The Magic Of Mathematics-slicer.pdf

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Discovering the Spell of othematics

M*L«*

lJ

,z

author of The Mathematics Calendars The Joy of Mathematics

THE

MAGIC Discovering the Spell of Mathematics

THEONI PAPPAS WIDE WORLD PUBLISHING/TETRA

Copyright © 1994 by Theoni Pappas. rights reserved. No part of this work may be reproduced or copied in any form or by any means without written permission

All

from Wide World

Publishing/Tetra.

Portions of this book have

appeared

in

previously published

works, but were too essential to not be included.

Wide World

Publishing/Tetra

P.O. Box 476 San Carlos, CA 94070

Printed in the United States of America. Second Printing, October 1994.

Library of Congress Cataloging-in-Publication

Data

Pappas, Theoni, The

magic of mathematics Theoni Pappas.

:

discovering

the

spell of mathematics /

cm.

p.

Includes

bibliographical

references and index

,

ISBN 0-933174-99-3 1.

Mathematics--Popular works.

QA93

,

P368

510-dc20

I. Title.

1994

94-11653 CIP

This book is dedicated to mathematicians

who have created and are

the magic

creating

of mathematics.

CONTENTS

PREFACE

1

MATHEMATICS IN EVERYDAY THINGS

3

MAGICAL MATHEMATICAL WORLDS

33

MATH EMATICS & ART

63

THE MAO IC OF NUMBERS

97

MATHEMATICAL MAGIC IN NATURE

119

MATHEMATICAL MAGIC FROM THE PAST

143

MATHEMATICS PLAYS ITS MUSIC

173

THE REVOLUTION OF COMPUTERS

189

MATHEMATICS & THE MYSTERIES OF LIFE

223

MATHEMATICS AND ARCHITECTURE

243

THE SPELL OF LOGIC, RECREATION & GAMES

265

SOLUTIONS

311

BIBLIOGRAPHY

315

INDEX

321

THE MAGiCOf MATHEMATICS

1

PREFACE You don't have to solve

be

problems

or

mathematician to discover the

a

magic of mathematics. This book with not

underlying

an

expect

Is

collection of Ideas

a

mathematical theme. It is not

to become

exhausted. The Magic

proficient of

in

topic

a

ideas

—

textbook. Do

a

find

or

an

idea

delves into the world of

Mathematics

ideas, explores the spell mathematics casts

on our

lives, and helps

you discover mathematics where you least expect it.

Many

think of mathematics

as a

fixed curriculum.

rigid

could be further from the truth. The human mind mathematical ideas and

creates

independent

of

world

our

world almost

as

which

from

new

objects

point

can

If

worlds

new

these ideas connect to

our

magic wand had been waved. The way

one

always

operate, equations solves

a

fascinating

and presto

—

Nothing

continually

dimension

can

disappear

be found between any two

into another,

in a

points, numbers

solved, graphs produce pictures, infinity

are

problems, formulas

generated

are

—

all

seem

to possess

a

magical quality. Mathematical ideas

are

in alien worlds and its

A

creativity. world,

while

perfect our

figments of the imagination. Its ideas exist objects are produced by sheer logic and

square

world

or

has

circle exists in

a

mathematical

of

only representations

things

mathematical. The topics and concepts which

by

no means

can

it

mentioned in each

easily cross over the arbitrary boundaries

were

possible,

idea to and

are

can

a

area.

Each topic

enjoyed independently.

I

stepping stone into mathematical worlds.

are

examples

of chapters. Even if

it would be undesirable to restrict

specific

be

chapter

confined to that section. On the contrary,

a

mathematical

essentially self-contained, hope this book will be a is

Print Gallery by M.C. Escher. © 1994 M.C. Escher/Cordon Art-Baarn-Holland, All

rights reserved.

MATHEMATICS IN EVERYDAY THINGS THE MATHEMATICS OF FLYING

THE MATHEMATICS OF A TELEPHONE CALL PARABOLIC REFLECTORS & YOUR HEADLIGHTS COMPLEXITY AND THE PRESENT MATHEMATICS & THE CAMERA

RECYCLING THE NUMBERS

BICYCLES, POOL TABLES & ELLIPSES THE RECYCLING NUMBERS

LOOKOUT FOR TESSELLATIONS

STAMPING OUT MATHEMATICS MOUSE'S TALE A MATHEMATICAL VISIT THE EQUATION OF TIME

WHY ARE MANHOLES ROUND?

4

THE MAGIC OF MATHEMATICS

There is

no

branch of mathematics, however abstract,

which may not someday be applied to of the real world.

So

many

things

routines have

from one

taking least

a

a

with which

we

come

mathematical basis

plane flight

expects,

one

to the

or

—

Nikolai

phenomena

Lobachevsky

into contact in

our

daily

connection. These range

shape of a manhole. Often when

finds mathematics is involved. Here is

random sampling of such

cases.

a

MATHEMATICS IN EVERYDAY THINGS

5

THE MATHEMATICS

OF FLYINC

The grace and

ease

human's desire to

of the

fly. Ancient

interest in various

realizes that the

just

a

Greek

flight

of birds have

always

tantalized

stories from many cultures attest to

creatures.

flying flight of Daedalus

myth. Today enormous

Viewing hang gliders,

and Icarus

was

probably

one

not

sized alrcrafts lift themselves

and their cargo into the domain of the bird. The historical steps to achieve

downs.

flight,

as we

Throughout

mathematicians and other

of flying and have

now

know It, has

had Its up and

literally

the years, scientists. Inventors, artists,

professions

have been

Intrigued by

developed designs, prototypes,

the Idea

and experiments

in efforts to be airborne.

Here Is

a

condensed outline of the

•Kites

were

invented

•Leonardo da Vinci

history of flying:

by the Chinese (400-300 B.C.).

scientifically studied

sketched various flying machines (1500).

the flight

of birds and

6

THE MAGIC OF MATHEMATICS

Italian mathematician Giovanni Borelli

•

muscles •

were

too weak to support flight

proved

that human

(1680).

Frenchmen Jean Pilatre de Rozier and

Marquis d! Arlandes

made the first hot air balloon ascent (1783). •

British inventor. Sir

George Cayley, designed the airfoil

(cross-section) of a wing, built andflew (1804) thefirst model

glider, andfounded the science of aerodynamics. •

Germany's

Otto Lilienthal devised

a

system to measure the lift

produced by experimental wings and made thefirst successful manned gliderflights between 1891-1896. •

In 1903 OrviUe and Wilbur

powered propeller

driven

with wind tunnels and

Wright made the first engine

airplane flights. They experimented

to measure the lift of designs. They perfected theirflying techniques and machines to the point that by 1905 their flights had reached 38

weighing systems

and drag

length covering a distance of20 miles!

minutes in

Here's how

we

In order to

fly,

get offthe ground: there

be balanced.

are

vertical and horizontal forces that must

Gravity (the downward vertical force) keeps

earthbound. To counteract the

pull of gravity, lift (

a

upward force) must be created. The shape of wings and the

airplanes

Is essential In

to

flight

is

always

a

think of air

medium,

as

airplane itself,

as a

never

or

that

as

the

speed of gas

or

seems

flight of birds,

of

almost

but without

of the components of

ground.

One does

since it is invisible. Yet air

an

slices the air

Swiss mathematician Daniel Bernoulli

Bernoulli's

It

have left the

substance,

The wing of

water.

divides

of the

key.

physical analyses

flying, today's airplanes would not

holds the

quantify the elegance

the mathematical and

design

creating lift. The study of nature's design

of wings and of birds In

sacrilegious

us

vertical

airplane, as

as

it passes

well

as

the

through

it.

(1700-1782) discovered

fluid increases its pressure decreases.

principle1 explains

the lift force. The top of the

how the

wing

shape of a wing

is curved. This

curve

creates increases

MATHEMATICS IN EVERYDAY THINGS

the

of air and

speed

passing curve,

speed of the

thus its air pressure is

wing

and thus lifts the

gravity)

air

passing under the wing

higher.

pushes toward

moves or

the air pressure of the air

Since the bottom of the wing does not have this

over it.

the

thereby decreases

7

plane

The

high

is slower and

air pressure beneath the

the low pressure above the wing,

The

into the air.

weight

(the pull of

is the vertical force that counteracts the lift of the

plane.

