4D Maths

 

FINITE AND INFINITE WAYS The Ost Ostberge bergerr Story to ry By Bob Beaumont Edited and illustrated over a five year period  by Bob Beaumont & Steven & Pat Burnand

Converted at a later date to web pages by Josef Karthauser  

Copyright © 1971, 1991, 2002 by The Educational Trust Company This is a dynamic publication. Unlike most books that go through a proce process ss of revision and republication from time to time this book is constantly being refined. This does not compromise the permanent foundation of the content. The terms of copyright are that any person may use or add to the work in any way,  providing that they make two, obvious references, within each separable piece of work, to the source name Ostberger . In such case there will be no charge for the use of this work. “

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If, however, no such reference is made or their is no willingness to do so royalties will be chargeable within the accepted international laws and in that case no person may copy,  place in a data retrieval system, transfer by internet, publish, print or reproduce reproduce in any way any part of this work; nor no r may they use the words Ostberger , extravariant , intravariant or other words unique to the work in any public place for performance or speech or other audio expression such as broadcasting without the written permission of a legally authorized person from the Educational Trust Company. “

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Table of Contents Synopsis.................................................................................................................................................................... Synopsis ....................................................................................................................................................................ii The Ways and the Truths .....................................................................................................................................iii .....................................................................................................................................iii I. The Chapters....................................................................................................................................................... Chapters.......................................................................................................................................................v v

1. Einstein and all that ....................................................................................................................................1 ....................................................................................................................................1 1.1. Introducing the reasons for this book   ............................................................................................ ............................................................................................ 1 1.2. Descartes........................................................................................................................................ Descartes ........................................................................................................................................2 2 1.3. Imaginary Time.............................................................................................................................. Time ..............................................................................................................................4 4 2. The elements of new geometry  geometry  .................................................................................................................. .................................................................................................................. 6 2.1. Rotations ........................................................................................................................................6 ........................................................................................................................................6 2.2. Three Dimensions ..........................................................................................................................6 ..........................................................................................................................6 2.3. Standard Forms ............................................................................................................................ ............................................................................................................................12 12 2.4. More Than Two Dimensions  Dimensions .......................................................................................... ....................................................................................................... ............. 14 3. A story and a theorem  theorem  .............................................................................................................................. .............................................................................................................................. 19 3.1. A Story. ........................................................................................................................................ ........................................................................................................................................19 19 3.2. And now for a theorem ................................................................................................................20 ................................................................................................................ 20 3.3. The first rope trick  ......................................................................................... ....................................................................................................................... .............................. 21 4. Orthogonality  Orthogonality ............................................................................................ ........................................................................................................................................... ...............................................23 23 4.1. The rules of Fleming.................................................................................................................... Fleming ....................................................................................................................23 23 4.2. Deflection of an electron beam .................................................................................................... ....................................................................................................25 25 4.3. Vectors of light............................................................................................................................. light .............................................................................................................................25 25 4.4. The vectors of the gyro ................................................................................................................26 ................................................................................................................ 26 4.5. Other examples ............................................................................................................................ ............................................................................................................................30 30 5. More than geometry ................................................................................................................................. .................................................................................................................................31 31 5.1. Studying direction........................................................................................................................ direction ........................................................................................................................31 31 5.2. Growing geometry ....................................................................................................................... .......................................................................................................................34 34 6. The development of standard forms (rank one vectors) ...........................................................................38 ........................................................................... 38 6.1. Standard forms............................................................................................................................. forms .............................................................................................................................38 38 6.2. The trigonometric forms ..............................................................................................................38 .............................................................................................................. 38 6.3. The trigonometric standard forms ............................................................................................... forms ............................................................................................... 39 6.4. Applying Euclid over Ostberger .................................................................................................. ..................................................................................................42 42 6.5. Amplitudes................................................................................................................................... Amplitudes ...................................................................................................................................44 44 6.6. Powers of trigonometric functions............................................................................................... functions ...............................................................................................44 44 6.7. Half angle tangent relations .........................................................................................................45 ......................................................................................................... 45 6.8. Algebraic equations .....................................................................................................................47 .....................................................................................................................47 6.9. Parametric equations.................................................................................................................... equations ....................................................................................................................47 47 6.10. Imaginary geometry................................................................................................................... geometry ...................................................................................................................47 47 7. Four dimensions are here  here ............................................................................................ ......................................................................................................................... ............................. 49 7.1. Four dimensions........................................................................................................................... dimensions ...........................................................................................................................49 49 7.2. Observational platform ................................................................................................................ ................................................................................................................52 52 7.3. Quarter points  points ................................................................................................. .............................................................................................................................. ............................. 52 7.4. Standing in another place............................................................................................................. place.............................................................................................................55 55 8. The development of Law Fields (rank two vectors) ................................................................................. .................................................................................59 59 8.1. Law Fields  Fields  ................................................................................................................................... ................................................................................................................................... 59 8.2. The Newton Law Field ................................................................................................................59 ................................................................................................................ 59 8.3. The origin..................................................................................................................................... origin.....................................................................................................................................61 61 8.4. The relations  relations ................................................................................................................................ ................................................................................................................................ 61 8.5. The magnetic Law Field .............................................................................................................. ..............................................................................................................61 61 8.6. The electric Law Field ................................................................................................................. .................................................................................................................62 62 8.7. The general Law Field ................................................................................................................. .................................................................................................................63 63

