1990), p. 98. Cf. also Bernstein [3.20]. Weinberg, S.: The First Three Minutes. Basic Books: New York (1993). Guth, A.H.: The Inflationary Universe. Addison-Wesley: Reading Mass. (1997). Steigman, G.: Observational tests of antimatter cosmologies. In: Annual Review of Astronomy and Astrophysics 14 (1976), pp. 339–372; Toussaint, D. Treiman, S.B. Wilczek, F. Zee, A.: Matter-antimatter accounting, thermodynamics, and blackhole radiation. In: Physical Review D 19 (Febr. 1979), pp. 1036–1045. Prigogine, I.: From Being to Becoming — Time and Complexity in Physical Sciences. Freemann: San Franciso (1980); Introduction to Thermodynamics of Irreversible Processes. John Wiley: New York (1961); Boccara, N. (ed.): Symmetries and Broken Symmetries in Condensed Matter Physics. IDSET: Paris (1981). Mainzer, K.: The Little Book of Time. Copernicus: New York (2002), chapt. 5. Lorenz, E.N.: Deterministic flow. In: J. Atmos. Sci. 20 (1963), p. 130; Schuster, H.G.: Deterministic Chaos. An Introduction. Physik-Verlag: Weinheim (1984), p. 9.

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3.34 Fig. 44 from Feynman, R.P. Leighton, R.B. Sands, M.: The Feynman Lectures of Physics. Vol. II. Addison-Wesley: Reading Mass. (1965). 3.35 Haken [2.21], p. 5. 3.36 The experiment dates back to J. Scott Russell: Report on waves. 14th meeting of the British Asociation for the Advancement of Science (BAAS) (1844) and is described by Scott, A.: Nonlinear Science. Oxford University Press: Oxford (2003), p. 2. 3.37 Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer: Berlin (1978); Davies, P.C.W.: The Physics of Time Asymmetry. Surrey University Press: London (1974); Penrose R.: The Emperor’s New Mind. Oxford University Press: Oxford (1989), p. 181. 3.38 Lichtenberg, A.J. Libermann, M.A.: Regular and Stochastic Motion. Springer: Berlin (1982); Schuster [3.33], p. 137. 3.39 Poincar´e, H.: Les M´ethodes Nouvelles de la M´ecanique C´eleste. Gauthier-Villars: Paris (1892). 3.40 Arnold, V.I.: Small denominators II. Proof of a theorem of A.N. Kolmogorov on the the preservation of conditionally-periodic motions under a small perturbation of the Hamiltonian. In: Russian Mathematical Surveys 18 (1963) p. 5; Kolmogorov, A.N.: On conservation of conditionally-periodic motions for a small change in Hamiltonian function. In: Doklady Akademy Nauk USSR 98 (1954), p. 525; Moser, J.: Convergent series expansions of quasi-periodic motions. In: Mathematische Annalen 169 (1967), p. 163. 3.41 Birkhoff, G.D.: Nouvelles recherches sur les syst`eme dynamiques. In: Mem. Pont. Acad. Sci. Novi Lyncaei 1 (1935), p. 85. 3.42 Eckmann, J.P.: Roads to turbulence in dissipative dynamical systems. In: Rev. Mod. Phys. 53 (1981), p. 643; computer simulation of Fig. 47 from Lanford, O.E.: Turbulence Seminar. In: Bernard, P. Rativ, T. (eds.): Lecture Notes in Mathematics 615. Springer: Berlin (1977), p. 114. 3.43 Landau, L.D. Lifshitz, I.M.: Course of Theoretical Physics. Vol.

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5: Statistical Physics. Pergamon Press: London/Paris 1959. Landau only considered phase transitions in thermal equilibrium; cf. also Brout, R.: Phase Transitions. Benjamin: New York (1965); Domb, C. Green, M.S. (eds.): Phase Transitions and Critical Phenomena. 5 vols. Academic Press: London (1972–1976). Cf. Haken [2.21]. Haken, H.: Encyclopedia of Physics. Vol XXV/2c. Light and Matter. Springer: Berlin (1970). Cf. also Satter, D.: Topics in Stability and Bifurcation Theory. Springer: Berlin (1973). For linear-stability analysis cf. also Guggenheimer, J. Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer: Berlin (1983); Nicolis, G. Prigogine, I.: Self-Organization in Non-Equilibrium Systems. John Wiley: New York (1977).

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Biggs, N.L. Loyd, E.K. Wilson, R.J.: Graph-Theory 1736–1936. Clarendon: Oxford (1990), p. 55. Partington, J.R.: A Short History of Chemistry. Macmillan: London (1965), pp. 300 ff. K´ekul´e, A.: Lehrbuch der organischen Chemie. Enke: Erlangen (1859a). Cf. Le Bel, J.A. van’t Hoff, J.H.: Die Lagerung der Atome im Raume. Vieweg: Braunschweig (1877). Cf. Mainzer [1.8], p. 135. Bader, R.F.W.: Atoms in Molecules. A Quantum Theory. Clarendon Press: Oxford (1990), p. 30. Primas, H.: Chemistry, Quantum Mechanics and Reductionism. Springer: Berlin (2nd ed. 1983). Bader [4.6], p. 6. Cf. also Flurry, R.L.: Symmetry Groups. Theory and Chemical Applications. Englewood Cliffs: New Jersey (1980); Donaldson, J.S. Ross, S.D.: Symmetry and Stereochemistry. Inter-

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text Books: London (1972); Hollas, J.M.: Die Symmetrie von Molek¨ ulen. De Gruyter: Berlin/New York (1975). Pasteur, L.: Lecons sur la dissym]’etrie mol´eculaire. Hachette: Paris (1861); P. Curie also spoke of “dissym´etrie” in: Sur la sym´etrie dans les ph´enom`enes physiques, sym´etrie d’un champ ´electrique et d’un champ magn´etique. In: Journal de Physique 3 (1894), p. 393. “Chirality” goes back to Lord Kelvin: Baltimore Lectures (1884/1893), Clay ans Sons: London (1904). H¨ uckel, E.: Zur Quantentheorie der Doppelbindung. In: Zeitschrift f¨ ur Physik 60 (1930), pp. 423–456; Die Elektronenfiguration des Benzols und verwandter Verbindungen. In: Zeitschrift f¨ ur Physik 70 (1931), pp. 204–286; Quantentheoretische Beitr¨age zum Problem der aromatischen und unges¨attigten Verbindungen. In: Zeitschrift f¨ ur Physik 76 (1932), pp. 628–648. The H¨ uckel model is the prototype of a semi-empirical method. In this case, the Hamiltonian operator is assigned free parameters which do not necessarily have any physical significance and are determined empirically. It is therefore not defined by first principles of physics, so that there is no strict reductionism to quantum mechanics. Woodward, R.B. Hoffmann, R.: The Conservation of Orbital Symmetry. Verlag Chemie: Weinheim (1970). Mandelkern, L.: Introduction to Macromolecules. Springer: New York (1983). Heisenberg, W.: Die Plancksche Entdeckung und die philosophischen Grundlagen der Atomlehre. In: Heisenberg, W.: Wandlungen in den Grundlagen der Naturwissenschaft. Hirzel: Stuttgart (1959), p. 183. Heisenberg speaks about the symmetries of arabic friezes with regard to the structure of the elementary particles. But they become immediately apparent in the illustrated structures of polymerization. Sund, H.: Evolution und Struktur der Proteine. Universit¨atsverlag: Konstanz (1968); Fasold, H.: Die Struktur der Proteine. Verlag Chemie: Weinheim (1973); Dickerson, R.E. Geis, I.: Struktur und Funktion der Proteine. Verlag Chemie: Weinheim (1975).

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4.16 Ingram, V.M.: The Hemoglobins in Genetics and Evolution. Columbia University Press: New York/London (1975). 4.17 See also Eisenberg, D.: X-ray crystallography and enzyme structure. In: Boyer, P.H. (ed.): The Enzymes. Structures and Control I. Academic Press: New York/London (1970), pp. 7 ff. 4.18 Cf. also Quack, M.: Die Symmetrie von Zeit und Raum und ihre Verletzung in molekularen Prozessen. In: Akademie der Wissenschaften (Berlin) Jahrbuch 1990–1992. De Gruyter: Berlin (1993), p. 469; Quack, M.: Molecular spectra, reaction dynamics, symmetries and life. In: Chimia 57 (2003), pp. 147–160. 4.19 Quack, M.: On the measurement of parity violating energy difference between enantiomers. In: Chemical Physics Letters 132 No. 2 (1986), pp. 147–153. 4.20 See also Morozov, L.L. Goldanskii, V.I.: Violation of symmetry and self-orgranization in prebiological evolution. In: Krinsky, V.I. (ed.): Self-Organization. Autowaves and Structures far from Equilibrium. Springer: Berlin (1984), pp. 224–232. 4.21 Curie [4.10], p. 393. 4.22 Mainzer, K.: Symmetry and complexity — fundamental concepts of research in chemistry. In: Hyle. An International Journal for the Philosophy of Chemistry 3 (1997), pp. 29–49. 4.23 Prigogine (1980) [3.31]. Prigogine’s work on the time operator and irreversibility is based on research of Misra, B.: Nonequilibrium Entropy, Lyapunov Variables, and Ergodic Properties of Classical Systems. Proceedings of the National Academy of Sciences of the United States of America 75 (1978), pp. 1627– 1631; Misra, B. Prigogine, I. Courbage, M.: P.N.A.S. (1979), pp. 4768–4772; B. Misra, M. Courbage (1978): Physica 104A (1980), pp. 359–377; B. Misra, I. Prigogine: Letters Mathe. Phys. 7 (1983), pp. 421–428. 4.24 Cf. Scott, S.C.: Chemical Chaos. Oxford University Press: Oxford (1961); Schneider, F.W. M¨ unster, A.F.: Nichtlineare Dynamik in der Chemie. Spektrum Akademischer Verlag: Heidelberg/Berlin/Oxford (1996). 4.25 M¨ uller, A.: Struktur-Vielfalt im Nanokosmos. In: Akademie-

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Journal. Magazin der Union der deutschen Akademien der Wissenschaften 1 (2003), p. 20; cf also M¨ uller, A. Shah, S.Q.N. B¨ogge, H. Schmidtmann, M.: Molecular growth from a Mo176 to a Mo248 cluster. In: Nature 48 (1999), p. 397. Whitesides, G.M. Mathias, J.P. Seto, C.T.: Molecular selfasembly and nanochemistry: A chemical strategy for the synthesis of nanostructures. In: Science 254 (1991), pp. 1312–1319. Feynman, R.: There’s plenty of room at the bottom. In: Miniaturization 282 (1961), pp. 295–296. Drexler, K.E.: Nanotechnology summary. In: Encyclopedia Britannica’s Science and the Future Yearbook 162 (1990); cf. also Drexler, K.E.: Nanosystems Molecular Machinery, Manufacturing, and Computation. John Wiley & Sons: New York (1992). Newcome, G.R. (ed.): Advances in Dendritic Macromolecules JAI Press: Greenweech Conn. (1994). Curl, R.F. Smalley, R.E.: Probing C60 . In: Science 242 (1988), pp. 1017–1022; Smalley, R.W.: Great balls of carbon: The story of Buckminsterfullerene. In: The Sciences 31 (1991), pp. 22–28. M¨ uller, A.: Supramolecular inorganic species: An expedition into a fascinating rather unknown land mesoscopia with interdisciplinary expectations and discoveries. In: J. Molecular Structure 325 (1994), p. 24; M¨ uller, A. Mainzer, K.: From molecular systems to more complex ones. In: M¨ uller, A. Dress, A. V¨ ogtle, F. (eds.): From Simplicity to Complexity in Chemistry — and Beyond. Vieweg: Wiesbaden (1995), pp. 1–11. Fig. 58 with drawings of Christie, B.: Spektrum der Wissenschaft Spezial 2 (2002), p. 22.