The wing's shape makes the distance over the top longer, which means air must travel over the top faster, making the pressure on the top of the wing lower than under the wing. The greater pressure below the wing pushes the

wing

up.

When the wing is at

a

steeper angle,

the distance

the top is

over

even

longer, thereby increasing the

liftingforce.

Drag and thrust

picture.

Thrust

backwards. relies

pushes

the

plane

A bird creates thrust on

its

another, i.e. be thrust and

greater the

the horizontal forces which enter the

flying

forward while

drag pushes

it

its

wings, while

a

by flapping For

to maintain

plane propellers jets. straight flight all the forces acting on it must equalize

plane and

are

zero.

or

a

The lift and gravity must be

drag must balance. During take off the

than the

drag,

but in

flight they

must be

zero,

a

level one

while the

thrust must be

equal, otherwise

plane's speed would be continually increasing.

Viewing birds swooping and diving reveals

two other

flying factors.

8

THE MACIC OF MATHEMATICS

When the will

also

speed of air over the top of the wing Increase.

approaching of the

wing

approximately bird

or

15

on

angle of attack,

be further Increased. If this or more

plane begins

it is called the

vortexes

By Increasing

air, called the

can

the

degrees,

the lift

can

Increased, the lift

wing's angle the

speed

angle

over

to

the

the

top

Increases to

stop abruptly and the

to fall Instead of rising. When this takes

angle of stall. The angle of stall

the

Is

place

makes the air form

top of the wing. These vibrate the wing causing the

lift to weaken and the force of gravity to overpower the lift force. Not

having

been endowed with the

flying equipment

humans have utilized mathematical and themselves and other

physical principles

to lift

off the ground. Engineering designs continually adapted to Improve an

things

and features2 have been aircraft's

of birds,

performance.

lLaws governing

the flow of air for

airplanes apply

to many other aspects certain computer

lives, such as skyscrapers, suspension bridges, disk drives, water and gas pumps, and turbines.

In

our

^The flaps and slots are changes adapted to the wing which enhance lift. The flap Is a hinged section that when engaged changes the curvature of the wing and adds to the lift force. Slots are openings In the wing that

delay

the stall for

a

few

degrees.

MATHEMATICS IN EVERYDAY THINGS

S

THE MATH EMATI C time

Every

telephone send

a

you

pick-up

the

place

call,

receiver to

fax,

a

OFATELEPHONI CALI

modem Information

or

you are entering a phenomenally complicated and enormous network. —

encompasses the

many calls How does

are a

globe

is

The communication net that

amazing. It

system

which is

call find its way to

day

over

"broken-up" by

different countries and bodies of water

phone

is difficult to

fielded and directed each

someone

varied

operate?

in your

Imagine how this network.

systems

How does

city,

state

or

a

of

single

another

country?

In the

early years

cranked the

of the

phone

to

telephone,

get

an

one

picked-up the

operator.

A local

the line from the local switch board and said and from there connected you with the reach.

Today

operator "number

party you

the process has mushroomed

as

receiver and

were

came on

please",

trying

to

have the various

10

THE MAGIC OF MATHEMATICS

methods used to convert and direct calls. Mathematics

Involving sophisticated types of linear programming, coupled with binary systems and codes, make

sense

out of a

potentially precarious

situation.

How does your voice travel? Your voice

converted In the receiver to electrical electrical

impulses

produces sounds which are

signals. Today

these

can

40

be carried and

converted In

3 ^l^.

J976

16'

a

variety of ways.

They may be changed

/*U*X~W*A~*-u^ft

to laser

light signals which

then

are

carried

along

fiber optics

cables1, they maybe N

converted to radio

signals

and transmitted over

radio

or

microwave links

from tower to tower

country,

or

kX**4-u4

u-

fcwv^U^

-fri,

tf^.c-

4L~-~i

Tt-t.

e^4r*y**f

**7

they

may remain electrical

along

*****

across a

the

as

signals

4tt~ZE^ce.

i

?k7\ lf~«Zio-tpr«ttkta<

a+bl, where

l=y^-l) the 16th

century.

are a means

of

as

organization of

this line Is linked to

The

has

only one only one

Imaginary numbers

line. So 2i is located and the

Imaginary

complex

use

numbers

location

on

they

on

other

the ordered

Combining

we

get

a

the

complex number

number 4.

organize —

only complex number has

one

This coordinate

and picture the

(4,0)

system

complex

was

chapter

The Magic

plane

number

imaginary axis

1

*f.

an

+

21^

(-31/2,2 i)

<

-?2i

.—! i

i i*i

-5 -31/2

complex

number

equals

the

ingenious way to

numbers. The questions

ofNumbers has

0

¦"-.

¦f—l

So

4+Oi which

Are there any numbers which do not appear

bet! The

-1

-4

-31/2

are

is matched with

means

r-r-r-^

complex

In the

complex

real axis

-31/2

picture

that location.

point (-4, 3)

-4 +31.

the real

way to

origins. Every point

pair for the

Every point on

this line.

what Is called the

form Is associated with

no

—

at their

zero.

imaginary number

an

shown.

as

number lines,

The real

real number and vice

number plane. The real and Imaginary lines

perpendicular

during

all real numbers and

their respective distances and sizes relative to

-31/2

real and

complex.1

picturing mathematical objects.

number line shows the the

versa—so

are

introduced

With their Introduction, all numbers Invented

thus far could be classified

Graphs

and b

a

were

on

some

this

now

is

plane? You

examples.

Any real number can be thought of as a complex number whose imaginary part is 0, and any imaginary number is a complex number with real part 0.

r -1

r-i

plane^

MAGICAL MATHEMATICAL WORLDS

43

THE WORLDS Ol

DIMENSION! • Let's look at the worlds which

created

are

by the Idea of dimensions. A

mathematical world

single point,

on a

exist

on a

single line,

on a

can

In space. In

plane.

(tesseract). Each

a

hypercube

higher dimension

encompasses those beneath It, each lower dimension In Itself.

your life

can

be

a

yet

world

Imagine your world and

on a

flat plane. You cannot

look up or down. Three dimensional creatures

without you

can

even

Invade your world

knowing by simply

entering your domain from above

or

below. Mathematicians, writers, and artists have used various Ideas to

to

try capture the essence of different dimensions In their works.

Dimensions

beyond the third have

alway been Intriguing. one

The cube

of the first 3-D

objects

was

to be

introduced Into the fourth dimension

hypercube. a

The

by becoming a

stages for arriving at

hypercube are illustrated.

Computer programs devised to derive fourth dimension

have

even

glimpses

been

of the

by picturing 3-D

perspectives of the various facets of the hypercube.

44

THE MAGIC OF MATHEMATICS

THE WORLDS OF

INFINITIES

To

see

the world in

a

grain of sand, And

a

heaven in

a

wildflower; Hold

And

has stimulated

Infinity an

Idea drawn upon

infinity eternity in

an

palm

of your hand,

hour. —William Blake

imaginations for thousands of years.

It is

by theologians, poets, artists, philosophers,

writers, scientists, mathematicians and

in the

—

an

idea that has

perplexed

idea that remains illusive.

Infinity has taken on different Identities In different fields of thought. In early times, the idea of Infinity was, rightly or wrongly, linked to large Intrigued

numbers.

by gazing Ancient

—

an

People of antiquity experienced at stars and

philosophers

planets and

or

at

argued

Aristotle

argued

the ideas that

feeling of the

grains of sand

mathematicians

Anaxagoras, Democrltus, Aristotle, and

a

on a

beach.

as

Zeno,

such

Archimedes

infinite

pondered, posed

infinity presented.

proposed the Ideas of potential and only potential infinity existed. 1

actual infinities. He

that

In The Sand Reckoner Archimedes

number of

determining

grains of sand a

method for

on

a

dispelled

beach

calculating

are

the idea that the

infinite

by actually

the number

on

all the

beaches of the earth.

Infinity has been the culprit In many paradoxes. of Achilles and the tortoise and the readers for centuries.

Galileo's

Zeno's

paradoxes

perplexed Dichotomy2 paradoxes3 dealing with have

segments, points, and infinite sets should also be noted.