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8.8. Continuity and discontinuity........................................................................................................ discontinuity........................................................................................................65 65 8.9. Matrices  Matrices ....................................................................................... ....................................................................................................................................... ................................................65 65 8.10. The Law Fields of number .........................................................................................................65 ......................................................................................................... 65 8.11. The Law Fields of thermodynamics ..........................................................................................68 .......................................................................................... 68 9. The omnipotent laws ................................................................................................................................ ................................................................................................................................74 74 9.1. Opposites ..................................................................................................................................... .....................................................................................................................................74 74 9.2. Absorption ...................................................................................................................................74 ...................................................................................................................................74 9.3. Reciprocity................................................................................................................................... Reciprocity...................................................................................................................................78 78 9.4. Conversion ...................................................................................................................................80 ...................................................................................................................................80 10. Law Worlds of the first kind................................................................................................................... kind ...................................................................................................................82 82 10.1. Vector as pictures ....................................................................................................................... .......................................................................................................................82 82 10.2. Separating direction from magnitude ........................................................................................ magnitude ........................................................................................ 83 10.3. Ordinary space ...........................................................................................................................83 ...........................................................................................................................83 10.4. The parallel principle................................................................................................................. principle .................................................................................................................83 83 10.5. Direction only ............................................................................................................................ ............................................................................................................................84 84 10.6. Curvatures.................................................................................................................................. Curvatures ..................................................................................................................................84 84 10.7. Assembling law fields................................................................................................................ fields ................................................................................................................85 85 10.8. Minkowski’s worlds................................................................................................................... worlds ...................................................................................................................86 86 10.9. The grav-electromagnetic world ................................................................................................87 ................................................................................................ 87 10.10. The thermodynamic world....................................................................................................... world .......................................................................................................90 90 10.11. The World of number............................................................................................................... number ...............................................................................................................92 92 10.12. Other Worlds............................................................................................................................ Worlds ............................................................................................................................93 93 11. Worlds of the second kind ...................................................................................................................... ......................................................................................................................98 98 11.1. Worlds of the second kind  kind ......................................................................................................... ......................................................................................................... 98 11.2. Bosons and Fermions................................................................................................................. Fermions .................................................................................................................98 98 11.3. GEM World of the second kind ...............................................................................................102 ............................................................................................... 102 11.4. The atomic elements ................................................................................................................ ................................................................................................................102 102 11.5. Computer Modelling................................................................................................................ Modelling ................................................................................................................107 107 12. Using the process for Sommerfeld’s fine-structure constant................................................................ constant ................................................................108 108 12.1. The first application .................................................................................................................108 ................................................................................................................. 108 12.2. Sommerfeld’s fine-structure constant ...................................................................................... ......................................................................................114 114 13. New wine into new skins...................................................................................................................... skins ......................................................................................................................118 118 II. The Appendices ............................................................................................................................................. .............................................................................................................................................121 121 A. The first rope trick ................................................................................................................................. trick .................................................................................................................................122 122 A.1. The note 104b  104b ........................................................................................................................... ........................................................................................................................... 122 A.2. Concluding remarks .................................................................................................................. ..................................................................................................................125 125 A.3. The 1  1//4π  connection ................................................................................................................126 ................................................................................................................ 126 A.4. Another theorem .......................................................................................................................127 .......................................................................................................................127 B. Geometric representations ..................................................................................................................... .....................................................................................................................129 129 B.1. Presentation stages .................................................................................................................... ....................................................................................................................129 129 B.2. Generation................................................................................................................................. Generation .................................................................................................................................129 129 B.3. Orthogonality ............................................................................................................................ ............................................................................................................................129 129 B.4. Density ......................................................................................................................................129 ......................................................................................................................................129 C. The case of two zeros  zeros ............................................................................................................................ ............................................................................................................................ 131 C.1. Something that actually happened in a company ......................................................................131 ...................................................................... 131