Chapter 5 5.1

Monod, J.: On symmetry and function in biological systems. In: Engstr¨ om, A. Strandberg, B. (eds.): Symmetry and Function of Biological Systems at the Macromolecular Level. Proceedings of the 11th Nobel Symposium 1968. Almqvist and Wiksell: Stockholm (1968), pp. 15–27. Cf. also Jenken, M.:

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The biological and philosophical definitions of life. In: Acta biotheoretica 24 (1975), pp. 14–21. 5.2 See Sitte, P.: Symmetrien bei Organismen. In: Biologie in unserer Zeit 6 (1884), p. 165; Klug, A.: Point groups and the design of aggregates. In: Engstr¨ om [5.1], p. 427. 5.3 Cf. also Gierer, A.: Physik der biologischen Gestaltbildung. In: Naturwissenschaften 68 (1981), pp. 245–251. 5.4 Weberling, F.: Symmetrie der Pflanzen. In: Stork, H. (ed.): Symmetrie. Aulis: Cologne (1985), pp. 175 ff.; cf. the older publications, e.g., Frey, A.: Geometrische Symmetriebetrachtungen. In: Flora 120 (1926), pp. 87–98; Troll, W.: Symmetriebetrachtungen in der Biologie. In: Studium generale 2 (1949), pp. 240–259. 5.5 See also Weberling, F.: Morphologie der Bl¨ uten und der Bl¨ utenst¨ ande. Eugen Ulmer: Stuttgart (1981). 5.6 Steiner, G.: Spiegelsymmetrie der Tierk¨orper. In: Naturwis¨ senschaftliche Rundschau 32 (1979), pp. 481–485; Uber die Symmetrie bei Tieren. In: Stork [5.4], pp. 187–211. 5.7 See also Doczi, C.: The Power of Limits. Shambhala Publications: Boulder Colorado (1981). 5.8 Haeckel, E.: Kunstformen der Natur. Verlag des Bibliographischen Instituts: Leipzig/Vienna (1899). 5.9 Thompson, D.W.: On Growth and Form. Birkh¨ auser: Basel (1973). 5.10 For a survey cf. Depew, D.J. Weber, B.H. (eds.): Evolution at a Crossroads. The New Biology and the New Philosophy of Science. M.I.T. Press: Cambridge Mass. (1985); Eberling, W. Feistel, R.: Physik der Selbstorganisation und Evolution. Akademischer Verlag: Berlin (1982); Haken, H. Haken-Krell, M.: Entstehung von biologischer Information und Ordnung. Wissenschaftliche Buchgesellschaft: Darmstadt (1989); Hofbauer, L.: Evolutionstheorie und dynamische Systeme. Mathematische Aspekte der Selektion. Springer: Berlin (1984). 5.11 Eigen, M. Schuster, P.: The Hypercycle: A Principle of Natural Self-Organization. Springer: Berlin (1979). 5.12 Levin, S.A.: Complex adaptive systems: Exploring the known,

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the unknown and the unknowable. In: Bulletin of the American Mathematical Society 40 (2002), pp. 3–19; Scott [3.36], chapt. 4.7.4. Eigen. M. Winkler, R.: Das Spiel. Naturgesetze steuern den Zufall. Piper: Munich/Z¨ urich (1975), pp. 272 ff.; cf. also K¨ uppers, B.-O.: Der Ursprung biologischer Information. Piper: Munich/Z¨ urich (1986), pp. 214 ff. Eigen, M.: Self-organization of matter and the evolution of biological macromolecules. In: Naturwissenschaften 58 (1971), p. 465; K¨ uppers [5.13], pp. 216 ff. For simplification the  sum term j ϕij xj has been left. It designates the population contribution made by all species xj=i on account of reverse mutations to the original sequence xi . Wright, S.: “Surfaces” of selection value. In: Proceedings of the National Academy of Sciences of the United States of America 58 (1967), p. 165; Die Entstehung des Lebens. In: Natur 3 (1983), p. 68; Eigen, M.: Homunculus im Zeitalter der Biotechnologie — Physikochemische Grundlagen der Lebensvorg¨ ange. In: Gross, R. (ed.): Geistige Grundlagen der Medizin. Springer: Berlin (1985), pp. 24 ff. For a survey on nonlinear models of DNA cf. Yakushevich, L.V.: Nonlinear Physics of DNA. John Wiley: Chichester (1998); Krumhansl, J.A.: The intersection of nonlinear science, molecular biology, and condensed matter physics. Viewpoints. In: Peyrard, M. (ed.): Nonlinear Excitations in Biomolecules. Springer: Berlin (1995), pp. 1–9. Watson, J.D.: Molecular Biology of the Gene. Benjamin: New York (1970). For a survey cf. Kauffmann, A.S.: Origins of Order. SelfOrganization and Selection in Evolution. Oxford University Press: Oxford (1992). Baas, N.A.: Emergence, Hierarchies, and Hyperstructures. In: Langton, C.G. (ed.): Artificial Life III. Addison-Wesley: Reading Mass. (1994). Cf. also Nicolis, J.S.: Dynamics of Hierarchical Systems. An Evolutionary Approach. Springer: Berlin (1986); Voorhees,

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B.H.: Axiomatic theory of hierarchical systems. In: Behavioral Science 28 (1983), pp. 24–34; Scott [3.36], p. 427. Cf. also Rosen, R.: Dynamical System Theory in Biology. Wiley-Interscience: New York (1970). Meinhardt, M.: Models of Biological Pattern Formation. Academic Press: London (1982). Meinhardt, H. Gierer, A.: Application of a theory of biological pattern formation based on lateral inhibition. In. Journal of Cell Science 15 (1974), p. 321. Bassingthwaite, J.B. van Beek, J.H.G.M.: Lightning and the heart: Fractal behavior in cardiac function. In: Proceedings of the IEEE 76 (1988), p. 696. Actually, heart research was one of the first medical applications of chaos theory. Cf. also Goldberger, A.L. Bhargava, V. West, B.J.: Nonlinear dynamics of the heartbeat. In: Physica 17D (1985), pp. 207–214; Winfree, A.T.: When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. Princeton University Press: Princeton (1987). Churchland, P.S. Sejnowski, T.J.: Perspectives in cognitive neuroscience. In: Science 242 (1988), pp. 741–745. The subset of visual cortex is adapted from van Essen, D. Maunsell, J.H.R.: Two-dimensional maps of the cerebral cortex. In: Comparative Neurobiology 191 (1980), pp. 255–281. The network model of ganglion cells is given in Hubel, D.H. Wiesel, T.N.: Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. In: Journal of Physiology 160 (1962), pp. 106–154. An example of chemical synapses is shown in Kandel, E.R. Schwartz, J.: Principles of Neural Science. Elsevier: New York (1985). Hodgekin, A.L. Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. In: Journal of Physiology 117 (1952), pp. 500–544. Hodgekin [5.27]; Huxley, A.F.: Can a nerve propagate a

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subthreshold disturbance? In: Journal of Physiology 148 (1959), pp. 80P–81P; cf. also Scott [3.36], p. 126. Cf. Scott, A.C.: Neuroscience. A Mathematical Primer. Springer: New York (2002). Hebb, D.O.: The Organization of Behavior. John Wiley: New York (1949). von der Malsburg, C.: Self-Organization of orientation sensitive cells in the striate cortex. In: Kybernetik 14 (1973), pp. 85–100; Wilshaw, D.J. von der Malsburg, C.: How patterned neural connections can be set up by self-organization. In. Proceedings of the Royal Society series B 194 (1976), pp. 431–445. Barlow, H.B.: Single units and sensatioperceptual psychology. In: Perception 1 (1972), p. 371. Singer, W.: The role of synchrony in neocortical processing and synaptic plasticity. In: Domany, E. van Hemmen, L. Schulten, K. (eds.): Model of Neural Networks II. Springer: Berlin (1994). For the emergence of consciousness cf. Mainzer, K.: Philosophical concepts of computational neuroscience. In: Eckmiller, R. Hartmann, G. Hauske, G. (eds.): Parallel Processing in Neural Systems and Computers. North-Holland: Amsterdam/New York (1990), pp. 9–12; Gehirn, Computer, Komplexit¨ at. Springer: Berlin (1997); Flohr, H.: Brain processes and phenomenal consciousness. A new and specific hypothesis. In: Theory & Psychology 1 (1991), p. 248; Scott, A.C.: Stairway to the Mind. Copernicus: New York (1995). Freeman, W.: Noise-induced first-order phase transitions in chaotic brain activity. In: International Journal of Bifurcation and Chaos 9 No. 11 (1999), pp. 2215–2218; A proposed name for aperiodic brain activity: stochastic chaos. In: Neural Networks 13 (2000), pp. 11–13. Freeman, W. et al.: A neurobiological theory of meaning in perception I–V. In: International Journal of Bifurcation and Chaos 13 (2003); How Brains make up their Minds. Columbia University Press: New York (2001). Rettenmeyer, C.W.: Behavioral studies of army ants. In: Uni-

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versity of Kansas Science Bulletin 44 (1963), p. 281; Prigogine, I.: Order through fluctuation: Self-organization and social system. In: Jantsch, E. Waddington, C.H. (eds.): Evolution and Consciousness. Human systems in transition. Addison-Wesley: London (1976), p. 108. 5.38 Cf. Johnson, L.: The thermodynamic origin of ecosystems: A tale of broken symmetry. In Weber, B.H. Depew, D.J. Smith, J.D. (eds.): Entropy, Information, and Evolution. New Perspectives on Physical and Biological Evolution. M.I.T. Press: Cambridge Mass. (1988), pp. 75–105; Schneider, E.D.: Thermodynamics, ecological succession, and natural selection: A common thread. In: Weber [5.38], pp. 107–138. Chapter 6 6.1

6.2 6.3

6.4

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6.6

Hobbes, T.: The Elements of Law Natural and Politic (1640). Ed. by F. T¨ onnies. Simpkin, Marshall, and Co.: London (1889); Leviathan, or the Matter, Form and Power of a Commonwealth, Ecclesiastical and Civil (1651). Ed. by C.B. MacPherson. Penguin: Harmondsworth (1968); Gough, J.W.: The Social Contract. A Critical Study of its Development. Clarendon Press: Oxford (1957). Meek, R.L.: The interpretation of the “Tableau ´economique”. In: Economica 27 (1960), pp. 322–347. Smith, A.: An Inquiry into the Nature and Causes of the Wealth of Nations. Ed. By E. Cannan. The University of Chicago Press: Chicago (1976). Mayr, O.: Adam Smith und das Konzept der Regelung. In: Troitzsch, U. Wohlauf, G. (eds.): Technikgeschichte. Suhrkamp: Frankfurt (1980), p. 245. Cf. Walras, L.: Principe d’une th´eorie math´ematique de l’echange. In: Journal des Economistes 33 (1874), pp. 1–21; Hicks, J.: L´eon Walras. In: Classics and Moderns. Collected Essays on Economic Theory. Vol. III. Basil Blackwell: Oxford (1983), pp. 85–95. von Neuman, J. Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press: Princeton (1943).