MAGICAL MATHEMATICAL WORLDS

This field of sunflowers in the Spanish

countryside gives

the illusion

The list of mathematicians with their discoveries and

of infinity.

uses

or

misuses of

300

Infinity extends through the centuries. Euclid (circa B.C.) showed that the prime numbers were Infinite by showing

there made

was no

by

last prime.

Headway in

the realm of the infinite

(1646-1716), and J.W.R. Dedeklnd (1831-1916). phenomenal work of

Georg Cantor (1845-1918)

major breakthrough. Building, creating found set.

a new

a

transflnlte numbers

Using

-

assigned

these topics

are

a

theory

equivalent

sets and same

by developing

cross

the realm of

countabillty,

number of

transflnlte number. His work and

ingenious.

was a

refining Ideas, Cantor by use of the notion of a

numbers that dared to

the idea of

them

But the set

way to compare Infinite sets

determined which Infinite sets had the and

on

and

way to organize mathematics

He determined

the finite.

was

Bernhard Bolzano (1781-1848), Gottfried W. Leibniz

he

objects

proofs

on

45

46

THE MACIC OF MATHEMATICS

In addition to

teasing

our

mathematical tool. It has

minds, Infinity

played

a

mathematical discoveries. We find it used In: volumes

for

and

calculus

—

e

and

—

Indispensable

infinite sets

—

areas

and

calculating approximations numbers—trigonometryself-perpetuating geometric

and more. dynamic symmetry Other parts of this book explore various notions of Infinity, such Ideas

objects

—

limits

—

other irrational

half-lives

an

determining the

both in geometry and calculus

7i,

Is

crucial role In many

infinite

as

search for ad

—

series

—

—

generating fractals, the chaos theory, the continual a

larger prime number, transfinite

numbers and others

Infinitum.

^The counting

numbers are potentially infinite, since one can be added get the next, but the entire set cannot be actually

to any number to

attained.

Dichotomy Paradox Zeno argues that a traveler walking to a destination will never reach the destination because the traveler must first walk half the distance. Reaching this halfway point, the traveler then has to walk half the remaining distance. Then half of the Since there will always be half of the part that part that remains. remains to walk and an infinite number of halfway points to pass, the traveler will never reach the destination. In The

specific

3 In Galileo's 1634 work, Dialogues Concerning Two New Sciences, he discusses infinity in relation to the positive integers and the squares of the positive integers. He even deals with one-to-one correspondence between these two infinite sets. But he reaches the conclusion that the concepts of equality, greater than, and less than were only applicable to Galileo believed the principle that the whole is always finite sets. greater than its parts had to apply to both finite and infinite sets. Three hundred years later, Cantor showed this principle did not hold for infinite sets and used the idea of one-to-one correspondence to revise the traditional notions of equality, greater than, and less than when dealing with infinite sets. Cantor's modifications did away with many paradoxes involving infinite sets and the whole is always greater than its parts.

MAGICAL MATHEMATICAL WORLDS

47

FRACTAL

WORLDS

/ coined fractal from the Latin

adjective fracfus. The

corresponding Latin verb fragere means 'to break': to create irregular fragments. how appropriate for our needs! that, in addition to 'fragmented' (as in fraction or refraction) fractus should also mean 'irregular1, both meanings preserved in fragment. —Benoit Mandelbrot ...

—

Fractals

are

shapes.

Ernesto Cesaro

magnificent objects

this about the strikes

me

which

come

In

Infinitely

geometric fractal, the Koch snowflake

above all about the

whole. To try to imagine it

as

curve

triangle

whole

an

reduced version

curve—

What

is that any part is similar to the

completely as possible,

realized that each small

shape reduced by

many

(Italian mathematician 1859-1906) wrote

it must be

in the construction contains the

appropriate factor. And this contains

of each small

triangle

a

which in turn contains the

II II

II II

ii ii

ii ii

llll llll

iiii mi

mi mi

mi nn

In 1883, Cantor constructed this fractal called the Cantor set Starting with the segment of length the unit interval on the number line, Cantor removed

the middle one third and got stage 1. Then to each remaining 3rds, removed the middle one-third, thereby creating the 2nd stage. Repeating the process ad infinitum, the infinite set of points that remains is called the Cantor set Here

are

the first stages

of the

Cantor set

48

TH E MAC IC O F MATH EMATICS

whole

reduced

shape

further

and

Infinity...

It

so

is

even

to

this

self-

its

part,

all

in

similarity

on

the

curve

wondrous.

reality

a

€

however small, that makes so

seems

If it appeared

tr

in

it would not be

possible to

destroy

it without

removing it altogether, for otherwise it would

ceaselessly the

rise up

depths of

like the

itself.

its

life of the

universe

This is the

essence

regenerate itself.

by repeatedly applying

triangles

that portion retains the

have

The first three stages of the Peano Curve. The Peano curue voas mode in. the 1890's,

again from

generation to of fractals.

essence

So what Is

a

a

If

successive

segment. a

portion

of it remains,

of the fractal—which in turn

fractal?

Perhaps

purposely avoided giving a definition

can

mathematicians

to not restrict

or

inhibit

The first four stages of the Koch snowjlake. The Koch snowjlake is generated by starting with an equilateral triangle. Divide each side into thirds, delete the middle third, and construct a point off that length out from the deleted side.

the

creativity of fractal

this very as

new

creations and ideas that

field of mathematics. With this

fractional dimensions, iteration

theory,

are

new

formulating

In

field, Ideas such

turbulence

appllca-

MAGICAL MATHEMATICAL WORLDS

self-similarity have evolved. Applications for tions,

?A A^\

fractals

astronomy to medicine, from

cartography

to economics, and

on

and

on.

Mathematically speaking, fractal is

as

triangle—

Divide it into Jour congruent triangles as shown and remove the middle one. Repeat this process to the smaller trianglesformed ad infinitum. The resultingfractal has infinite perimeter and zero areal

object

an

segment,

a

being

a

that is

altered

—such

point,

rule

can

by reapplying

be described

The previous

words.

a

constantly

rule ad infinitum.

a

a

form which

a

begins with

four stages of the Slerplnksi triangle. Begin with an equilateral triangle.

The first

by

to

cinematography

A^J^ /K?^

or

acid

rain to zeolites, from

^K

mathematical formula

from

range

The

by

a

diagrams

illustrate four of the earliest fractals made. One

can

think of

a

fractal

as an ever

fractal, you must really view it

developing. Today

we are

in

growing motion.

curve.

It

is

fortunate to have computers

generating fractals before

our

Benoit Mandelbrot, in the

same

eyes. It

spirit

was

of the

equally

To view

a

constantly capable of

fortunate that

early mathematicians,

almost

expanded the ideas and applications of fractals singlehandedly from 1951-75. In fact, he coined the word

fractal

How astonished the adventurous mathematicians * of the

studied and

century, who first dared to look at these ideas most monstrous2 and psychopathic, would be to see the

19th

considered wondrous

geometry of fractals

When

view

we

seeing of its

it for

an

one

growth.

links fractals

illustration

in motion.

or

moment in time

In

essence

photograph

—

it is frozen at

it is this idea of

dramatically

of

a a

growth

fractal,

we are

particular stage or

change

that

to nature. For what is there in nature

49

50

THE MAGIC OF MATHEMATICS

that Is not

Fractals

changing? Even a rock is changing on a molecular level. be designed to simulate almost any shape you can

can

imagine. Fractals

series of rules and own

fractal. Pick a

necessarily confined to one rule, but a stipulations can be the rule. Try creating your

are

not

simple object and design a rule to apply to It.

The firstjive stages

of a computer generated geometric fractal

1 Mathematicians Georg Cantor, Helge von Koch, Karl Welerstrass, Dubois Reymond, Gulseppe Peano, Waclaw Slerplnskl, Felix Haussdorff, A.S. Besicovitch (Haussdorff and Besicovitch worked on fractional dimensions), Gaston Julia Pierre Fatou (Julia and Fatou worked on Iteration theory), Lewis Richardson (worked on turbulence and self-similarity) spanning the years from 1860's to early 20th century—explored Ideas dealing with the "monsters". ,

—

2These

"monsters" were neither accepted or considered worth exploring conservative mathematicians of the time. It was felt that fractals contradicted accepted mathematics because some were continuous functions that were not dlfferentlable, some had finite areas and infinite

by

perimeters, and

some

could

completely

fill space.