C.2. Accounting errors...................................................................................................................... errors ......................................................................................................................131 131 D. Uses of the word  “Dimension” .............................................................................................................132 ............................................................................................................. 132

D.1. Note 137 Summary page........................................................................................................... page...........................................................................................................132 132 E. Connectedness and disconnectedness .................................................................................................... ....................................................................................................134 134 F. Newtonian cases of Magnitudes with Directions ...................................................................................137 ................................................................................... 137 F.1. Direction Z g : Directional case, Potentials orthogonal to Force ................................................ Force  ................................................ 137

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F.2. Direction Z g : Magnitudinal case, Directions parallel to Velocity ............................................. .............................................137 137 F.3. Momentum: Directional case - the satellite, Momentum orthogonal to direction ..................... 137 F.4. Momentum: Magnitudinal case - the rocket, Momentum parallel to direction ......................... 138 F.5. Force: Directional case - the gyro, Force orthogonal to precession/direction ........................... 139 F.6. Force: Magnitudinal case, Force parallel to direction  ............................................................... 140 G. Grav-electromagnetic relations  relations  ................................................................................................. ............................................................................................................. ............ 141 H. Spherical surface or sphere.................................................................................................................... sphere ....................................................................................................................143 143 I. Linear algebra relations........................................................................................................................... relations...........................................................................................................................144 144 J. Table of delta values  values ............................................................................................................................... ............................................................................................................................... 145 K. The magnitudinal conics  conics  ....................................................................................................................... ....................................................................................................................... 147 Bibliography............................................................................................................................................... Bibliography ...............................................................................................................................................149 149 Ostberger notes........................................................................................................................................... notes ...........................................................................................................................................149 149 Glossary...................................................................................................................................................... Glossary ......................................................................................................................................................151 151 Colophon............................................................................................................................................................. Colophon .............................................................................................................................................................170 170

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List of Tables 8-1. The Work Law Field (firstness) ......................................................................................................................69 ......................................................................................................................69 8-2. The Organisation Field (secondness)  (secondness)  ............................................................................................................. ............................................................................................................. 70 8-3. The Reaction Field (thirdness)  (thirdness)  ........................................................................................ ....................................................................................................................... ............................... 71 9-1. From the notes on social activity .................................................................................................................... ....................................................................................................................79 79 G-1. The effects of the rotations (Curls) that take place in the fields ..................................................................141 .................................................................. 141 G-2. The magnetic, electric and gravitic inverse square laws.............................................................................. laws ..............................................................................141 141 G-3. Potential gradents of the three fields............................................................................................................ fields ............................................................................................................142 142 I-1. Algebraic Structure  Structure  .......................................................................................... ....................................................................................................................................... .............................................144 144 I-2. Comparing Law Fields with Linear Algebra................................................................................................. Algebra .................................................................................................144 144 J-1. Definition of the delta functions.................................................................................................................... functions ....................................................................................................................145 145 J-2. The delta values for  q  =  = 1.............................................................................................................................145 .............................................................................................................................145 J-3. The delta values for  q  = .............................................................................................................................145 145  = 2............................................................................................................................. J-4. The delta values for  q  =  = 3............................................................................................................................. .............................................................................................................................145 145 J-5. The delta values for  q  = .............................................................................................................................146 146  = 4.............................................................................................................................