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Shubik, M.: Game Theory and Related Approaches to Social Behavior. John Wiley: New York (1964). Nash, J.: Noncooperative games. In: Annals of Mathematics 54 (1951), pp. 289–295. Cf. also van Damme, E.: Stability and Perfection of Nash Equlibrium. Springer: Berlin (1987); Bicchieri, C.: Game theory: Nash equilibrium. In: Floridi, L. (ed.): Philosophy of Computing and Information. Blackwell: Oxford (2004), pp. 289–304. Harsanyi, J. Selten, R.: A General Theory of Equilibrium Selction in Games. M.I.T. Press: Cambridge Mass. (1987). Hofbauer, J. Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press: Cambridge (1998). Maynard Smith, J.: Evolution and the Theory of Games. Cambridge University Press: Cambridge (1982). Cf. Mainzer [3.32], chapt. 8. Weidlich, W.: Synergetic modelling concepts for sociodynamics with application to collective political opinion formation. In: Journal of Mathematical Sociology 18 (1994), pp. 267–291. Henderson, J.M. Quandt, R.E.: Microeconomic Theory — A Mathematical Approach. McGraw-Hill: London (1980); Nicholson, W.: Microeconomic Theory — Basic Principles and Extensions. The Dryden Press: Fort Worth Texas (1992); Varain, H.R.: Microeconomic Analysis. W.W. Norton: New York (1992). The concept of “bounded rationality” was introduced by Simon, H.A.: Models of Bounded Rationality. 2 vols. M.I.T. Press: Cambridge Mass. (1982). Cf. Sterman, J.D.: Business Dynamics. Systems Thinking and Modeling for a Complex World. McGraw-Hill: Boston (2000); Weidlich, W.: Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences. Taylor & Francis: London (2002); Auyang, S.Y.: Complex-System Theories in Economics, Evolutionary Biology, and Statistical Physics. Cambridge University Press: Cambridge (1998); WeiBin Zhang: Synergetic Economics — Time and Change in Non-

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linear Economics. Springer: Berlin (1991); Goodwin, R.M.: Chaotic Economic Dynamics. Oxford University Press: New York (1990); Anderson, P.W. Pines, D. (eds.): The Economy as an Eolving Complex System. Addison-Wesley: Redwood City (1988). Weidlich, W. Braun, M.: The master equation approach to nonlinear economics. In: Journal of Evolutionary Economics 2 (1992), pp. 233–265. Mayntz, R.: The influence of natural science theories on contemporary social science. In: Dierkes, M. Bievert, B. (eds.): European Social Science in Transition. Campus: Frankfurt (1992), pp. 27–79. T¨onu, P.: Nonlinear Economic Dynamics. Springer: Berlin (1992); Barnett, W.A. Geweke, J. Shell, K. (eds.): Economic Complexity. Chaos, Sunspots, Bubbles, and Nonlinearity. Cambridge University Press: Cambridge (1990); Brian Arthur, W. Durlauf, N.S. Lane, D.L. (eds.), The Economy as an Evolving Complex System II. Proceedings Volume of the Santa F´e Institute. Vol. XXVII. Addison-Wesley: Reading Mass. (1997); Lorenz, H.-W.: Nonlinear Dynamical Economics and Chaotic Motion. Springer: Berlin (1989). Weidlich [6.17], p. 224, Fig. 7.1. Dendrinos, D.S. Sonis, M.: Chaos and Socio-Spatial Dynamics. Springer: Berlin (1990); Pumain, D.: Spatial Analysis and Population Dynamics. John Libbey Eurotext: Montrouge/London/Wien (1991); Lee, E.S.: A theory of migration. In: Demography (1966), pp. 47–57. Haag, G. Dendrinos, D.S.: Toward a stochastic dynamical theory of location: (A) A nonlinear migration process. In: Geographical Analysis 15 (1983), pp. 269–286. Weidlich, W. Haag, G. (eds.): Interregional Migration — Dynamic Theory and Comparative Analysis. Springer: Berlin (1988); Weidlich [6.17], p. 92, Fig. 4.2. Weidlich [6.17], p. 95, Fig. 4.4. Allen, P.M.: Self-organization in the urban system. In: Schieve, W.C. Allen, P.M. (eds.): Self-Organization and Dissipative

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Structures. Applications in the Physical and Social Sciences. University of Texas Press: Austin (1982), pp. 142–146. Batty, M. Longley, P.A.: Fractal Cities. Academic Press: London (1994); Frankhauser, P.: La fractalit´e des structures urbaines. Anthropos: Paris (1994). Frankhauser, P. Sadler, R.: Fractal analysis of urban structures. In: Natural Structures — Principles, Strategies and Models in Architecture and Nature. Proceedings of the Internernational Symposium SFB 230. Vol. II. (1992), pp. 57–65. White, R. Engelen, G.: Cellular automata and fractal urban form. A cellular modelling approach to the evolution of urban land-use oatterns. In: Environment and Planning A 25 (1993), pp. 1175–1199; Cellular automata as the basis of integrated dynamic regional modelling. In: Environment and Planning B 24 (1997), pp. 235–246; The use of constrained cellular automata of high-resolution modelling of urban land-use dynamics. In: Environment and Planning B. Planning and Design 24 (1997), pp. 323–343. Helbing, D.: Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes. Kluwer Academic Publishers: Dordrecht (1995). Beynon, J. Dunkerley, D. (eds.): Globalization: The Reader. Athlone: London (2000); O’Rourke, K. Williamson, J.: Globalization and History. M.I.T. Press: Cambridge Mass. (1999); Lechner, F. Boli, J. (eds.): The Globalization Reader. Blackwell: Oxford (2000). Cf. also Boswell, T.: Hegemony and Bifurcation Points in World History. In: Bornschier, V. Chase-Dum, C. (eds.): The Future of Global Conflict. Sage: London (1999), pp. 263–284. Therborn, G.: European Modernity and Beyond. The Trajectory of European Societies 1945–2000. Sage: London (1995). Cf. Smith [6.3]. Cf. also Kaufmann, S.A.: The Origins of Order — SelfOrganization and Selection in Evolution. Oxford University Press: Oxford (1993).

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6.36 Deacon, B.: Global Social Policy: International Organizations and the Future of Welfare. Sage: London (1997). 6.37 Marx, K. Engels, F.: Das Kommunistische Manifest. Studientexte 4. Fink: Munich (1969). 6.38 Mainzer [6.13], chapt. 8; Novotny, H.: Eigenzeit. Entstehung und Strukturierung eines Zeitgef¨ uhls. Suhrkamp: Frankfurt (1989). 6.39 Cf. also Friedman, M.: A Theory of the Consumption Function. Princeton University Press: Princeton (1957). 6.40 Cf. also von Weizs¨acker, C.C.: Logik der Globalisierung. Vandenhoeck & Ruprecht: G¨ ottingen (2003). 6.41 Dunning, J.H.: The Globalization of Business. Routledge: London (1993). 6.42 Von Hayek, F.A.: Notes on the evolution of systems of rules of conduct. In: von Hayek, F.A.: Studies in Philosophy and Economics. Routledge: London (1967), pp. 66–81; The Road of Serfdom. Routledge: London (1945). 6.43 O’Briek, R. Goetz, A.M. Scholte, J.A. Williams, M.: Consulting Global Governance. Cambridge University Press: Cambridge Mass. (2000); Weiss, T.G.: Governance, good governance and global governance. Conceptual and actual challenges. In: Third World Quarterly 21 (2000), pp. 795–814. 6.44 Easterly, W.: The Effect of IMF and World Bank Programs on Poverty. Oxford University Press: Oxford (2001); Dunkley, G.: The Free Trade Adventure: The WTO, GATT and Globalism. A Critique. Zed: London (1999). 6.45 Meadows, D.H. et al.: The Limits to Growth. Universe Books: New York (1972); Randers, J. Meadows, D.: System Simulation to Test Environmental Policy I: A Sample Study of DDT Movement in the Environment. M.I.T. Press: Cambridge Mass. (1971). ¨ 6.46 Siebert, H.: Okonomische Theorie der Umwelt. Mohr: ¨ T¨ ubingen (1978); Mainzer, K. (ed.): Okonomie und ¨ Okologie/Economie et Ecologie. Haupt: Bern (1993). 6.47 Castles, F.G.: Ethnicity and Globalization. Sage: London (2000).

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6.48 Chua, A.: World on Fire. Doubleday: New York (2003). 6.49 Kant, I: Zum ewigen Frieden (1795). In: Kants gesammelte Schriften. Ed. by K¨ oniglich Preußische Akademie der Wissenschaften. Bd. VIII. De Gruyter: Berlin (1912/1923). 6.50 Cf. also Holden, B. (ed.): Global Democracy: Key Debates. Routledge: London (2000); Beck, U.: World Risk Society. Polity Press: Cambridge Mass. (2000). 6.51 Cf. also chapt. 7.1 of this book. 6.52 Dierkes, M. Berthoin Antal, A. Child, J. Nonaka. I.: Handbook of Organizational Learning & Knowledge. Oxford University Press: Oxford (2001); Mainzer, K.: Vom Komplexit¨ atszum Kreativit¨atsmanagement. Auf Talentsuche in der Wissensgesellschaft. In: G¨ otz, K. (ed.): Personalarbeit der Zukunft. Managementkonzepte DaimlerChrysler. Vol. 27. Hampp Verlag: Munich (2002), pp.13–25. Chapter 7 7.1

7.2 7.3 7.4 7.5

7.6 7.7 7.8

Shannon, C.E. Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press: Chicago (1949); cf. also Cover T. Thomas, J.: Elements of Information Theory. John Wiley & Sons: New York (1991). Cf. also Carrington, G.: Basic Thermodynamics. Oxford University Press: Oxford (1994). Cf. Deco, G. Sch¨ urmann, B.: Information Dynamics. Foundations and Applications. Springer: Berlin (2001), chapt. 3. Cf. the early analysis of Jaynes, E.T.: Information theory and statistical mechanics. In: Physical Review 106 (1957), p. 620. Cf. Ebeling, W. Freund, J. Schweitzer, F.: Komplexe Strukturen: Entropie und Information. Teubner: Stuttgart/Leipzig (1998). Benett, C.H.: Quantum information and computation. In: Physics Today 10 (1995), pp. 24–30. von Weizs¨acker, C.F.: Aufbau der Physik. Dtv: Munich (3rd ed. 1994). Cf. also Feistel, R. Eberling, W.: Evolution of Complex