MAC ICAL MATHEMATICAL WORLDS

THE PARABLE OF THE FRACTAL

'Wake-up Fractal!

You must

get

to

work," the voice prodded the

sleeping Fractal. "Not again and

so

early," pleaded Fractal "IJust got my dimensions

in order."

"Wake-up Fractal! Come downfrom that cloud sleeping Fractal.

'You're needed at the to be

you made," the voice prodded the

Geological Survey— another coastline needs

described," the voice continued.

"When will I get a break?' questioned Fractal 'YouVe had it easy for centuries voice

replied.

—

now

it's time to

get to work," the

51

52

TH E MAC \C O F MATH EMATICS

'Work, u>ork, work. Why don't they call on Square, Circle, Polygon, or

any other Euclideanfigure?

"First you

Whyme7' asked Fractal

and that they called you a understanding you, you want to finally they're

complained at being ignored,

monster. Now that

retire. Just be thankful you

"Popular is one thing,

but

are so

popular," the voice rebutted.

they won't let me rest.

Its

never

been the

since that Mandelbrot christened me and gave me my

same

replied Fractal.

"Mathematicians

were

debut,"

tediously struggling with

Tm sure my fractional dimensions threw them offfor a whUe.

me.

Those poor souls from the 19th century had

no

computers to help

them. Most mathematicians would not accept me, for I did

follow

their mathematical rules. But

stubborn. Now here I am, areas—

some

being designed and used

computers certainly

were a

notfit or

mathematicians in

so

were

many

boon. One moment the screen

displays afragment or beginning part of a fractal and the next moment the

screen

is

being filled with its generations—

ever

growing. They are now using me in almost everything— describe roots,

vegetables,

must say its is very

exciting to stretch my limits.

coastlines because it still

enclosed a region whose Fm

Icon

trees, popcorn, clouds, scenery... I I love to do

baffles many people to learn lean area

Is finite while my perimeter Is

serving for modeling many of the world's phenomena.

infinite.

For

example—population with Peano curves, fractal curvesfor creating scene

in movies, fractalsfor describing

economics,

ecology,

involved that since

they

astronomy, meteorology,

et cetera, et cetera, et cetera. Tm so

busy and

things are beginning to get a bit chaotic, especially

mixed me Into the chaos

theory," Fractal said sounding

very tired.

The voice started you

some

again. "Stop complaining! Chaos theory offers

variety. Without it you'd Just be continually repeating

the same rule and generating the

same

old shape

over

and over, at

MAC ICAL MATH EMATICAL WO RLDS

The

beginning stages of a fractal cloud.

least when

different

some

can

"I suppose

slightly varied, something totally

evolve."

you're right," Fractal sighed.

"Of course I'm same

initial input is

right.

Just think how boring it would be to be the

shape forever, like a poor square or a circle," the voice

asserted.

"Well, at least there are no surprisesfor a square or circle." Fractal countered. "That's

precisely it.

Life is full of surprises; that's

why they are

calling on you so often. You are more like life." The voice seemed complementing Fractal. 'You

mean

Fm human?' Fractal asked.

"I wouldn't go that far. And besides all life isn't human. Let's say

you're just different, and you're non-Euclidean!" And with comment the voice drifted off.

that

53

54

TH E MAC \C O F MATH EMATICS

FINDING THE AREA OF A SNOWFLAKE CURVE This beautiful

geometric fractal was created

Koch. To generate

Koch snowflake

a

curve,

equilateral triangle. Divide each side middle third, and construct deleted side.

a

in 1904

by Helge von

begin with

an

into thirds. Delete the

point of that length out from the

Repeat the process for each resulting point ad

infinitum. A

Two

fascinating properties, which

contradictory, •the the

area

are—

of the snowflake that

original triangle

curve

an

snowflake

informal curve

I. Assume the

is

area

Now concentrate

8/5 of the area of

curve

is infinite,

of equilateral AABC is k.

a, as on

congruent equilateral

shown. Thus k=9a.

determining the limit of the area

of one of the 6 initial points of the snowflake

triangles.

from it have

triangle

area

g

8/5 of its generating triangle.

We know the area of the

of the nine

j

proof that the area of the

II. Divide AABC into nine

triangles of area,

is

generates it; »the

perimeter of the snowflake Here is

seem

large point

curve.

is a, since its

one

*.,"

The next set of points

(a) (1 /9) each, Just like

had been divided into 9

generated the original

congruent triangles

it also is. In

fact, each successive point is broken down into nine congruent

triangles with two triangles springing from it. STEP III shows the summation of the various STEP IV: Now,

points plus

triangle,

get expression IV.

STEP IV is

of this

point.

by adding up the areas created by each of the 6 hexagon in the interior of the original generating

the

we

areas

changed to STEP V.

The

resulting series

in the

MAC ICAL MATHEMATICAL WORLDS

III.

,9*9

^•9»9, Notice there

Notice there are

8

of this

are

IV.

a +

2

+2

-

9) 2 H +

-

+

—Z-

+

brackets is

a

93

geometric

so we can

STEP VI.

9.9.9.9 n-2

-r-

+

6a

.

9l9

94

series with ratio

calculate its limit.

4/9 and 2/9

as

=

of the

6a

its initial

we

get

72a/5

Now we need to express the area

+

(2/9)/(l-(4/9))=2/5.

Substituting the 2/5 for the limit of the series,

[l+2/5]6a+6a

k, the

9.9.9

!•¥

5—+

92

of this

42 + 21

14 + 2

2*4Z

2*4

9

term,

19*9

32

stage -points.

stage points.

area

of the snowflake

original generating triangle.

a=k/9. Substituting this for

a

in

72a/S,

(8/5)k.

we

curve

in terms of

Since k=9a,

get get (72/S)(k/9) we

=

THE MAGIC OF MATHEMATICS

56

MONSTER CURVES

The stages of the Sierpinskt triangle. Suppose the area of the initial generating is 1

equilateral triangle

square unit. The sum of the areas of the black and white indicated

triangles are through the first

five generations. Suppose the black triangle represents removal of area. Notice how the value for the white triangles is

continually decreasing, meaning the white

area

is

approaching zero. Thus the area for the Sierpinskt triangle approaches 0, UJhlle its perimeter

approaches infinity. 64

64

XS6*

*56

Until Benolt Mandelbrot coined the term "fractal" In the late referred to

1970's, these

curves were

conservative

mathematicians

pathological.

They

neither

as

monsters.

19th

century

considered these monster

accepted

or

curves

considered them worth

exploring because they contradicted accepted mathematical ideas. For

example,

any

gaps)

some were

that

were

not

infinite perimeters, and

Sierpinski triangle perimeter and out how the

Earned

a

J

continuous functions

dlfferentlable, some

(also called

finite

area.

some

could

(functions without

had finite

areas

fill space.

completely Sierpinskt gasket)

has

an

and The

Infinite

The illustration above tries to point

Sierpinskt triangle's area is zero.

after mathematician Waclaw

Sierpinski (1882-1920).

MAGICALMATHEMATICAL WORLDS

MANDELBROT SET CONTROVERSY In the 17th

century

number of prominent

a

mathematicians (Galileo, Pascal, Torricelli, Descartes, Fermat, Wren, Wallis, Johann Bernoulli, Leibniz, intent

cycloid. Even at this

as

period

there

were

many discoveries

of time, there

arguments about

were

the properties of the

discovering

on

Huygens,

Newton)

were

also many

who had discovered what

first, accusations of plagiarism, and minimization of one another's work. As

the

cycloid

result,

apple of of geometry. 20th century mathematicians

discord and the Helen to have

now seem

a

has been labeled the

a new

Helen of geometry— the Mandelbrot set.

Who first discovered the Mandelbrot

question among present day

set1?

This is

mathematicians.

a

very heated

The contenders

are:

—

Benoit Mandelbrot is

work

on

fractals

the Mandelbrot set

as a

pioneer because of his initial

Mandelbrots work

showing

variants

of

December 26, 1980 in Annals of the of Sciences. His work on the actual Mandelbrot set

was

New York was

often described

in the 1970s.

published

Academy published in 1982.