List of Figures 2-1. Cartesian frame of reference............................................................................................................................. reference.............................................................................................................................6 6 2-2. The origin is numerically zero in every direction............................................................................................. direction .............................................................................................7 7 2-3. Polar Cartesian.................................................................................................................................................. Cartesian..................................................................................................................................................7 7 1 2-4. The pen stand ................................................................................................................................................... ...................................................................................................................................................8 8 2-5. An example curved geometry ...........................................................................................................................9 ...........................................................................................................................9 2-6. An important basic geometry  geometry ........................................................................................................................... ........................................................................................................................... 9 2-7. Multiples of trigonometric functions ..............................................................................................................10 .............................................................................................................. 10 6 2-8. The first sets of integer surds .........................................................................................................................10 .........................................................................................................................10 6 2-9. Elements of the form ..................................................................................................................11 11 a(a + 1) ..................................................................................................................

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2-10. Two curves straightmeet linesorthogonally...................................................................................................................... meet at right......................................................................................................................12 angles .........................................................................................................12 ......................................................................................................... 12 12 2-11. Two orthogonally 2-12. We can apply directions along line elements................................................................................................ elements ................................................................................................13 13 2-13. The shortest path between two orthogonal points is a quarter rotation ........................................................13 ........................................................ 13 2-14. Different types of orthogonal elements ........................................................................................................14 ........................................................................................................ 14 2-15. There are eight possible orthogonal curvatures of the single line element; four cylindrical and four annular 14

2-18. The front page of a note on sine and cosine addition and subtraction 8 ........................................................ ........................................................16 16 2-19. A larger curved geometry .............................................................................................................................17 .............................................................................................................................17 3-1. The Wun-man’s universe ................................................................................................................................19 ................................................................................................................................19 3-2. A rope around the Earth.................................................................................................................................. Earth..................................................................................................................................21 21 3-3. A rope around Jupiter .....................................................................................................................................21 .....................................................................................................................................21 3-4. one unit of overlap ..........................................................................................................................................22 ..........................................................................................................................................22 4-1. Left hand rule for the motor  motor  ........................................................................................................................... ........................................................................................................................... 23 4-2. Right hand rule for the generator.................................................................................................................... generator ....................................................................................................................23 23 4-3. The unidirectional representation ...................................................................................................................24 ................................................................................................................... 24 4-4. The bidirectional representation .....................................................................................................................24 .....................................................................................................................24 4-5. The orthogonal vectors of light  light  ......................................................................................... ...................................................................................................................... ............................. 25 4-6. Using a bicycle wheel as a gyro  gyro ..................................................................................................................... ..................................................................................................................... 26 4-7. For the left hand holding the spindle; the magnitudinal part of the spin. .......................................................27 ....................................................... 27 4-8. Gravity pulling down creates a torque ............................................................................................................27 ............................................................................................................ 27 4-9. For the right hand holding the spindle; the directional part of the vectors. .................................................... ....................................................28 28