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Systems. Kluwer: Dordrecht (1989); Klimontovich, Y.: Statistical Theory of Open Systems. Kluwer: Dordrecht (1995). Adamii, C.: Learning and Complexity in Genetic AutoAdaptive Systems. In: Physica D 80 (1995), pp. 154–170. Eberling, W. Volkenstein, M.V.: Entropy and the evolution of biological information. In: Physica A 163 (1990), p. 398. Sagan, C.: Die Drachen von Eden. Das Wunder der menschlichen Intelligenz. Droemer Knaur: Munich/Zurich (1978): cf. also Gould, S.J.: The Book of Life. W.W. Norton: New York (1993). Churchland, P.S. Sejnowski, T.J.: The Computational Brain. M.I.T. Press: Cambridge Mass. (1992); Mainzer [5.34]. Cf. H¨olldobler, B.: Vom Verhalten zum Gen. Die Soziobiologie eines Superorganismus. In: Nova Acta Leopoldina NF 76303 (1997), pp. 205–223. Historically, the concept of a “superorganism” was introduced by Wheeler, W.M.: The antcolony as an organism. In: Journal of Morphology 22 (1911), pp. 307–325. Cf. Mainzer, K.: KI — K¨ unstliche Intelligenz. Grundlagen intelligenter Systeme. Wissenschaftliche Buchgesellschaft: Darmstadt (2003). Mainzer [6.13], p. 81. Hawking, S.W.: The Universe in a Nutshell. Bantam Books: New York (2001). The path integral was suggested in a paper of Hawking during the 17th International Conference on General Relativity and Gravitation: July 25, 2004 at Dublin. Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem.” In: Proceedings of the London Mathematical Society Series 2 42 (1936), pp. 103–105; Teuscher, C. (ed.): Alan Turing: The Life and Legacy of a Great Thinker. Springer: Berlin (2004). Wegener, I.: Complexity Theory. Limiting Factors on the Efficiency of Algorithms. Springer: Berlin (2004); Hromkovic, J.: Theoretical Computer Science. Introduction to Automata, Computability, Complexity, Algorithmics, Randamization, Communication, and Cryptography. Springer: Berlin

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(2004); Homer, S. Selman, A.L.: Computability and Complexity Theory. Springer: New York (2001). Chaitin, G.J.: On the length of programs for computing finite binary sequences. In: Journal of the ACM 13 (1966), pp. 547– 569; On the length of programs for computing finite binary sequences: statistical considerations. In: Journal of the ACM 16 (1969), pp. 145–159. Chaitin, G.J.: The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning. Springer: New York (1998); Calude, C.S.: Information and Randomness. An Algorithmic Perspective. Springer: New York (2nd ed. 2002). Cf. Bennett [7.6]; Hirvensalo, M.: Quantum Computing. Springer: New York (2nd ed. 2004). Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. In: Proc. Roy. Soc. A 400 (1985). pp. 97–117. Bouwmeester, A.E. Zeilinger, A. (eds.): The Physics of Quantum Information. Springer: Berlin (2000); Nielsen, M.A. Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press: Cambridge (2001); Lo, H.-K. Popesku, S. Spiller, T. (eds.): Introduction to Quantum Computation and Information. World Scientific: Singapore (1998). Audretsch, J. Mainzer, K. (eds.): Wieviele Leben hat Schr¨ odingers Katze? Zur Physik und Philosophie der Quantenmechanik. Spektrum: Heidelberg (1996). Adleman, L.M.: Molecular computation of solutions to combinatorical problems. In: Science 266 (1994), pp. 1021–1024; Lipton, R.J.: DNA solutions of hard computational problems. In: Science 268 (1995), pp. 542–545. von Neumann, J.: The Computer and the Brain. Yale University Press: New Haven (1958); Burks, A.W. (ed.): Essays on Cellular Automata. University of Illinois Press: UrbanaChampaign Ill. (1970). Wolfram, S.: Universality and complexity in cellular automata. In: Farmer, D. Toffoli, T. Wolfram, S. (eds.): Proceedings of

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an Interdisciplinary Workshop. North-Holland: Amsterdam (1984). Mainzer, K.: Komplexit¨ at in der Natur. In: Nova Acta Leopoldina NF 76 No. 303 (1997), pp. 165–189. Wolfram, S.: A New Kind of Science. Wolfram Media Inc.: Champaign, Ill. (2002), p. 443. Wolfram [7.29], p. 737. McCulloch, W.S. Pitts, W.: A logical calculus of the ideas immanent in nervous activity. In: Bulletin of Mathematical Biophysics 5 (1943), pp. 115–133. Rosenblatt, F.: The perceptron: A probabilistic model for information storage and organization in the brain. In: Psychological review 65 (1958), pp. 386–408. Minsky, M. Papert, S.A.: Perceptrons. M.I.T Press: Cambridge Mass. (1969) (expanded ed. 1988). Hopfield, J.J.: Neural Network and physical systems with emergent collective computational abilities. In: Proceedings of the National Academy of Sciences 79 (1982), pp. 2554–2558. Rummelhart, D.E. Smolensky, P. McClelland, J.L. Hinton, G.E.: Schemata and sequential thought processes. In: McClelland, J.L. Rumelhart, D.E. (eds.): Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Vol.2: Applications. M.I.T Press: Cambridge Mass. (1986); Churchland, P.S.: Neurophilosophy. Toward a Unified Science of the Mind-Brain. M.I.T. Press: Cambridge Mass. (1988), p. 465. Chua, L.O.: CNN: A Paradigm for Complexity. World Scientific: Singapore (1998); Mainzer, K.: CNN and the evolution of complex information systems in nature and society. In: Tetzlaff, R. (ed.): Cellular neural networks and their applications. Proceedings of the 7th IEEE International CNN Workshop. World Scientific: Singapore (2002), pp. 483–497. Chua, L.O. Yang, L.: Cellular neural networks: Theory. In: IEEE Transactions on Circuits and Systems 35 (1988), pp. 1257–1272; Cellular neural networks: Applications. In: IEEE Transactions on Circuits and Systems 35 (1988), pp. 1273–1290. Chua, L.O. Roska, T.: Cellular Neural Networks and Visual

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Computing: Foundations and Applications. Cambridge University Press: Cambridge (2002), p. 7 (Fig. 2.1), p. 8 (Fig. 2.2); Mainzer, K.: Cellular neural networks and visual computing: Foundations and applications according to the book of Leon O. Chua and Tam´ a Roska. In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 13 1 (2003), pp. 1–6. Chua [7.36], Fig. 2.6.24, p. 74. Mainzer [7.14]; Mainzer, K.: Computerphilosophie zur Einf¨ uhrung. Junius Verlag: Hamburg (2003). Dadam, P.: Verteilte Datenbanken und Client/Server-Systeme. Springer: Berlin (1996), p. 26; Mainzer, K.: Computernetze und virtuelle Realit¨ at. Springer: Berlin (1999), p. 31. Itoh, M. Chua, L.O.: Star cellular neural networks for associative and dynamic memories. In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 14 5 (2004), pp. 1725–1772. Itoh [7.42], Fig. 3, p. 1730. Christos, G.: Memory and Dreams. Rutgers University Press: New Brunswic/New Jersey/London (2003). Itoh [7.42], Fig. 19, p. 1754. Bollob´as, B.: Random Graphs. Academic Press: London (1985); Albert, R. Barab´ asi, A.-L.: Statistical mechanics of complex networks. In: Reviews of Modern Physics 74 1(2002), pp. 47–97. Watts, D.J.: Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton University Press: Princeton NJ (1999). Wasserman, S. Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press: Cambridge (1994). Albert [7.46], Fig. 1, p. 51. Halabi, B.: Internet-Routing-Architekturen: Grundlagen, Design und Implementierung. Carl Hanser Verlag: Munich (1998); Calvert, K.M. Doar, M.B. Zegura, E.W.: Modeling Internet

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topology. In: IEEE Transactions Communication 1 (1997), p. 160. Fukuda, K. Takayasu, H. Takayasu, M.: Spatial and temporal behavior of congestion in internet traffic. In: Fractals 7 1 (1999), p. 67. Mainzer, K. B¨ uchs, M. Kundish, D. Pyrka, P.: Physical and virtual mobility: Analogies between traffic and virtual highways. In: Mayinger, F. (ed.): Mobility and Traffic in the 21st Century. Springer: Berlin (2001), pp. 243–318. Takayasu, M. Takayasu, H. Sato, T.: Critical behaviors and 1/f noise in information traffic. In: Physica A 233 (1996), Fig. 1, p. 825. Crestani, F. Pasi, G. (eds.): Soft Computing in Information Retrieval: Techniques and Applications. Physica Verlag (Springer): Heidelberg (2000). Weiser, T.: The computer for the 21st century. In: Scientific American 9 (1999), p. 66; Norman, D.A.: The Invisible Computer. Cambridge University Press: Cambridge (1998). Mainzer, K.: Self-organization and emergence in complex dynamical systems. Interdisciplinary perspectives for organic computing. In: 34. Jahrestagung der Gesellschaft f¨ ur Informatik. INFORMATIK 2004 Ulm. Hofmann, P.E.H. et al.: Evolution¨ are E/E-Architekturen. Vision einer neuartigen Elektronik-Architektur f¨ ur Fahrzeuge. DaimlerChrysler: Esslingen (2002).

Chapter 8 8.1 8.2

8.3

Aristotle: The Categories, On Interpretation, Prior Analytics. Harvard University Press: Cambridge Mass. (1938). Cf. Bochenski, I.M. Church, A. Goodman, N.: The Problem of Universals. A Symposium. Notre Dame, Indiana (1956); Quine, W.V.O.: Ontological Relativity and other Essays. Columbia University Press: New York/London (1969). Bourbaki, N.: Elements of Mathematics: Theory of Sets. ´ ements de Math´ematique: Th´eorie des Ensembles]. Her[El´ mann: Paris (1968); Ebbinghaus [1.11].

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Cf. Bourbaki [8.3], note 61, chapt. IV; Scheibe, E.: Invariance and covariance. In: Agassi, J. Cohen, R.S. (eds.): Scientific Philosophy Today. Kluwer: Boston (1981), pp. 311–331. Carnap, R.: Der logische Aufbau der Welt. Weltkreis: Berlin (1928), repr. Meiner: Hamburg (1974). For the structuralistic view of symmetry cf. van Fraassen, B.C.: Laws and Symmetry. Clarendon Press: Oxford (1989) and Mainzer, K.: Symmetry of Nature. De Gruyter: Berlin/New York (1996) [German: 1988], chapt. 5.3. The semantic view dates back to P. Suppes [e.g., What is a scientific theory? In: Morgenbesser, S. (ed.): Philosophy of Science Today. Basic Books: New York (1967)] and decribes the mathematical practise of modeling. Cf. also Suppe, F.: The Semantic Conception of Theories and Scientific Realism. University of Illinois Press: Urbana Ill. (1988); Giere, R.: Understanding Scientific Reasoning. Holt, Rinehart, and Winston: New York (1979). Stegm¨ uller emphasized structuralism as “non-statement view” in Stegm¨ uller, W.: The Structuralistic View of Theories. Springer: Berlin (1979); cf. also Moulines, C.U.: Approximate application of theories. In: Erkenntnis 10 (1976), pp. 201–227. Cf. also Scheibe, E.: Struktur und Theorie in der Physik. In: Audretsch, J. Mainzer, K. (eds.): Philosophie und Physik der Raum-Zeit. B.I. Wissenschaftsverlag: Mannheim (2nd ed. 1996), pp. 103–120. Ludwig, G.: Die Grundstrukturen einer physikalischen Theorie. Springer: Berlin (1978), §10, §12. Ludwig [8.9], §8. Cf., e.g., Derrida, J. Rorty, R.: Objectivity, Relativism and Truth. Philosophical Papers I. Cambridge University Press: Cambridge (1991). Mainzer [8.6], chapt. 5.3 and chapt. 5.42; cf. also Emch, G.G.: Mathematical and Conceptual Foundations of 20th Century Physics. North-Holland: New York/Oxford (1984). Mainzer, K.: System. An introduction to systems science. In:

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8.17 8.18 8.19

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Floridi, L. (ed.): Philosophy of Computing and Information. Blackwell Publishing: Oxford (2004), pp. 28–39. Cf. Mainzer [3.32]. Bergson, H.: Dur´ee et simultan´et´e. A propos de la th´eorie d’Einstein. F´elix Alcan: Paris (1922). Bergson related his concept of duration to Einstein’s concept of relativity which is misleading. Prigogine interpreted Bergson’s “dur´ee” for his operator approach in Prigogine (1980) [3.31]. The concept of relative computability is analyzed in my Ph.D. thesis “Mathematical Constructivism” (University of M¨ unster 1973) and later in Mainzer [7.40], chapt. 3. Peter of Spain [Petrus Hispanus]: Summulae Logicales. Ed. by I.M. Bochenski. Marietti: Turin (1947). Mainzer [7.41], chapt. 3. Sowa, J.F.: Knowledge Representation. Logical, Philosophical, and Computational Foundations. Brooks/Cole: Pacific Grove (2000). Broy, M./Steinbr¨ uggen, R.: Modellbildung in der Informatik. Springer: Berlin (2004). Petri, C.A.: Kommunikation mit Automaten. [Diss. University of Bonn]. Bonn (1962); Best, E. Devillers, R. Koutny, M.: Petri Net Algebra. Springer: Berlin (2001). My definition of actual and potential models is logicalmathematical and depends on theoretical structures. But, in a heuristical sense, it can also be related to Leibniz’s distinction of “possible worlds” and Aristotle’s distinction of “actualization” and “potentiality”. Cf. also Huntley, H.E.: The Devine Proportion: A Study in Mathematical Beauty. Dover: New York (1970). Cf. Hambidge, J.: Dynamic Symmetry. The Greek Vase. Yale University Press: New Haven (1920); for pyramid construction cf. Stecchini, L.C.: Notes on the relation of ancient measure to the great pyramid. In: Tomkins, P.S. (ed.): The Secrets of the Great Pyramid. Harper & Row: New York (1978), p. 368; Dominguez, M.A.: La Architectura Precolumbina en Mexico. Mexico D.F., Orion (1956).

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8.25 Diels-Kranz [1.19], I 293 c. 27; Diels, H.: Antike Technik. Teubner: Leipzig/Berlin (1924), pp. 15f. 8.26 von Steuben, H.: Der Kanon des Polyklet. Doryphoros und Amazone: T¨ ubingen (1973). 8.27 Vitruvius: De architectura. Ed. by C. Fensterbusch. Wissenschaftliche Buchgesellschaft: Darmstadt (1964), I, 2.4; cf. also Knell, H.: Grundz¨ uge der griechischen Architektur. Wissenschaftliche Buchgesellschaft: Darmstadt (1980). 8.28 In that connection also Kepes, G. (ed.): Symmetry, Rhythm. George Braziller: New York (1966). 8.29 Michell, G.: Der Hindu-Tempel. Bauformen und Bedeutung. DuMont: Cologne (1979), p. 71 ff.; likewise Brown, P.: Indian Architecture. Vol. 1: Buddhist and Hindu Periods. Taraporevala: Bombay (5th ed. 1965); for Indian sacral geometry, cf. also Michaels, A.: Beweisverfahren in der vedischen Sakralgeometrie. Ein Beitrag zur Entstehungsgeschichte von Wissenschaft. Franz Steiner: Wiesbaden (1978). 8.30 Cf. also Doczi, G.: The Power of Limits. Shambhala: Boulder Colorado (1981), pp. 134 ff. 8.31 Vogt-G¨ognil, U.: Die Moschee-Grundformen sakraler Baukunst. Artemis: Zurich (1978), pp. 111 f. 8.32 In that connection also, Golvin, L.: Essai sur l’architecture religieuse Musulmane. 3 vols. Klincksieck: Paris (1970–1974). 8.33 Panofsky, E.: Architecture and Scholasticism. Faber and Faber: Latrobe (1951). 8.34 Cf. Siebenh¨ uhner, H.: Deutsche K¨ unstler am Mail¨ ander Dom. Bruckmann: Munich (1944); Bascape, G. Mezzanotte, P.: Il Duomo di Milano. Bramante: Milan (1965); likewise Ackermann, J.S.: Ars sine scientia nihil est. In: The Art Bulletin XXXI (1949), pp. 84–111. 8.35 Cf. de Bruyne, E.: Etudes d’esth´etique m´edi´evale. 3 vols. De Tempel: Brugge (1946). 8.36 Cf. Willis, R. (transl. and ed.): Fascimile of the Sketchbook of Villard de Honnecourt. John Henry and James Parker: London (1859). The original west rose window is portrayed, e.g., in

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Swaan, W.: Die großen Kathedralen. DuMont: Cologne (1969), p. 97. Alberti, L.B.: De re aedicatoria. Nicolaus Laurentii Alamani: Florence (1485), IX, p. 5. Vasari, G.: Le vite de pi` u eccelenti pittori scultori ed architettori. Florence (1550). Ed. by G. Milanesi. Sansoni: Florence (1878), I, pp. 168 f. L¨ ucke, Th.: Leonardo da Vinci. Tageb¨ ucher und Aufzeichnungen. Asmus: Leipzig (1940), p. 480. Cf. also Steiner, R.A.: Theorie und Wirklichkeit der Kunst bei Leonardo da Vinci. Fink: Munich (1979); on Leonardo’s accomplishments as scientist, engineer and inventor, cf., e.g., Clagett, M.: Leonardo da Vinci and the Medieval Archimedes. In: Physis 11 (1969), pp. 100–151; Feldhaus, F.M.: Leonardo der Techniker und Erfinder. Eugen Diederichs: Jena (2nd ed. 1922); Giustini, P.A.: Da Leonardo a Leibniz. La rivoluzione scientifica. Trevisini: Milan (1976); Mittelstraß, J.: ¨ Leonardo-Welt. Uber Wissenschaft, Forschung und Verantwortung. Suhrkamp: Frankfurt (1992). Pedretti, C.: A Chronology of Leonardo da Vinci’s Architectural Studies after 1500. E. Droz: Geneva (1962), pp. 143 ff. Leonardo da Vinci: Trattato della pittura. Ed. by R. du Fresne. Jacques Langlois: Paris (1651). Milan (1939). Sketch of a human anatomy in the canon of ideal proportions (Royal Library). Pacioli, L.: De divina proportione. Venice (1509), repr. Officina Bodoni: Milan (1956). Cf. Mainzer [1.8], pp. 70 ff., pp. 134 ff. Cf. also Pirenne, M.H.: The Scientific Basis of Leonardo da Vinci’s Theory of Perspective. In: British Journal for the Philosophy of Science 3 (1952), pp. 169–185. Cf. Schr¨ oder, E.: Kunst und Geometrie. D¨ urers k¨ unstlerisches Schaffen aus der Sicht seiner “Underweysung”. AkademieVerlag: Berlin (1980), pp. 34 ff. For the break-up of the artistic canon after the Renaissance, cf. Wittkower, R.: Architectural Principles in the Age of Hu-

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manism. Alec Tiranti: London (1962); likewise, Kambartel, ¨ W.: Symmetrie und Sch¨ onheit. Uber m¨ ogliche Voraussetzungen des neueren Kunstbewußtseins in der Architekturtheorie Claude Perraults. Fink: Munich (1972). Cf. Werker, W.: Studien u ¨ber Symmetrie im Bau der Fugen und die motivische Zusammengeh¨ origkeit der Pr¨ aludien und Fugen des “Wohltemperierten Klaviers” von J.S. Bach. Breitkopf & H¨ artel: Leipzig (1922). Cf. also Werner, E.: Grunds¨ atzliche Betrachtungen u ¨ber Symmetrie in der Musik des Westens. In: Studia Musicologia 11 (1969), p. 486; Mehner, K.: Beitr¨ age zum Symmetriebegriff in der Musik. In: Beitr¨ age der Musikwissenschaft 13 (1971), p. 11. Rameau, J.P.: Trait´e de l’harmonie. Ballard: Paris (1722) [English: Dover: New York 1971]. Cf. for the following exposition, Eimert, H.: Lehrbuch der Zw¨olftontechnik. Breitkopf & H¨ artel: Wiesbaden (1966). Cf. Budden, F.J.: The Fascination of Groups. Cambridge University Press: Cambridge (1972); Claus, R.: Symmetrie in der Musik. Zur Anwendung gruppentheoretischer Methoden. In: Preisinger, A. (ed.): Symmetrie. Springer: Vienna/New York (1980), p. 70, p. 76. Cf. also Solomon, L.J.: Symmetry as a Determinant of Musical Composition [Diss. West Virginia State University]. Morgantown West Virginia (1973). In that connection, Claus [8.53], pp. 31 ff. Cf. also the discussion by M. Eigen in Eigen [5.13], p. 357. Cf. also Claus [8.53], pp. 101 ff. van Beethoven, L.: Klaviersonate op. 53. 1st movement, measures 27–29; in that connection also Claus [8.53], p. 109. Sch¨onberg, A.: Herz und Hirn in der Musik. In: Sch¨ onberg, A.: Gesammelte Schriften. Ed. by I. Voitech. Vol. 1. Suhrkamp: Frankfurt (1976), pp. 104–122. Sowa [8.19], p. 15 Cf. Metzinger, J.: Le cubisme ´etait n´e. Souvenirs. Pr´esance: Paris (1972): Apollinaire, G.: Le peinture cubiste. Paris (1913).

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References

8.62 8.63 8.64

8.65 8.66

8.67

8.68 8.69

8.70 8.71 8.72

8.73

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The C´ezanne memorial exhibition of 1907 influenced Picasso and Braque. The theory of cubism was developed by Metzinger and Gleizes in their 1912 essay “Du cubisme”. Klee, P.: Das bildnerische Denken. Schriften zur Form- und Gestaltlehre. Ed. by J. Spiller. Schwabe: Basel (1964), p. 79. Klee, P.: Exakte Versuche im Bereich der Kunst. In: Bauhaus. Zeitschrift f¨ ur Bau und Gestaltung 2 No. 2/3 (1928). Kandinsky, W.: Punkt und Linie zu Fl¨ ache. Beitrag zur Analyse der malerischen Elemente. Bauhaus Book No. 9. Albert Langen: Munich (1926), repr. Benteli: Bern-B¨ umplitz (7th ed. 1973). Kandinsky, W.: R¨ uckblicke. Klein: Baden-Baden (1955), p. 25. Klee, P.: Beitr¨age zur bildnerischen Formlehre. Ed. by J. Glaesemer. Paul Klee-Stiftung [Kunstmuseum Bern]: Basel (1979). Cf. also Kagan, A.: Paul Klee: Art and Music. Cornell University Press: Ithaca N.Y. (1983); D¨ uchting, H.: Paul Klee. Malerei und Musik. Prestel: Munich (2001). Gropius, W.: Der stilbildende Wert industrieller Bauformen. In: Jahrbuch des Deutschen Werkbundes (1914), p. 29. Le Corbusier: Vers une Architecture. In: Conrads, U. (ed.): Programme und Manifeste zur Architektur des 20. Jahrhunderts. Bauwelt Fundamente. Vol. 1. Berlin (1964). Le Corbusier: Urbanisme. In: Conrads [8.69]. Schumacher, F.: Sozialer St¨ adtebau. In: Kulturpolitik. Jena (1919). Cf. also Frampton, K.: Die Architektur der Moderne. Deutsche Verlagsanstalt: Stuttgart (1983), Jencks, C.: Die Sprache der postmodernen Architektur. Deutsche Verlagsanstalt: Stuttgart (1978). Cf. Lyotard, J.-F.: Das postmoderne Wissen. Ein Bericht. Impuls & Association: Bremen (1982); Habermas, J.: Die Moderne — ein unvollendetes Projekt. Theodor W. Adorno Preis 1980. In: Habermas, J.: Kleine politische Schriften I–IV. Suhrkamp: Frankfurt (1981).