Hubbard of Cornell University and Adrien Douady of the University of Paris named the set Mandelbrot in the 1980s while working on proofs of various aspects of the set In 1979, Hubbard says

—John H

he met with Mandelbrot, and showed Mandelbrot how to program a to plot iterative functions. Hubbard admits that Mandelbrot

computer later —

developed

Robert

Brooks

a

superior methodfor generating the images of the and

J.

Peter

Matelski

claim

they

discovered and described the set prior to Mandelbrot work was not published until 1981.

set

independently although their

—Pierre Fatou described Julia sets' unusual Gaston Julia's work

on

Julia sets

properties around 1906, and predates Fatou's. (Julia sets acted as

springboards for Mandelbrot sets.) Perhaps all.

Who gets the credit?

iThe illustration above is the most familiar fractal form from the Mandelbrot set. The Mandelbrot set is a treasure trove of fractals, which contains an infinite number of fractals. The set is generated by an iterative equation, e.g. z^+c, where z and c are complex numbers and c produces values than are confined to a certain boundary.

57

58

THE MAGIC OF MATHEMATICS

MATHEMATICAL WORLDS IN

There is

astonishing imagination even in

LITERATURE

the science of mathematics.

Is the tesseract the

only

an

figment of a mathematical imagination?

Is the

"real" dimension the 3rd dimension? We learn in Euclidean that

geometry

since it has

point only shows location, and dimension. Yet

of these invisible

composed such

a

zero

a

in

asymptotic lines of

geometry;

numbers.

transfinite

complex these

number

can

concepts

their

in are

our

world.

Many writers,

point thick.

our

of

pseudosphere

the

Madeleine

the

circle. One wonders

there

is no

mathematical

if

doubt of their

these

systems,

world.

artists and mathematicians have

as

a

hyperbolic

imaginary numbers,

ingeniously

these concepts to describe worlds where these ideas Such writers

What is

functions, infinities of

even a

Although

respective

only models in

one

exponential

Consider

plane, fractals, and

exist in

existence

only

world? Consider the

our

seen

segment

length, yet does What about a plane?

exist in the realm of our lives?

figure

it cannot be

line

A line is infinite in

points.

Infinite in two dimensions and

plane

we can see a

come

used

to life.

Dante, Italo Calvino, Jorge Luis Borges, and

L'Engle have

drawn

on

mathematics to enhance their

creations.

In the

19th

model of

a

century,

hyperbolic

Here to all

things

mathematician Henri Poincare created world contained in the interior of

and inhabitants,

never

was

everything would shrink

it moved away from the center of the circle, while

approached

a

circle.

their circular world

infinite. Unbeknownst to these creatures, as

a

the center. This meant the circle's

to be reached, and hence their world

growing

boundary

as

it

was

appeared infinite

to

MAGICAL MATHEMATICAL WORLDS

Circle Limit IV (Heaven Si Hell) by M.C. Escher depicts a world reminscent of Henri. Poincare's hyperbolic wodd. © 1994 M.C. Escher/Cordon Art-Baam-HoEand. AR rights reserved.

them. In 1958, artist M.C. Escher created

a

series of woodcuts,

entitled Circle Limit I, II, III, IV which convey Poincare had described. Escher described

this

infinite

For her

a

a

world

feeling as

"the beauty

of

world-in-an-enclosed plane." 1

novel, A Wrinkle in Time, Madeleine L'Engle

ract and

of what

multiple

dimensions

as means

uses

the tesse-

of allowing her characters

"...for the 5th dimension you'd through square the fourth and add that to the otherfour dimensions and you to travel

can

outer space,

travel through space without having to go the long way

59

60

THE MAGIC OF MATHEMATICS

around...In other words

a

straight line

is not the shortest distance

between two points." Italo Cavino describes short

believe such we were

been? what

a

world that exists in

story AH At One Point.

"packed

did

we

have for time,

exists.

even

space to

pack

we

Looking back to the find Euclidean The cone's

a

us

of

one

"Naturally, we

have

Or time either:

packed in there like sardines? I say

like sardines," using

which was where

hell.

actually

In his

makes

Qfwfg said,—where else could

coincided with every point

we

single point

ingenious creativity

dimensional world

all there, —Old

a

Nobody knew then that there could be space.

use

wasn't

a zero

His

literary image:

into.

each

of

in

reality

there

Every point of each of the others in

a

us

single point,

all were."

Middle

Ages

and Dante's The Divine

geometric objects

shape

was

were

used to hold

Comedy,

the bases for Dante's

people

in

stages of hell.

Within it, Dante had nine circular cross-sections that acted

as

platforms which grouped people by sins committed.

From Dante's The Divine Comedy. The plan of concentric spheres, which shows the Etarth in the sphere (bearing the epicycle) of the Moon, and these are also enclosed in the sphere (bearing the epicycle) of Mercury.

MAGICAL MATHEMATICAL WORLDS

In the 1900's

of Sand.

infinity was

featured in

Jorge

Luis

Here the main character acquires

"The number None is the

of pages

first

numbered in this

in this book is none

page,

arbitrary

Borges'

The Book

"marvelous" book.

no more or

less than infinite.

I don't know

the last.

way.

a

why they're

Perhaps to suggest the terms of an

infinite series admit any number."This book adversely changes his life and his outlook to

dispose

infinite

on

book

might

until he realizes he must find

things,

of it— "J thought

a

way

offire, but Ifeared that the burning of an

likewise prove

infinite

and

with smoke." What would your solution be?

suffocate You

the planet

might

want to

read the book to find how the hero resolved his dilemma. Science fiction writers have utilized

mathematical Ideas to

example,

Next Generation, the

being pulled by an "invisible" force Only when the ship's schematic monitor

toward

a

starship

black hole.

changes perspective

does the

in

an

episode of

help

create their worlds. For

Star Trek —The

is

crew

realize the unknown force is

a

2-dimensional world of minute life forms. Mathematics Is full of Ideas that make one's and wonder— Are

Mathematicians reside

—

they real?

are

perhaps

imagination churn

To mathematicians

they

are

real.

familiar with the worlds in which these Ideas

not within

our

realm, but real in their

nonetheless!

M.C.Escher, Harry N. Abrams.Inc, New York, 1983.

own

61

MATHEMATICS &ART ART,THE4TH DIMENSIONS. NON-PERIODIC TILING

MATHEMATICS & SCULPTURE MATHEMATICAL DESIGNS & ART MATHEMATICS & THE ART OF M.C.ESCHER

PROJECTIVE GEOMETRY & ART MIXING MATHEMATICS & ART OF ABRECHT PURER COMPUTER ART

64

THE MAGIC OF MATHEMATICS

The most beautiful It is the

mysterious.

thing

source

we can

experience is the

of all true art and science. —Albert Einstein

Linking mathematics and art may seem alien to many people. mathematical

worlds

of

computers have and

simplify,

provided perfect their work.

dimensions,

algebra,

geometries,

tools for artists to

But

explore, enhance,

Over the centuries, artists and

their works have been Influenced

mathematics. The used the

golden

Albrecht Dfirer

by the knowledge and use of ancient Greek sculptor, Phidias, Is said to have

mean In

the proportions of many of his works.

employed concepts from projective geometry

achieve perspective, and

geometric

fj^JN^A WH*il»'rHHfi «{»$*«$*

constructions

played

a

to

vital

role In his

fc3h

typography of Roman letters.

Since

religious doctrine

¦&

prohibited the use

A sketch

from one of Leonardo da VincCs notebooks Illustrating lines converging to a vanishing point

Moslem art, Moslem artists had to avenue

for their artistic

wealth of tessellation

inquiry

can

rely

on

as

an

to create

a

Leonardo da Vinci felt "...no human

be called science unless it pursues its

mathematical exposition and demonstration." sculptures and

objects In

mathematics

expression, thus leading them

designs.

of animate

path through

Leonardo's

paintings illustrate his study of the golden rectangle,

proportions, and projective geometry, while his architectural

designs

show his work in geometric structures and

symmetry.

illustrating

The

topics

in

this

section

are

a

knowledge

few

of

examples

the connection between mathematics and art.