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4-10. The orthogonal surface vectors of the gyro .................................................................................................. ..................................................................................................28 28 4-11. The gyro surface produces a Law Field ........................................................................................................29 ........................................................................................................ 29 5-1. A line minimum element ................................................................................................................................31 ................................................................................................................................31 5-2. A page from Ostberger’s notes on tensor calculus .........................................................................................32 ......................................................................................... 32 5-4. Orthogonal curvatures in a World geometry .................................................................................................. geometry .................................................................................................. 33 5-5. The interior Euclidean chord theorem ............................................................................................................ ............................................................................................................34 34 5-6. The magnitude powers of  p  p ............................................................................................................................ ............................................................................................................................35 35 5-7. The field of an electric dipole ......................................................................................................................... .........................................................................................................................35 35 5-8. The field of a pair of parallel wires................................................................................................................. wires .................................................................................................................36 36 6-1. The directional form; the functions arise as directional ratios .......................................................................39 ....................................................................... 39 6-2. The magnitudinal form; the functions arise as line length ............................................................................. .............................................................................39 39 6-3. The magnitude form of tangent ......................................................................................................................39 ......................................................................................................................39 6-4. The magnitude form of secant ........................................................................................................................40 ........................................................................................................................40 6-5. Magnitudes of circular functions of the outer angle....................................................................................... angle .......................................................................................40 40 6-6. Magnitudes of circular functions of the the inner angle ................................................................................. .................................................................................41 41 6-7. Directions of the magnitudes for the outer angle in the first quadrant ...........................................................41 ........................................................... 41 6-8. Applying Euclidean theorems to the trigonometric standard forms............................................................... forms ...............................................................42 42 6-11. The inside powers of circular functions belonging to the outer angle.......................................................... angle ..........................................................45 45 6-12. The outside powers of circular functions belonging to the inner angle........................................................ angle ........................................................45 45 6-13. A copy of the half tangent study from the Ostberger notebook  ...................................................................  ................................................................... 46 6-14. The normal form ........................................................................................................................................... ...........................................................................................................................................47 47 7-1. A World point ................................................................................................................................................. .................................................................................................................................................49 49 7-2. Cylindrical elements ....................................................................................................................................... .......................................................................................................................................49 49 7-3. Annular elements............................................................................................................................................ elements ............................................................................................................................................50 50 7-4. Measuring the directions of the surface.......................................................................................................... surface ..........................................................................................................50 50 7-5. Numbered octants ........................................................................................................................................... ...........................................................................................................................................51 51 7-6. Rotations associated with the Lorentz tranformation .....................................................................................53 ..................................................................................... 53 7-7. Rotating the World.......................................................................................................................................... World ..........................................................................................................................................55 55  “straight” 7-8. of  “straight”  nulls.........................................................................................................  nulls .........................................................................................................56 56 7-9. Curves Curves in notthe in planes orthogonal planes .....................................................................................................................56 ..................................................................................................................... 56 7-10. Curves in the moving and rotating orthogonal planes .................................................................................. ..................................................................................57 57

8-1. The generalised Law Field  Field  ............................................................................................................................. ............................................................................................................................. 59 8-2. The Newton Law Field ................................................................................................................................... ...................................................................................................................................60 60 8-3. The magnetic Law Field ................................................................................................................................. .................................................................................................................................61 61 8-4. The electric Law Field .................................................................................................................................... ....................................................................................................................................62 62 8-5. The general properties of a Law Field............................................................................................................ Field ............................................................................................................63 63 8-7. The number operations Law Field (secondness) ............................................................................................ ............................................................................................67 67 8-8. The interpretation of number Law Field (thirdness) ...................................................................................... (thirdness) ...................................................................................... 67 8-10. The organisation law field (thermodynamic secondness)............................................................................. secondness) .............................................................................69 69 8-11. The reation Law Field (thermodynamic thirdness) ...................................................................................... thirdness)  ...................................................................................... 70 9-1. An absorption matrix ......................................................................................................................................74 ......................................................................................................................................74 9-2. An associative matrix with missing elements is still completable ................................................................. completable  ................................................................. 75 9-3. An associative matrix with a quadrant magnitude (determinant) of 10.......................................................... 10..........................................................75 75 5 9-4. The absorption bottle ..................................................................................................................................... .....................................................................................................................................76 76 9-5. The absorption identity matrix  matrix .......................................................................................... ....................................................................................................................... ............................. 77 9-6. The Law Fields of Matrices ............................................................................................................................77 ............................................................................................................................77 9-7. Cardinal and Ordinal numbers........................................................................................................................ numbers ........................................................................................................................78 78 10-1. Equal vectors are parallel, and of the same magnitude ................................................................................ magnitude ................................................................................ 84 10-2. Curving an element....................................................................................................................................... element .......................................................................................................................................85 85 10-7. The extravariant Thermodynamic World...................................................................................................... World ......................................................................................................91 91 10-8. The Extravariant Number World  World  ...................................................................................................... .................................................................................................................. ............ 92 10-9. A copy of a page of the notebook showing a sketch of the World of intravariant Number .......................... 93