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¨ 8.74 Adorno, T.W.: Asthetische Theorie. Suhrkamp: Frankfurt (1970), p. 237. 8.75 Peitgen [2.32], map 33, p. 80. 8.76 Cf. also Steller, E.: Computer und Kunst. Programmierte ¨ Gestaltung: Wurzeln und Tendenzen neuer Asthetik. B.I. Wissenschaftsverlag: Mannheim (1992). 8.77 Picture of the theme issue “Cognition and Complex Brain Dynamics” (ed. by P. beim Graben, D. Saddy, J. Kurths) in: International Journal of Bifurcation and Chaos 14 No. 2 (2004).

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Subject Index

α-helix, 183 absolute space, 337 absolute time, 337 activator, 224, 225 actual information, 277, 283 aerodynamics, 357 aesthetics, 370, 381 agents, 324 algebra, 173 algorithmic complexity, 289, 347 algorithmic information theory, 289 algorithms, 285, 347 anharmonic oscillator, 86, 87 antigravity, 151 antimatter, 152 Antique-Medieval philosophy, 358 Antique-Medieval astronomy, 3, 38, 46 Antique-Medieval mathematics, 2 Antiquity, 22, 368, 371 antisymmetry, 181, 183, 218, 219 aperiodic crystals, 193 Aphrodite of Cyrene, 360, 361 architecture, 1 Aristotelian-Thomistic philosophy, 364 artificial intelligence, 348 artificial life, 325, 348 arts, 1, 21, 76, 329 associative memory, 316 astronomy, 4 asymmetry, 14, 135, 152, 188, 199, 210, 262 atomic model, 128–130 atoms, 47, 53, 54

attractors, 4, 13, 16, 17, 19, 158, 162, 192, 236, 254, 257, 272, 296, 345, 385 autocatalytic reaction, 236 automobile traffic, 321 automorphism, 64, 72, 180, 339 automorphism group, 4, 8, 66 autonomous nonlinear differential equations, 252 axiomatic set theory, 335 β-decay, 140, 152 B´enard experiment, 155, 158, 163 B´enard-effect, 277 balance, 1, 13, 18, 54, 359, 381 Bauhaus, 21, 378, 379, 381 beauty, 1, 2, 20, 21, 25, 357, 360, 386 bifurcation, 5, 13, 83, 88, 95, 97, 99, 167, 195, 252, 254 bifurcation tree, 87, 190, 211, 220, 221, 236, 253, 274, 354 Big Bang, 126, 148, 151 Big Picture, 97, 98 bilateral symmetry, 16, 210 bilateralia, 206 biochemistry, 199, 339 biodiversity, 223, 260 biological evolution, 347 biology, 1, 15, 154, 171, 247, 273, 283, 329 bits, 273, 290 black holes, 12, 283 Boltzmann machine, 303 Boltzmann-constant, 275 425

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Symmetry and Complexity

Born–Oppenheimer procedure, 176 bosons, 143 bounded rationality, 250 brain, 17, 210, 281 brain dynamics, 234 Buckminsterfullerenes, 195, 297 Buddhist, 362, 363 Buddhist diagrams, 26 bus topologies, 313 butterfly effect, 272, 324 BZ-reaction, 190, 191, 230, 277 Calabi–Yau spaces, 145 canon of proportions, 371 canonical invariance, 338, 340 Cantor set, 101, 106 Cantor’s diagonal procedure, 288 Cartesian geometry, 78, 111 categories, 348 causa efficiens, 55 causa finalis, 55 causa formalis, 55 causa materials, 55 causal loop, 160, 248 cell-assemblies, 231, 232, 281 cellular automata (CA), 20, 294, 296, 297, 307, 312 cellular neural networks (CNN), 306–308 cellular nonlinear networks (CNN), 20, 310 cellular self-organization, 302 cellular symmetries, 202 central nervous system (CNS), 227 central symmetry, 26 chaos, 1, 2, 12, 13, 19, 20, 48, 157, 163, 191, 249, 254, 307 chaos attractor, 98, 191, 297 chaos theory, 105 charge, 188 charge conjugation, 135 chemistry, 1, 14, 154, 171, 273, 306, 329, 339 China, 260 Chinese astronomy, 39 Chinese mirrors, 26

Chinese philosophy, 52, 60 chirality, 15, 138, 180, 184, 185 Christian Middle Ages, 364 Church’s thesis, 285, 293 classical mechanics, 118, 378 classical physics, 5, 240 cognitive hierarchies, 281 commensurability, 38 communication, 274, 282, 320, 321 competition, 261, 263 complex dynamical systems, 1, 13, 14, 17, 20, 148, 222, 224, 266, 268, 275, 282, 385 complexity, 48, 96, 99, 158, 163, 166, 188, 190, 199, 249, 256, 269, 271, 307, 313, 318, 337 complexity classes, 286 complexity theory, 286 computability, 287 computational complexity, 289, 290 computational dynamics, 284 computational ecologies, 318 computational irreducibility, 300 computational mathematics, 20 computational network, 320 computational parallelism, 291 computational symmetry, 386 computational systems, 2, 20, 312, 316 computer experiments, 20 computer science, 286 congestion, 322 consciousness, 312 conservation quantity, 116 conservative self-organization, 12, 196, 197 constructibility, 31 continuous groups, 4, 6, 75, 107, 118 continuous phase transitions, 166 control parameter, 13, 87–89, 96, 98, 167, 168, 264 Copernican revolution, 123 cortex, 227, 234 cosmic arrow of time, 12 cosmological principle, 124, 125, 127 CP -symmetry, 153 CP T -symmetry, 140, 143, 185, 188

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Subject Index

CP T -theorem, 11, 139 creativity, 387 crystallography, 4, 15, 54, 63, 374 cube, 34 cultural symmetry, 22 Curie-point, 165 curvature, 7, 78, 79, 126 cyclic groups, 66 D-tartaric acid, 172–174 data traffic, 323 deferents, 41 democracies, 263 deterministic chaos, 5, 163, 234, 254 dialectics, 268 differential geometry, 4, 77 differential topology, 175 diffusion-reaction processes, 191, 277, 310 digital information, 280 dihedral group, 67, 205 discrete group, 66 dissipation, 13, 57, 154, 190 dissipative self-organization, 13, 16, 190, 192, 197 dissipative system, 157, 190 dissymmetry, 14, 180, 185, 188, 199 distributed intelligence, 317 distribution functions, 251, 258, 319, 345, 346 diversity, 48, 154, 199, 256, 261, 329 DNA, 15, 213, 217, 218, 220, 281, 293 dodecahedron, 34, 46, 71 Doppler effect, 123 double helix, 218 duration, 346 dynamical systems, 5, 20, 158, 317, 344 dynamical systems theory, 175 ecological balance, 384 ecological dynamics, 267 ecological equilibria, 237 ecological systems, 17, 237 ecology, 237, 339 economic equilibrium, 239, 242

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economic order, 18 economic self-organization, 19 economics, 252, 267, 339 EEGs, 234 Egypt, 240 eigenvalue equation, 14, 168, 346 Einstein’s space-time, 343 Eleatic philosophy, 51 electrodynamics, 6, 9, 107, 118 electromagnetic field, 6, 10 electron, 7 elementary particle physics, 48 elementary particles, 1, 7 emergence, 2, 5, 15–18, 88, 99, 127, 128, 153, 154, 158, 159, 167, 169, 213, 252, 256, 260, 261, 279, 309, 319, 344, 353, 386 emergent cognitive structures, 233 emergent structures, 223, 228 enantiomers, 180, 185 energy, 51, 153 enlightenment, 363, 383 Entity-Relationship (ER)-diagram, 350, 352, 353, 355 entropy, 154, 275, 277, 278, 283, 284, 299 epicycle-deferent technique, 41–45, 58 EPR-experiment, 292 equilibrium, 54, 88, 240, 241, 296 equilibrium of fright, 260 ethics, 38 Euclidean geometry, 2, 4, 5, 32, 46, 65, 73, 74, 81, 110, 111, 121, 338, 356 Euclidean group, 112, 113 Europe, 240 evolution, 17 evolutionary architecture, 325, 327 evolutionary game theory, 247 evolutionary stable strategy (ESS), 247 evolutionary tree, 220 exponential computational time, 286 extensive variables, 250, 252 extremal principle, 215 face-vase illusion, 309

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Symmetry and Complexity

feedback loops, 266 Feigenbaum-constant, 95 Fermat prime numbers, 31 fermions, 143 ferromagnet, 12, 153, 165 Fibonacci sequence, 37, 204, 205, 363 field equations, 144 fitness degrees, 220, 247 fixed point attractor, 13, 19, 162, 233, 296, 297, 303 fluctuations, 257, 260, 261, 321 fluid dynamics, 13 form invariance, 64 fractal dimension, 101, 164 fractal geometry, 5, 106 fractals, 5, 99, 256 frieze groups, 67, 68, 182, 375 fright of nonequilibrium, 260 functional symmetry, 201 functionalism, 382 Galilean invariance, 5, 108, 121, 133 Galileo group, 112, 114, 117, 118 Galois theory, 4, 76 game of life, 295, 301 game theory, 243, 246 game tree, 244 gauge fields, 6, 139 gauge groups, 141, 189 gauge invariance, 108 gauge symmetries, 276, 339 General Agreement on Tariffs and Trade (GATT), 265 general relativity theory, 9, 121, 277 genetic algorithms, 324 genetic information, 239, 279, 280, 283, 302 genetic information systems, 280 genetic self-organization, 220, 278, 293, 294 geometry, 173, 382 gestalt-psychology, 304 global governance, 239, 265 global Lorentz invariance, 123 global networking, 325 global symmetry, 6, 7, 135, 136

globalization, 239, 259, 260, 263, 264, 267, 269, 272, 329, 372, 385 Golden Rectangle, 36, 37 Golden Section, 35–37, 46, 204, 206, 358–360, 363, 365, 368, 384 Golden Spiral, 36, 75 GPS, 325 gravitation, 11, 284, 337 gravitational field, 121, 122 graviton, 11, 143 Greek astronomy, 40 Greek mathematics, 29, 358 Greek philosophy, 60, 358 group theory, 76, 178, 338, 371 groups, 5, 336 H¨ uckel model, 181 hadrons, 10 halting problem, 287, 288 Hamiltonian dynamics, 161 Hamiltonian function, 160, 291 Hamiltonian mechanics, 130, 131 Hamiltonian operator, 131, 133, 139, 181, 291 harmonic oscillator, 84 harmony, 1, 2, 23, 25, 34, 36, 38, 40, 51, 127, 240, 371, 382 Hartree-Fock method, 177 Hebb-type learning, 303 Helmholtz–Lie requirement, 81 Heraclitean philosophy, 51 hierarchy, 18, 222, 330, 348, 365 Higgs mechanism, 149 Hilbert space, 8, 82, 83, 108, 131, 176, 291, 334, 339 Hinduism, 358, 362, 363 Hodgkin–Huxley equation, 229 homo economicus, 249 homogeneity, 53, 79, 110, 111, 117, 123 Hooke’s law, 84, 85 Hopf-bifurcation, 89, 97 Hopfield system, 303 hydrodynamics, 159 hypercycles, 15, 212–214 I Ching, 27