MATH EMATICS & ART

65

ART,THE4TH Mathematics takes

us

DIMENSIONS, NON-PERIODIC TILING

into the

region of absolute necessity, to which not only the actual world, but every possible world,

must conform. —Albert Einstein

On

a

the artist Is restricted to

canvas,

communicate the

two-dimensions to

feeling of other dimensions.

Icon artists of the

Byzantine period depicted three-dimensional religious only two-dimensions, giving the subject matter appearance.

a

scenes

in

mystical

During the Renaissance, artists using the concepts of

projective geometry transformed their flat canvas into the threedimensional world they wanted to convey. Today, mathematics

plays

an

active role In

mathematical

hypercube1,

the

adapted

dimensional

Salvador unfolded

Artists

higher dimensions. by artists to take

example, has been used

Into the fourth-dimension.

Bragdon

artist's Ideas.

an

Ideas to escape into

for

and tools for the

providing Inspiration

creation and communication of

hypercube

early

hypercube

in his

designs Dali3 delved

In the

use

The

step

a

1900's architect Claude

along

with

other

four-

work2. Intrigued by the hypercube,

Into mathematics

which is the focal

for his model of

point

in his

painting

an

The

Crucifixion4 (1954). Today, with

there

are a

number of artists pursuing art In connection

mathematical

Ideas

in

—

particular,

mathematics

non-periodic tiling, multi-dimensions and computer renditions. fact,

computer

renditions

of

the

hypercube,

created

of In

by

mathematician Thomas Brancroft and computer scientist Charles Strauss of Brown

University, produce visualization of the

hypercube moving in and out of

hypercube

a

in the 3-D world are

3-D space. Various

thereby captured

on

images of the the

computer

66

THE MAGIC OF MATHEMATICS

>¦*'

The unfolded hypercube was the Inspiration for Salvador Dall's The Crucifixion (1954). Metropolitan Museum of Art, Gift of the Chester Dale Collection, 1955. (55.5)

MATH EMATICS & ART

monitor. Introduced to this

part of mathematics,

bin has created 3-D

representations of the hypercube with

the

canvas

acting

which

have aided Robbln in

One moment

intertwined in

to the

pentagonal

non-periodic tiling unusual

an

Tony Rob-

creating fascinating structures,

views a series of

one

interlaced

position

combination of

artist

plane intersecting the hypercube. Nontiles, quasicrystal geometry and fivefold

change dramatically according

viewer. next

as a

Penrose

periodic tilings,

symmetry5,

canvas

perspective of the

triangles

stars

while In the The

appear.

of both 2 and 3-D forms

type of symmetry

create

almost

an

contradictory image.

* Also known 2

as

the tesseract— a 4-dimensional representation of a cube.

same time Bragdon used magic lines graphic designs of books and textiles.

At the

and

3 Dali contacted the mathematics

in architectural ornaments

department

at Brown

University for

further information. *

Jesus

Christ

5Non-periodic tiling designs which have n-Jold symmetry

nailed

is

fourth-dimensional

:

to

a

cross

represented by

the

unfolded

hypercube. tessellating

is no

If

with tiles

or

shapes

which

create

pattern.

a

pattern is preserved when rotated 360'/n, it is Therefore, a pattern has fivefold symmetry pattern.

said to have n-fold symmetry. if a rotation of 72' retains the

Quasicrystals are a newly discovered state of solid matter. Solid matter thought to exist only in two states, amorphorous or crystalline. In amorphorous the atoms (or molecules) are arranged randomly, while in crystalline the arrangement is the periodic repetition of a unit cell building block. The discovery of quasicrystals revealed a new state in which the arrangement of units is non-periodic and with an unusual symmetry, e.g. fivefold, not present in amorphorous or crystalline matter. was

67

68

THE MAGIC OF MATHEMATICS

MATHEMATICS & SCULPTURE Dimensions,

gravity,

geometric

symmetry, and

objects,

are

sets

complementary

all mathematical ideas

which when

role in

into

come

sculptor

a

Space plays a

play

creates.

prominent sculptor's works.

Some works

a

simply

occupy space in the we

of

center

space,

and other

same

way

living things

do. In these works the center of

gravity1

within

These

the are

the

to

For

Greek The Discobolus (circa 450 B.C.)

bronze, captures

a

a

example the

by artist,

the ancient

Myron,

or

Bufano's

St.

Horseback

all

Beniamino

by Myron,

in

we are

or

Michelangelo's David, Discobolus

are

ground

with which

comfortable

in

that

occupy space

manner

accustomed.

point

a

sculpture.

objects

anchored and

is

cast

moment in motion.

Francis

on

have their center of gravity within the

play with These

mass

use

as

uchi, the

Some modern art

sculptures

space and its three dimensions in unconventional ways. space

center of gravity mass,

of their sculpture.

as an can

integral part of the work. Consequently the

be

a

point of space rather than

such works

illustrated

by

Eclpse by

Charles

Perry,

as

Red Cube

a

by

point of the Isamu

Nog-

and the Vaillancourt Fountain

MATH EMATICS & ART

by

69

Louis Vaillancourt.

Other

depend

sculptures their

on

interaction with space. Here

the space around the artwork

(the

complementary

set of

points

of the mass) is

as,

or

equally, important the sculpture.

as

Consider

Zinc Zinc Plain

by

Carl Andre. This sculpture

is

room

works

other

The

in

staged

a

devoid of any

objects.

or

is created

plane

small

36

forming

by

squares

a

square

which lies flat

on

the San Francisco's controversial Vaillancourt Fountain

_-

„

tloor.

ine room repre-

has

as

its center of gravity

a

point of space.

sents space, the set of

all

points, and he describes his work

works

seem

to

defy gravity.

as

"a cut of

These include such

space".2

Some

sculptures

as

the

mobiles of Alexander Calder with their exquisite balance and

symmetry

and

mysteriously

on

Isamu's

Noguchi's

its vertex. There

the Earth itself as

an

integral part

are even

mysterious geometric

grass theorems

possible.

sculptures

balancing which

Crista, Carl Andre's Secant, and

by

physical nature

mathematical

Cube

use

of the art and its statement, e.g.

The Running Fence

Often the

Red

appearing

in

the

England.

of an artist's conceptional work requires

understanding and knowledge to make the work mathematically analyzed most of his

Leonardo da Vinci

creations before

mathematically

undertaking

a

work.

If M.C. Escher had not

dissected the ideas of tessellation and

optical

70

THE MAGIC OF MATHEMATICS

Illusions, his works would not have evolved with the which he

was

able to undertake them

once

ease

with

he understood the

mathematics of these ideas.

Today there are many examples of sculptors looking at mathematical ideas to

expand their art. Tony Robbin

uses

the

study of

quasicrystal geometry, 4th dimensional geometry, and computer science to develop and

expand his art. In his giant sculpture

Easter egg

Ronald Dale Resch had to

use

Intuition, ingenuity,

mathematics,and the computer as

This sketch by Leonardo shows his analysis of the horse's anatomy.

well

as

his hands to

complete his creation. And artist-

mathematician Helaman R.P.

Ferguson uses

traditional sculpting, the

computer and mathematical equations to create such works

as

Wide Sphere and Klein Bottle with Cross-cap & Vector. is not

Consequently it

surprising

to find

mathematical models

doubling as artistic models. Author In front of Continuum by Charles Perry. National Air & Space Museum,

Washington

D.C.

Among these we

find the cube, the

polycube,

the

sphere,

the

MATH EMATICS & ART

torus,

the

trefoil

hemisphere,

knot,

knots,

M6bius

squares,

Mathematical

prisms

the

objects

strip, polyhedrons, the

circles,

triangles,

from Euclidean

pyramids,

geometry and

topology have

played important roles in the

sculptures

of such

artists

Isamu

as

Noguchi,

David

Smith, Henry Moore, Sol LeWitt.

Regardless of the

sculpture, mathematics is inherent in it. It may

have been

conceived and created without

a

mathematical

thought, nevertheless An Alexander Calder mobile. East

National

Building Gallery of Art, Washington, D.C.

of the

mathematics

.

exists in that

work, just as it exists in natural creations.

^The

center

of gravity Is the point

on which an object can be balanced. the center of gravity or centrold of a triangle can be determined by drawing that triangle's medians. The point where the three medians Intersect happens to be the center of gravity.