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10-10. The first issue of the three Law Fields of Fluids Fluids ........................................................................................  ........................................................................................ 94 10-11. The first issue of the incomplete Fluidic Law World ................................................................................. World  ................................................................................. 94 11-1. A minimum line element ..............................................................................................................................98 ..............................................................................................................................98 11-2. Two kinds of particle statistics  statistics  ..................................................................................................................... ..................................................................................................................... 98 11-3. The four fold infinite transformation which births the extravariant World from the intravariant ................ intravariant  ................ 99 11-4. Assembly of the representations of Fermions and Bosons......................................................................... Bosons. ........................................................................100 100 11-5. The anti-commuting assembly of the intra and extra Grav-electromagnetic Worlds that creates the representation of atomic elements 4.............................................................................................................102 ............................................................................................................. 102 11-6. The first set of orthogonal anticommuting elements on the intravariant surface ........................................103 ........................................ 103 11-7. Counting electrons, the triplet states........................................................................................................... states...........................................................................................................105 105 11-8. Counting electrons, the first multiplet state ................................................................................................105 ................................................................................................ 105 11-9. A sketch of a three electron state................................................................................................................ state ................................................................................................................106 106 12-1. Finding the new universe............................................................................................................................ universe ............................................................................................................................108 108 12-2. The one unit rope trick .................................................................................................................................109 ...............................................................................................................................109 12-3. The Planck Trick .........................................................................................................................................109 .........................................................................................................................................109 12-4. The complete set of Red geometries of the inner product .......................................................................... ..........................................................................110 110 12-5. The complete set of Blue geometries of the outer product......................................................................... product .........................................................................111 111 12-7. A magnitude essential to Quantum Mechanics ..........................................................................................112 .......................................................................................... 112 12-8. The delta geometry sets are part of the Hydrogen atom solutions ( q  =  = 1)14 .............................................. ..............................................113 113 A-1. A rope around the world .............................................................................................................................. ..............................................................................................................................122 122 A-2. For each meter of circumference ... ............................................................................................................. .............................................................................................................122 122 A-3. A rope around Jupiter ..................................................................................................................................123 ..................................................................................................................................123 A-4. The largest possible curvature of the rope ...................................................................................................123 ................................................................................................... 123 A-5. Curvatures at infinity  infinity  ................................................................................................................................... ................................................................................................................................... 123 A-6. One unit  unit ......................................................................................... ....................................................................................................................................................... ..............................................................123 123

A-7. Half a step to infinity ................................................................................................................................... ...................................................................................................................................124 124 A-8. One step to infinity....................................................................................................................................... infinity.......................................................................................................................................124 124 A-9. One extra unit of circumference internally .................................................................................................. ..................................................................................................127 127 A-10. One extra unit of circumference externally ............................................................................................... externally  ............................................................................................... 127 E-1. Finding the new universe  universe ............................................................................................................................. ............................................................................................................................. 135 F-1. A Schwarzchild gravitational field ...............................................................................................................137 ............................................................................................................... 137 F-2. The satellite operates in the momentum-static plane ...................................................................................138 ................................................................................... 138 F-3. The rocket is an exchange of momentum device ......................................................................................... device  ......................................................................................... 139 H-1. Elements of a sphere’s surface..................................................................................................................... surface .....................................................................................................................143 143 H-2. Elements of a spherical surface  surface  ................................................................................................................... ................................................................................................................... 143 K-1. Sketches of the conic sections expressed in Magnitudinal and Directional form  ....................................... 147