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Subject Index

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429

icosahedron, 34, 46, 71 incommensurable, 36 incompleteness theorem, 288, 289 India, 260 Indian, 26, 67 inertial system, 5, 112, 343 inflationary period, 150, 151 information, 2, 244, 274, 284, 316, 357 information content, 274 information dynamics, 273 information entropy, 275, 276 information flow, 275 information packet, 321 information processing, 317 information retrieval (IR), 324 information symmetry, 271 information systems, 278, 279, 311, 322 information theory, 273, 275 inhibitor, 224, 225 instability, 1, 5, 239, 248, 260, 268 intelligence, 325 intensive personal variables, 252 intentionality, 234, 387 Internet, 20, 320, 321, 385 invariance, 5, 72, 74, 338 inversion, 68, 179 invisible hand, 242 irreducibility, 300 irreversibility, 188, 189, 345, 346 Islamic culture, 362–364 Islamic ornaments, 67 isomerism, 173 isometry group, 81, 125 isospin symmetry, 10 isotropic space, 111 isotropy, 53, 58, 79, 110, 123

Lagrangian operator, 114, 115, 144 laissez-faire economics, 268 Laplacian spirit, 162, 256 laser, 157 Lausanne school, 19 LCAO method, 180, 181 learning algorithms, 311 left-right symmetry, 10, 15 Leibniz-Pauli-principle, 8 leptons, 11 Leviathan, 240 Lie groups, 4 life sciences, 15, 199 limbic system, 234 limit cycle, 17, 88, 90, 96, 98, 106, 162, 191, 233, 254, 259 linear computational time, 286 linear decision, 271 linear differential equations, 85, 159, 163, 168 linear system, 2, 85 linear-stability analysis, 14, 167, 169, 221, 222, 229, 252, 354 linearity, 84, 256, 277 Liouville’s theorem, 161, 162 Lobachevski geometry, 126 local activity principle, 306 local equilibria, 211 local Lorentz invariance, 123 local symmetries, 6, 7, 127, 135, 136 lock and key principle, 195 logarithmic spiral, 75 logos, 22, 35, 51, 360, 383 longitudinal symmetry, 203 Lorentz group, 6, 114, 120 Lorentz invariance, 6, 108, 120, 121, 128, 135

Judaism, 363 Julia sets, 103, 105, 385

M-theory, 110, 146–148, 339 macro-irreversibility, 345 macro-molecular chemistry, 182 macrocosm, 3 macrodynamics, 166, 222, 231, 232, 249, 250 macrolevel, 18, 213, 345 macromolecules, 184

KAM theorem, 162 knowledge representation, 350, 355 knowledge society, 327 L-tartaric acid, 172–174

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Symmetry and Complexity

macroscopic order, 153, 237 macroscopic patterns, 13, 166, 297 macroscopic point of view, 248 macrostate, 18, 232, 242, 252, 275 macrovariables, 19, 250, 251 Mandelbrot set, 19, 104, 105, 385 many-bodies problems, 344 markets, 19, 242, 263 master equation, 19, 252 mathematics, 3, 4, 63, 335, 369 matter, 152, 252 maximin-strategy, 244 Maya, 39 McCulloch-Pitt network, 302 mechanics, 242 MEMS (micro-electro-mechanical system), 306 Meru diagram, 26 mesh topologies, 314, 315 meso-tartaric acid, 174 mesocosm, 3 metabolism, 200, 214 metamorphosis, 205 meteorology, 17, 60 microcosm, 3 microdynamics, 222, 232, 248, 258 microeconomic model, 249 microlevel, 18, 212, 345 microreversibility, 12, 154, 292, 298, 299, 345 microstates, 18, 242, 275 Middle Ages, 21, 31, 55, 240, 368 migration dynamics, 19, 254, 255 mind, 18 minimax-strategy, 244 Minkowski geometry, 338, 343 Minkowskian geometry, 6, 118, 121 mixed strategy, 245, 247 models, 341, 349, 351, 353, 356, 357 modernism, 382 molecular computer, 293 molecular self-organization, 293 morphogenesis, 225 morphological symmetries, 16 multiuniverse, 151 music, 371–377

12-tone music, 371, 372, 379 mutation, 200, 220 mythologies, 21, 24, 39 nanocrystals, 193, 196 nanosystems, 190 nanotechnology, 293 Nash equilibrium, 246 natural philosophy, 2–4, 6, 127, 331 nature, 1, 21, 55 Navajo Indians, 23, 24 Navier–Stokes equation, 96 neo-classical economics, 242 nervous systems, 17 network typologies, 313 networks, 231, 294 neural information, 280 neural networks, 325 neural self-organization, 231, 239, 302 neurobiological evolution, 306 neurotransmitter, 311 neutrons, 10 Newtonian space-time, 343 Noether theorem, 118 noise, 12, 235 nominalism, 332–334 non-equilibrium chemistry, 310 non-Euclidean geometry, 74 noncooperative games, 246 nonequilibrium dynamics, 165–167, 169, 211, 217, 222, 230, 232, 239, 244, 248, 260 nonlinear complex systems, 2, 88, 210, 296, 301 nonlinear decisions, 271 nonlinear differential equations, 222, 230, 295 nonlinear diffusion equation, 229 nonlinear diffusion–reaction equation, 228 nonlinear dynamics, 18–20, 105, 199, 217, 222, 226, 252, 254, 263, 265, 268, 271, 273 nonlinear equations, 89, 159, 249, 252 nonlinear partial differential equation, 310

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Subject Index

nonlinearity, 13, 99, 128, 158, 163, 190, 199, 211, 267, 269, 344 normal-form games, 243, 244 NP-problem, 287 object-oriented programming, 348–350, 352, 353 octahedron, 32, 34, 71 OECD, 265 ontogenesis, 16 ontology, 330, 332, 348, 349, 351 operator, 160 optical activity, 172 optimization, 212 orbital symmetry, 181, 182 order parameters, 13, 14, 16, 18, 165, 166, 167, 169, 220, 231, 239, 248, 252, 258, 261, 268, 306, 354 ornament groups, 69, 72 ornament symmetries, 376 ornaments, 1, 180, 375 oscillations, 19, 162, 180 P, 286 parabolic manifolds, 81 Pareto-optimal, 243 parity, 10, 188 parity violating weak interaction, 15, 152, 185–187 partial differential equations (PDE), 12–14, 18, 307 particle–antiparticle symmetry, 134, 135 pattern formation, 309, 310 pattern recognition, 278, 304, 305, 309 pentagons, 33 pentagram, 65 Perceptron, 302, 307, 311 period doubling bifurcation tree, 95, 96, 105 permutation group, 8, 76 Petri nets, 355 phase portraits, 177, 254, 255, 301 phase space, 162 phase transition, 1, 12, 13, 16, 20–22, 109, 147–149, 153, 155–157, 165,

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192, 199, 210, 212, 217, 225, 234, 240, 250–254, 257, 260, 268, 269, 273, 280, 293, 296, 303, 304, 321, 353, 381, 386 philosophy, 23, 36, 329, 330, 332, 356 philosophy of science, 336, 341 photon, 9, 11 phyllotaxis, 37, 204 phylogenesis, 16 physics, 1, 4, 7, 154, 171, 273, 306, 329 physiocratic economy, 241 Planck’s constant, 142, 290 Planck’s length, 142 Planck-time, 148, 150 Platonic bodies, 33, 64, 71, 74, 180, 192, 195, 200 Poincar´e group, 120 Poincar´e maps, 91–94, 106 point groups, 178 Poisson distribution, 319 political equilibrium, 259 political phase transitions, 258 political systems, 263 politics, 18, 252 polygons, 29, 71 polyhedron, 33, 74 polymerization, 182, 218 polynominal computational time, 286 polyphony, 380 population dynamics, 239 positivism, 332, 383 positron, 9 postmodernism, 384 potential information, 276, 277, 283 pragmatic information, 281 pre-established harmony, 6 prebiotic evolution, 212 predicative logic, 355 presocratic philosophy, 4, 116 principle of computational equivalence, 20, 301, 326, 387 principle of relativity, 5 probabilistic distributions, 245, 252, 254, 255, 275 program-size complexity, 290 projective geometry, 74, 369

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Symmetry and Complexity

protein analysis, 183, 184 protons, 10 Pythagorean quadrivium, 2, 38, 46, 372 Pythagorean theorem, 119 Pythagoreans, 29, 31, 32, 34, 36, 372 quadratic computational time, 286 quantum bits, 21, 276, 291 quantum chemistry, 176, 178, 185, 187 quantum chromodynamics, 10 quantum computer, 21, 291 quantum dynamics, 283 quantum electrodynamics, 108, 134, 135, 141 quantum field theory, 134 quantum gravity, 284 quantum information, 21, 276, 277, 291, 292 quantum machine, 290 quantum mechanics, 7–9, 82, 83, 108, 128, 130–133, 137, 172, 290, 333, 338, 345, 357 quantum physics, 12, 298 quantum systems, 8, 9, 82, 131, 176, 184 quantum teleporting, 292 quantum theory, 276 quantum universe, 21 quantum vacuum, 151 quarks, 10, 11, 140 R¨ ossler attractor, 93, 94 Radiolaria, 208 random network, 318, 319, 320 random variables, 275 randomness, 1, 48, 249, 297, 299, 307 Rayleigh number, 155 reaction-diffusion processes, 211 recursion, 101 reflection, 5, 67, 69, 179, 202, 374 reflection symmetry, 24, 26, 66, 210 regular polygons, 30, 31, 180 regular solids, 4, 46, 59 relative computability, 347 relativistic cosmology, 123

religion, 21 Renaissance, 21, 60, 255, 366, 367, 370, 385 representation of knowledge, 349 reversibility, 299, 345 reversible cellular automaton, 298, 299 reversible structures, 12 Reynolds number, 155 Riemannian manifolds, 78, 79, 81, 122 ring topologies, 313 Robertson–Walker metric, 125 rotation, 5, 202, 374 rotation groups, 72, 111 rotational symmetry, 24, 66, 89, 208 routers, 321 scholastics, 365 Schr¨ odinger equation, 8, 131, 291, 338 second law of thermodynamics, 12, 154, 189, 275, 277, 299 selection, 200, 214, 261 self-consciousness, 232 self-organization, 12, 110, 154, 165, 167, 193, 213, 248, 258, 260, 277, 302, 326, 327 self-regulation, 241 self-similarity, 5, 99, 103, 106, 322, 323 semantic webs, 350, 351, 353, 355 sequence space, 215 similarity group, 64, 73 simplicity, 1, 58 singularity, 126, 217 small-world concept, 318, 320 social balance, 239, 262, 339 social evolution, 283 social self-organization, 273, 283, 329 social symmetry, 19, 239, 262 society, 18, 19, 259, 267, 381 socio-economic dynamics, 260 socio-economic transition, 248 sociobiology, 235, 317, 324 socioconfiguration, 19, 250–252, 258 sociodiversity, 259–261, 264 sociodynamics, 239, 252, 258, 278, 357 socioeconomic systems, 262 sociology, 18, 252