For

example,

2Art & Physics,

Leonard Shlaln, William Morrow & Co. NY, 1981.

71

72

TH E MAC IC O F MATH EMATICS

PUTTING MATH EMATICS INTO STONE Trefoil knots

spheres

—

—

—

torus

vectors

flow— movement

—

—

these

are some

of

the mathematical ideas inherent In the

sculptures of Helaman Ferguson. We have often heard

of artists

using

mathematical ideas to enhance their

work. Mathematician- artist

Helaman R.P. Ferguson

conveys the

beauty of mathematics in his Eine Kleine Rock Musik HI Photography by Ed Bernik. From Helaman Ferguson: Mathematics in Stone and Bronze by Claire Ferguson. Meridian Creative Group, Copyright © 1994.

phenomenal sculptures. art form and

believe it is feasible to communicate mathematics channels to the To create his

traditional

a

an

science...!

along

aesthetic

general audience."1

exquisite forms, Helaman utilizes methods from

sculpting,

the

His works bear such

Cross-cap

As he states

"Mathematics is both

computer, and mathematical equations. names as

Wild

Sphere;

Klein Bottle With

And Vector Field, Torus, Umbilical Torus With Vector

Field., WhaledreamU (horned sphere). 1 Ivars Peterson.

Equations

in Stone, Science News Vol. 138

September 8,

1990.

MATH EMATICS & ART

LAYING AN ECC MATHEMATICALLY When Ronald Dale Resch mission to

design for

sculpture

a

com

gigantic Easier egg he

Vegreville, Alberta,

discovered he would have

soon

develop

the

accepted

to

the mathematics for the task

virtually from scratch. Over the years Resch has refined

the art of

manipulating

into 3-D forms.

problems

2-D

objects

His work and the

he has solved

point

to

mathematics, yet he has had little formal mathematical training.

Working with sheets of such

as

various materials

aluminum,

or

paper

transforms them into works of art he has

he

by folding

He solves

developed. geometric problems using intuition, ingenuity, mathematics, the computer and his hands. techniques

His initial instincts about the

design

of the egg

were

that he could

ellipsoids for the ends and a bulging cylinder for the center. He quickly realized this would not work. Discovering that available mathematics for the egg was limited1, he realized he make two

would have to go it alone. His Easter egg

524

required 2,208 identical equilateral triangular tiles and

three-pointed

width varied

by varying

resulting design for the magnificent

stars

according

the

angle

tiles, which

were

to their location

on

equilateral

and whose

the egg. He found that

of placement of the tiles

ever so

slightly (from

less than 1" to 7'), the flat tiles gave the Impression of curving and the contour of the egg resulted.

long and 1

The final structure is 25.7 feet

18.3 feet wide, and weighs 5000 pounds.

Algebraic equations for egg-shaped curves were developed by French mathematician Rene Descartes (1596-1650). In 1842, as a youth, Scottish mathematician James Clerk Maxwell (1831-1879) devised a method for constructing an egg using a pencil, string and tacks.

73

74

THE MAGIC OF MATHEMATICS

MATHEMATICS DESIGN* & ART

The

following figures

can

and have

been used to create artistic and

graphic designs.

Their bases

mathematical, but that does take away from their beauty and simple

are

not

elegance.

MATHEMATICAL STARS The

illustrations

show

how

to

by using regular polygons

generate

stars

polygons.

For odd sided

simply join every other vertex of the polygon, the

and you will arrive back at

starting point.

For

even

pointed

stars, note that the star is made up of two rotated

polygons.

These will

have half the number of sides the

generating polygon.

as

MATH EMATICS & ART

75

MATHEMATICAL EMBROIDERY Math enthusiasts have been

mathematical It Is

curves over

to discover

always fascinating formed

curve

from

"embroidering"

the centuries

series

a

a

of

straight line segments. The "stitch es" (line segments) end up being 0K1AN?iV0*?—14 wuiHinmrntmuai ntaam an mm in

i

w

3988 Take the year you

were

born. To this add the year of an important

event in your life. To this 1994.

Finally,

sum

add to this

add the age you will be at the end of

sum

the number of years ago that the

important event took place. The answer Is always

3988!

THE MAGIC OF NUMBERS

117

PLAYING Every math enthusiast or

at

one

another has discovered

by

tricks

or

been

oddities

delighted involving

numbers. Here

for you to

explore and hopefully enjoy.

or

WITH NUMBER!

time

are

two

SUMMING *£ SQUARING

l+Z+3+4+3+2+l=42 •

••••••••••••

•

••••••••••••

•

•••••••••••a

THE 1'g PYRAMID

x*-x ii2=ui

1112=1Z3X1 IIIIX=IX343ZI

XXXXXX=X2345432X XXXXXX*=X2345*5432X IIIIIIIX=IX345«>7«>543XI IIIIIIIIX=IX345«»78710.4. The triangle's area

tortoise, in the beehive's

honeycomb,

would

come

would be

of

4y/3**6.9.

The

square's

area

is 9.

Since 1987

been

focusing much

attention

on

Magellanic Cloud, supernova

not

1987A

was

the

observed.

first time gas

the

Large

where It Is

bubbles

have been

seen

explosions,

but It Is the first time

following

stellar

the bubbles appear clustered in

honeycomb shape. The six lines

of symmetry of regular hexagon.

honeycomb, which

composed

measures

years. Wang suggests that a

cluster,

composed of similarly sized stars which have been

evolving at about same

the

Wang University of Manchester

England

approximately

rate for

several thou-

a

of in

discovered the

30x90

light

of about 20 bubbles whose diameters

light

the

a

Lifan

years, and Is are

about 10

MATHEMATICAL MAGIC IN NATURE

sand years, creates winds of such

magnitude they shape

the bubbles into the hexagonal

configuration.

Lastly,

look

a

nature's

at

snowflake illustrates

symmetry and fractal

hexagonal

geometry. The snowflake the

possesses

hexagon.

growth

of

shape In

the

addition, the

of a snowflake is

Koch snowflake

simulated

by the

curve.

This fractal is gener-

These

are

ated as

by an equilateral triangle snowflake indicated in the drawings f^fj1 into thirds. °

at the

Consequently,

the

first four stages of with

starting

curve,

an

^T9^: divi*:, e

who

are

able to

translate ancient Chinese writing Is difficult.

Finding experts

able to

translate manuscripts that deal with mathematical Ideas Is more

difficult. This

mathematical themes

technique algebraic

explains why

are scarce.

examples

even

of

Hsuan-thu, the piling of squares,

was a

that Chinese mathematicians used in order to arrive at

conclusions

particular

Chinese

using geometric and arithmetic

illustration is from the Chinese

The date of Chou Pel is

disputed,

with

means.

This

manuscript, Chou Pel.

possible dates ranging from

MATHEMATICAL MAC IC FROM THE PAST

1200 B.C. to 100 A.D.

If 1200 B.C. is accurate, it would be

.

the earliest known demonstrations of the and the

theorem has

in many civilizations

angle.

means

of

Pythagorean theorem,

predating Pythagoras

appeared architecture. It was one

one

The

Pythagoreans.

Pythagorean thoughout history. In

of assuring the formation of a

In mathematics is has been and is an

right

indespensible

tool

crossing many mathematical disciplines. The

theorem states that

Pythagorean

right triangle the sum of the squares of its two legs equals its the square of its hypotenuse (a2+ br=cP).

for

hypotenuse

leg=a

any

(The

converse

is also

true.)

left diagram below, the interior square's area is indicated as 5x5 or 52 right triangles, each of area (l/2)(3x4) and a square qf area lxl, totaling 25 square units. In the right diagram below, the square is divided into two smaller overlapping squares, one 3x3 and the other 4x4. The part they overlap has the same area as the vacant part qf the 5x5 square they do not occupy, which illustrates that the area qf larger square (5*) equals the sum qf the two smaller squares' area, namely 3* and 4*.