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Synopsis There is a star situated somewhere in the Milky Way which is the mother of our home, the Earth. It is our Sun. And ultimately it is the Sun’s energy falling upon the surface of this tiny satellite we call Earth, that feeds and enlivens the life that has been created over the millennia. Life which is agenerate, whose entropy is negative and with which, our tiny minds venture to see out into the universe, trying, desperately, to understand the order both below us and above us. The Sun is remarkable because it too is an agenerate part of nature. From Hydrogen it generates Helium in a frenzy of fusion. It is as if the hydrogen were coming in through some four dimensional hole in the universe to be fused into the beginnings of agenerate life in the Milky Way. Forming the lesser Hydrogen into the greater Helium and expending vast amounts of radiative energy in doing so. This Hydrogen is the beginning of this process and yet Hydrogen is the least of all the elements we know in the universe. It is an atom with a hole in it where the neutron is supposed to be. It consists of just one electron and one proton and because of its asymmetric form it likes to live with a partner which is similar to itself as the diatomic molecule  H 2 . But hydrog hydrogen en also also live livess within within us. During During the proces processs of digest digestion ion the Kre Krebb bb cyc cycle le ex extra tracts cts energy energy fro from m con conve versi rsion on of Adeninine-diphosphate into Adeninine-triphosphate and in doing so leaves a Hydrogen atom to be split into its component parts, the electron and the proton. Without this process, we die. And without the Sun’s energy, we die. Both are created out of the least of all elements, the Hydrogen atom.

Then, at the beginning of the twentieth century man formulated theories that began our understanding of the Universe Univ erse around us. Riemann, Riemann, Planck, Planck, Einst Einstein, ein, Mink Minkowsk owski, i, Hilbert, Hilbert, Somme Sommerfeld, rfeld, Dirac, Heise Heisenber nberg, g, Bohr Bohr,, Schroedinger, Feynman, Hawkin, Penrose all contributed to the mental pictures of the four dimensional world we live in. The Hydrogen atom was described and its properties exposed with ever increasing correlation to experimental evidence. Sommerfeld discovered a constant which was as universal to Hydrogen as the speed of light is to the universe. A constant that is the basic measure of the scale belonging to the Hydrogen atom. A constant which is so basic to the universe that we can calculate from it, the velocity of light. And then came Ostberger. Ostberger has produced a geometry which provides the natural constant which measures the scale of the Hydrogen atom. The geometry is a universal shape that exists for our understanding. And that is remarkable. For it says, that this man has produced with a paper and pencil (although now a computer) the Sommerfeld fine-structure constant which has hitherto been the subject of theoretical calculation and laboratory experiment throughout the last century. century. He has used the processes of geometry to reason. He calls them simply,  “Directions”  and says that they can be studied in just the same way that we do numbers. That an algebra of Directions can be formulated which leads to even bigger geometries which eventually provides the answers to things like the Hydrogen atom. He relates all his work to that of the ancient Chinese Taoism as well as to the modern Quantum Mechanics. He discovers a four dimension geometry and even n-dimensional ones too. He shows that there are at least a pair of  four dimensional geometries for every set that can be created, That one of this pair is stable and the other is not. And it is all done with   “pictures”. From the beginnings beginnings of Euclid, Euclid, Ostberger Ostberger creates a library library of geometric geometric solutions which event eventually ually build into a complete picture of the subjects that we study. He culminates with a model for the chemical elements in a deeply complex but nonetheless visible geometry from which he calculates the most basic of known physical constant, the Sommerfeld fine-structure constant for Hydrogen.  “Its source”, he says,  “comes from asking the geometry a question, when is your yin equal to your yang?”  The result is the selection of one number from a table of new constants he calls the  “Delta values”. This selection yields this Universal Constant to an asymptotically endless accuracy of decimal places. This book is one man’s attempt to tell the story of the creation of this remarkable theory which took Ostberger thirty years. It is sometimes difficult to comprehend but the geometries are clearly set down for us all to see. The