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Subject Index

software-engineering, 352, 353 solitary waves, 159, 160, 229 space groups, 184 space-time, 344 space-time symmetry, 6, 8, 114, 116, 124 special relativity theory, 6, 108, 121, 276 spectroscopy, 63 spherical symmetry, 41, 129 spiral symmetry, 209 spontaneous symmetry breaking, 186, 217 stability, 239 stable modes, 354 star topology, 313, 315 state space, 96, 344 state-transition diagrams, 354, 355 state-transition systems, 354, 377 stereochemistry, 4, 15, 173, 178 stochastic chaos, 234 stochastic machine, 290 stochastic nonlinear differential equation, 252 Stoic philosophy of nature, 61, 362 strange attractor, 164 strategy, 243 string theory, 143, 144 strong interactions, 10 structural species, 342 structural types, 342 structures, 331, 332, 334, 341, 349, 357 SU(2) × U(1) symmetry, 11, 109, 140, 149, 152, 188 SU(2) symmetry, 10, 139, 140, 150, 188 SU(3) symmetry, 10, 109, 140, 150 SU(5) symmetry, 11, 109, 152, 153 superposition, 9, 21, 291, 292 superselection rules, 9 superspace, 99 superstring theories, 11, 145, 147 supersymmetry, 12, 109, 142, 144, 339, 340 superunification, 140 supramolecular chemistry, 193, 194

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sustainable, 266, 267 swarm intelligence, 317 symmetrical spaces, 7, 82, 124 symmetry breaking, 2, 12, 14–17, 21, 49, 83, 84, 86, 88, 90, 96, 97, 99, 110, 147, 149, 150, 154, 155, 158, 184, 185, 187, 199, 201, 210, 248, 253, 256, 257, 273, 274, 284, 304, 305, 308, 309, 329, 340, 345, 353, 354, 360, 361, 382 symmetry groups, 67, 68, 107, 144 symmetry of time, 12 synergetics, 310 systems, 165 Tao, 60 Taoist philosophy, 60, 358, 362 telecommunication, 320 teleology, 16 tetrahedron, 34, 46, 71 theorema egregium, 78 thermal equilibrium, 12, 13, 16, 189, 192, 197, 211, 225 thermodynamic self-organization, 278 thermodynamical arrow of time, 12 thermodynamics, 239, 242, 283, 302, 339, 346 three-body problem, 162 time operator, 346 time reversibility, 132, 133 time violation, 188 time-series analysis, 385 topology, 74 Trade Related Aspects of Intellectual Property Rights (TRIPS), 265 trajectories, 161, 164, 176, 345 transformation groups, 73, 338 translation, 5, 67, 68, 180, 202, 374 translation group, 111, 112 translation symmetries, 67 tree topologies, 314 Turing machine, 285–287, 290, 291, 293, 300 ψ-oracle Turing machines, 347 U(1) symmetry, 9, 108, 140, 188

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Symmetry and Complexity

ubiquitous computing, 325 UN, 270 uncertainty, 244, 275, 278, 281 uncertainty principle, 142 undecidability, 289, 300 unification, 141 Unified Modeling Language (UML), 355 universal Turing machine, 285 universals, 332 unsolvability, 288 unstable modes, 354 urban dynamics, 257 urbanisation, 254 utility function, 243

virus, 200 von Neumann computers, 294 weak interaction, 10 welfare, 262 wholeness, 199 Woodward–Hoffmann rules, 182 World Trade Organization (WTO), 265, 266 World Wide Web (WWW), 318–323 yang, 27, 60 Yang–Mills theory, 149 yin, 27, 60 zero-sum games, 243, 245, 260

virtual reality, 325

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Name Index

Abel, N.H., 76 Adorno, T.W., 383 Alberti, L.B., 366 Alembert, J. d’, 76, 114, 372 Anaxagoras, 52, 53 Anaximander, 49, 50, 359 Anaximenes, 50 Andronov, A., 99 Apollonius of Perga, 41, 42 Archimedes, 75 Aristophanes, 359 Aristotle, 3–5, 16, 38, 54–57, 60, 61, 222, 329, 330, 332, 346, 348, 349 Arnold, V.I., 162 Augustine, 365 Bach, J.S., 371, 375, 380 Barlow, H.B., 231 Barlow, W., 72 Beltrami, E., 80 B´ enard, H., 155 Bergson, H., 346 Bernoulli, J., 75 Birkhoff, G., 89, 163 Bohr, N., 7, 8, 128–130, 331 Boltzmann, L., 12, 189, 275 Bonnet, C., 37, 205 Bonnet, O., 78 Born, M., 130 Braque, G., 378 Broglie, L. de, 130 Bruno, G., 123 Buckminster Fuller, R., 195 Cantor, G., 288 Carnap, R., 336, 352 Cartan, E., 7, 77, 81, 82, 108, 124

Cayley, A., 173 C´ ezanne, P., 378 Chaitin, G.J., 289 Chua, L.O., 306, 307, 310 Church, A., 285, 286 Conway, J.H., 295, 301 Copernicus, 58, 59 Copernicus, N., 3, 42 Crick, F.C., 218 Crum Brown, A., 173 Curie, P., 188 Darwin, C., 212, 220, 261 Democritus, 53 Descartes, R., 53, 255, 387 Dirac, P.A.M., 9, 108, 134, 331 Douady, A., 105 Drechsler, E., 194 D¨ urer, A., 370 Eddington, A.S., 121 Eigen, M., 212, 214, 215 Einstein, A., 7, 77, 108, 118, 120, 121, 125, 127, 128, 141, 144, 145, 159, 277, 331, 343, 344, 353, 378 Empedocles, 46, 52, 60 Euclid, 20, 30, 32, 35, 40, 65, 71, 74, 356 Eudemus, 41 Eudoxus of Knidos, 3, 41 Euler, L., 74, 114 Fatou, P., 103 Fedorov, E.S., 68, 72 Feynman, R.P., 9, 193, 194, 284 Fischer, E., 194, 195 Fludd, R., 370 Freeman, W., 234 435

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Symmetry and Complexity

Galenus, 359 Galilei, G., 57, 122, 332, 341 Galois, E., 76 Gauss, C.F., 30, 31, 76–78, 102, 104, 105, 108 Gerisch, G., 16 Gierer, A., 224 Glashow, S., 11, 109 G¨ odel, K., 288 Goethe, J.W. von, 37, 205, 331 Gropius, W., 21, 381 Ha¨ uy, R.J., 172 Haeckel, E., 206, 208, 209 Haken, H., 168, 310 Hamilton, W.R., 117 Hawking, S.W., 284 Hayek, F. von, 264 Hebb, D., 230, 300, 302, 312 Hegel, G.W.F., 268, 269 Heisenberg, W., 10, 48, 50, 142, 151, 183, 333 Helmholtz, H. von, 7, 65, 79–81 Heraclitus of Ephesus, 50, 51, 53, 55, 116 Hermes, J., 31 Hertz, H., 6 Hilbert, D., 7, 82, 100, 127 Hippasus of Metapontum, 36 Hippodemus , 359 Hobbes, T., 240 Hodgkin, A., 228 Hogarth, W., 370 Honnecourt, V. de, 366, 367 Hopf, E., 89 Hopfield, J.J., 306 Hubbard, J.H., 105 Hubble, E.P., 123, 124 Hume, D., 370 Huxley, A., 228, 378 Julia, G., 103, 105 Kaluza, T., 144, 146 Kandinsky, W., 379 Kant, I., 53, 258, 269, 270, 333, 356 Kekul´ e, A., 172, 173, 180 Kepler, J., 31, 34, 37, 41, 43, 58–60, 68, 141, 332 Keynes, J.M., 249, 284 Kirchner, A., 31

Klee, P., 378–380 Klein, F., 6, 72, 73, 107, 112, 117, 338 Klein, O., 144 Koch, H. von, 100 Kolmogorov, A.N., 162, 289 Lagrange, J.L., 76, 114 Landau, L.D., 165, 166 Laplace, P.S., 53, 162 Le Bel, J.A., 173 Le Corbusier, 381 Lee, T.D., 138 Leibniz, G.W., 6, 8, 64, 79, 333, 352 Leonardo da Vinci, 367–369 Leonardo of Pisa, 37 Lie, S., 7, 76, 80, 81, 107, 108, 113 Linde, A., 151 Liouville, J., 161 Locke, J., 241 Lorenz, E.N., 17, 163, 164 Lotka, A.J., 17 Lullus, R., 348 Malsburg, C. von der, 231 Mandelbrot, B.B., 5, 19, 100, 102–104 Mann, T., 15 Marx, K., 262 Maxwell, J.C., 6, 7, 9, 107, 128, 141, 144 Maynard Smith, J., 247 McClelland, J.L., 304 Meinhardt, H., 16, 224 Mie, G., 7, 127, 128 Minkowski, H., 118, 343 Minsky, M., 311 Morgenstern, O., 243 Moser, J., 162 Nash, J.F., 246, 247 Neumann, J. von, 8, 9, 108, 243, 245, 247, 285, 294, 307 Newton, I., 9, 65, 112, 114, 120, 141, 145, 241, 331, 337 Noether, E., 6, 117, 118 Occam, W., 334 Pacioli, L., 368 Parler, H., 365 Parmenides of Elea, 51, 53, 55, 116 Pasteur, L., 15, 172, 180, 185, 187

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Name Index Pauli, W., 8 Pauling, L., 183 Pericles, 359 Perutz, M., 184 Peter of Spain, 348 Picasso, P., 378 Planck, M., 129, 142, 146, 378 Plato, 3, 20, 29, 30, 40–42, 46–48, 51, 54, 91, 99, 329, 332 Poincar´ e, H., 5, 77, 89, 91, 93, 162, 163, 299 Polykletus, 359 Prigogine, I., 189, 310, 346 Ptolemy, 30, 41, 43, 58 Pythagoras, 35, 111, 127, 356 Rameau, J.P., 372 Rashevsky, N., 224 Richelot, F.J., 31 Riemann, B., 7, 77, 78, 82, 108 Robertson, H.P., 125 Rosenblatt, F., 302, 307, 311 Rumelhart, D., 304 Salam, A., 11, 109 Sch¨ onberg, A., 371, 372, 375, 376 Schoenflies, A., 72 Schr¨ odinger, E., 130, 178, 193, 331, 339, 342 Schumacher, F., 382 Schuster, P., 212, 214 Schwinger, J.S., 9 Shannon, C.E., 273, 327 Simon, R., 37 Singer, W., 231 Smith, A., 19, 241, 242, 261, 265, 327, 331

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Spencer, H., 16, 224 Stockhausen, K., 371 Thales of Miletus, 49 Theaetetus, 32 Thom, R., 97–99 Thomas Aquinas, 364 Thompson D’Arcy, 207, 208 Toynbee, A.J., 248 Turing, A.M., 16, 224, 284, 287, 288, 300, 301 van Beethoven, L., 371, 375 van’t Hoff, J.H., 173, 175, 185 Vasari, G., 367 Vitruvius, 359, 362, 367 Volterra, V., 17 Walker, H.G., 125 Watson, J.D., 218 Weinberg, S., 11, 109 Weizs¨ acker, C.F. von, 276 Weyl, H., 6, 8, 11, 81, 108, 135, 333 Wheeler, J.A., 142 Wigner, E.P., 8, 108, 133 Wilson, W., 270 Witten, E., 148 Wolfram, S., 295, 300 Wright, S., 215 Wu, C.S., 138 Yang, C.N., 138 Yang, L., 306 Zarathustra, 35 Zuse, K., 285