In the =

25 square units. It has been subdivided into 4

'

c/

^

o

_

C

Aw*

A

v

V

/\

b

>

\

"«

>

'

)

This diagram explains how to find the area of the interior shaded square by summing the areas of 4 triangles and the unit square in the middle. In general it shows—

c2=4(l/2)ab + (a-b)2 =2ab + (a2-2ab+b)2 =a2+b2

R

/

>

/v

The

sumofthe two shaded rectangles' areas equals the area of the small shaded square (this is the square created by two overlapping squares). Letting 5, 4, and 3 take b, it shows

on

a2

the variable +

b2=c2.

c, a

and

15J

154

THE MAGIC OF MATHEMATICS

ONEOFTHE EARLIEST RANDOM NUMBER GENERATORS

Although to as

a

during credit for

being

in the National

have

played

one

this die

was

not referred

random number generator ancient Greece, it still

gets

of the earliest remaining die. Today it appears

Archeological

many roles

foretell the future, to

over

Museum in Athens, Greece. Dice

the centuries.

implement

moves

They have been used

of various games such

to as

backgammon and

monopoly, as

or,

in craps,

they are the main elements

of the game.

Mathematicians have

long been intrigued with dice from the

viewpoint of

probability. fact, dice

can

be considered

and Pierre de Fermat to focus their attention

gambling,

Pascal was asked

if the game

were to

Fermat about the

new

probability.

While

by a friend how the pot should be split

stopped before it was over. Pascal wrote problem. In 1654 the two men worked out their their correspondences, and thus launched

branch of mathematics.

random number generators the

on

be

theory of probability in this

In

responsible for getting Blaise Pascal

theory of probability.

are

Today

dice and other shaped

used to teach various aspects of

MATHEMATICAL MACIC FROM THE PAST

155

EGYPTIAN The

of

method

Egyptian

and

civilizations. In

spread

ancient Greek schools it

the Middle

specific

Ages

names

Here is

halving.

its

fig

V

/]

\%\\f\f\ 111 fl/VVt Hi flfl^t

as

need

for

plication tables, on

In

and referred to

by

taught

duplatio for doubling

the Rhind papyrus of how

14 i#j

^**

OA

*iAA

give 24, which

1120

was

by

multiply on

how

doubled to give 48, which in turn is

doubled giving 96. Slashes next to the 4 and the

their

the answer, 144.

glving

drawn

Then

amounts

corresponding

are

8, indicating

twelve.

ls

sum

are

their

added,

The

Egyptian multiplication eliminated

of

memorizing

multi-

ftftftft

since it relied

mainly

ftftftft

OA



•¦,

c

r;rils/Qn

5

ft

3Djr

Sen n

siiw MILITRAM mimtran eati 31055 SAIL

nffMSDL

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n-^BECOd BAT LCLITA OIBOUllFLaG CPS = iCES BPl PILOT SPRIMT Pet CELIAC n^ja" = «TUTDR STROBES PL/1 dywiB FL5 GPL IPt'V U =

PAL

Some

E

True OOCUS Jotial GRAF IT TBJ!

computers languages developed

to communicate with

£P"

5 -

DllTlffTE computers.

THE COMPUTER REVOLUTION

191

& i©@K &1T ™^ P&S1T OBSOLETE •The ten digits our

ofour hands

CALCULATORS

were

^

earliest counting device.

•The Chinese devised box to

use

This box

a

II

with their rod numerals.

was

used to write systems

of equations. •The abacus

was

used in calculation

by many cultures including the Chinese, the Greeks, the Romans, and the Japanese. •The Incas used the knotted quipu

as

their accounting device.

•Napiefs rods Napier in the

were

invented

by John

1600s to aid

computation. •The slide rule 1620

was

invented around

by Edmound Gunter.

•Thefirst adding machine was invented

by Blaise Pascal in

And in 1673,

Gothfired

Leibniz invented

multiply •In the

one

1642.

Wilheim

von

that could also

and divide.

early

1800s Charles

Babbage's designs and work on the difference and analytical engine furnished thefoundationfor the modern computer.

1 iiiii

segmented

ill

III in

T JL ii ¥ 1 T

II 1

—

zs

mi

THE MAGIC OF MATHEMATICS

191

A D D 1 T 1 O N.

Matter,

This illustration Is

be eaficft toap in tljis arte ,»3 to acts but ttoo fummes at ones togrttjee: J>ott> be it, pou marc a&oc mc:e,as J toil tci rouanont* ttyctcfoit toxemic pou topllc BDDe ttoo fortunes,pou fl)flll fpitte fct Dottmc

^Y *•-*-

a

re-

productton/rom the early English arithmetic book The Orounde of Artes, In by Robert Recorde. addition to the lines representing Os, 10s, 100s, etc, the places between these lines also were used to represent 5s, 50s, 500s following the Roman numerals. The V on the line was initially used to mark the 1000s but later it was line, used to indicate a comma when writing such numbers as 23,650. When

of tbem^tfoicet¬tolncbc.anDttjcti it Dzato a Ipnc croffe tbc otber Ipjtte.tf no aftcttt>«t>c I'ettcDotincHKorbcr famine, fo one

bp

tljattuat Ipncmape

t»cbcttticnctUcw;aa if pou tooulfce aont t*S9tO 8*4* , POU mutt fct pouc funics

aBpoufecberc. &uo t$cu i( pou Ipft, pou map; aDDr

five

counters

accumulated on a line those beads would be removed and tDc oic to tijc otber m tlje famr place. 02 els one counter would be a :icto carried to the above the place: to&t'cl) \faap,bpraufc if xa molt plpiica space. Hence, possible origin of the term "carry". In addition to writing this book, Robert Recorde (circa 1510-1558) introduced the symbol "=" for equality, wrote an important algebra text book called The Whetstone qfWitte and the geometry book. Pathway to

•poumapafcortl):mborfjcro&itlKcin

Knowledge. 9

3

"*

9 3

9

* 4 3 6/4

1

4

T

1 I9|l!4l»

4/»

|z|5|ojij3

X9 3

l

6

1

%

9517&

These tables are

from

a

book that

*

I z

9

5

r/\o/

was

4 ¦

Treviso,

Italy.

o/|o/ o/ 9 /9I/5 /4 < 1

6

9

5

4

5

V i\

multiplying 934 and 314.

4 6

R9|\5 i

oX|o\ [\7l\9

*

*

0omms-

*

9

S

It

shows four methods for

T7\T7 X X

printed

in 1478 in

/

193

THE COMPUTER REVOLUTION

NAPIER'S CHESS BOARD The

uses

only

the

key

to

with

communicating electronic

CALCULATOR

two),

Os and 1 s to write

held

numbers,

its

(base

binary system

which

computers

since Os

—

-'

and Is could indicate the "ofi"

position of

and "on" The

electricity.

famous

Scottish

mathematician John

Napier

(1550-1617) utilized the advent of is

electricity.

Napier

for

known

best

74

two before the

concepts of base

99

46

rods (or

Napier's based

on

used

were

,

known

bones)

•

His

logarithms.

calculating rods,

• •

• ••

m

revolutionizing computation by his invention of

m

• ••

as

128 64 32

16

8

4

2

1

were

logarithms

and

1"

¦

by merchants to

j

perform multiplication, division, and could be used to

find square roots and cube roots. Less well known is his

chess board method of

calculating. Although he did use

binary

not

notation to write

numbers, the board does illustrate

how

he

expressed

the numbers in base two. For

example,

to add 74

+

99+ 46,

sum

%•

* •m •

128 64 32

16

8

4

2

1

sum's final form 128 64 32

16

8

4

2

1

each number is written out in a row

of the chess board

by

194

TH E MAC IC O F MATH EMATICS

placing sum

markers in the appropriate squares of the

of the markers values (indicated

the number

64+8+2=74. After each number is

on

the bottom

one one

added

are

row.

row so

the bottom

expressed

by gathering

Two markers

on

the chess board,

the markers'

sharing

the

vertically down

same

square

marker to their immediate left. So two "2" markers, "4" marker.

sharing in the

the

same

adjacent

square will have

from

Working square

that the

line) total

74 has markers at 64, 8 and 2 since

they represent.

the numbers

along

are

right

equal

produce

to left, any two markers

removed and

marker

replaced by one

square at the left. At the end of this process, more

than

one

marker. The

the remaining markers represents the

sum

sum

no

of the values of

of the numbers.

m

M

:¦-:

*

¦

.'.

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