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Synopsis

whole book is backed by the notebook of Telle Ostberger which it is said gives mathematical detail of all that is included in the book. Much of it is so simple that it cannot be denied and yet the rest is so enriched that it cannot be fully understood. We cannot but be surprised by this man’s remarkable discoveries. The world is invaded with directions of all kinds. We simply have not recognised them as such. But now, the numerical size of an object will take its place amongst its directional properties. There are many questions and only a few answers but even they are surprising. The universe is not where we think it is! The stars we see are distributed according to the laws of science and that includes their respective directions. So when we look at a star we are actually looking around the corner into a universe which is all curves like Ostberger’s geometries. He says  “goodbye” to Descartes Is homoeopathy the medical of directional phenomena in the body? Is the directional part of atoms what tells them where their next energy level is? Where are the Directions of money? Can we transport ourselves into an orthogonal universe? It may not be an international best seller but then that is not the object of this book. The object is to expose this

man’s work to the world so that a new science may be launched which will benefit all mankind. This book is only about the Ostberger notes which relate to Mathematics and Physics. There are other books to follow which relate to Finance and Social Order and other more controversial subjects. The notebook of Telle Ostberger is voluminous. The author and his helpers have only had time and money enough to present this book and many of the associated notes which support it. We hope you find it as exciting as we do. Bob Beaumont, for E.T.C.

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The Ways and the Truths I am mad. I am completely insane, round the bend, over the hill, half a coin and waiting for the yellow coat to even think about writing a book like this and expect anybody to read it. In fact, any publisher would also be mad to imagine he could find an audience for such a subject as this. I would even venture to suggest that I might do better to wait for the next extra terrestial arrivals and hand it to them. For the subject matter is about as cranky as some people imagine flying saucers to be. So why the h$#! have I done it?

For many years I have lived with people and served them as customers of a business who are of the opinion that they have been dropped off on this planet by accident. That their intergalactic bus is about to come back and pick  them up with an apology for making the mistake. I discovered, a long time ago that there were others waiting in the same queue. I read their work and realised that they are the would-be saviours of mankind. In Ostberger’s case, he had a new language that could be easily learned and applied to everyday activities in a way that would enabled every person on the planet to see the whole picture of the subject before discussing it in detail. Thinking globally and acting locally would become a reality. It was about to come to life and it was only just around the corner. Lending a hand seemed natural. This remarkable discovery, that directions can be studied in just the same way that numbers can, is a god-send to mankind. The Truths may be important but they vie with the Ways. Ostberger has found a few of the Ways. The Rope Trick of  Appendix   Appendix A is A  is just one example of many  directional  theorems. The only number involved is the number 1. Yet we can deduce from this theorem a great deal about our universe and the  way  we must observe it. There are no straight lines in our observations of the universe which leads to the idea that we should study only curvatures for our understanding of it. But do we? Have we told our children that that is what Riemann (mid 19th century mathematician) said and that that is what Einstein said too? Out of all the mathematicians from the   “Gottingen club”  and all the mathematicians from virtually every University, who in this millennium has understood the message of these old sages? Have they said that we must learn about curvature? Are we guilty of  betraying the next generation? Then there are the absorption matrices which give a complete picture of our accounting processes. They are so simple. How could we have missed them? They are fun for children like chequers or chess, yet they are about learning accounts and understanding the flows of money which will also turn out to be a god-send for mankind, for they are the flows of many other phenomena too. How could we miss the simple messages of this book? How could so many communications from Ostberger (his Notebook 1400 series) fall on so many deaf ears for thirty years? The World geometry appears in no literature that I could find except one,   The School Mathematics Mathematics Pr Project oject book Z   of the Nuffield Foundation (http://www.nuffieldfoundation.org) (http://www .nuffieldfoundation.org) for  O level  Students. There it is passed over as an example of a surface that cannot be defined by just two coordinate numbers. And what did Winston Churchill say? From time to time, throughout history, man has stumbled upon the truth. He has picked himself up and carried on.

I stumbled at the bus stop and I intend waiting because I know that the bus stops here. Ostberger Ostber ger is say saying ing that that our lives lives are perva pervaded ded with with magnit magnitude udess and their their dire directi ctions ons,, the Truth ruthss and the Ways ays.. The They y are implicit in all studies not just mathematics and physics. This is the  yang  and the  yin  of the Eastern wisdom. His discovery is that we can as easily study the yin or Directional process as we can the yang or Magnitudinal process. He has begun the study of the yin process and begun to show its relationship to the yang process. The Directions are studied as geometric forms. In particular orthogonal geometric forms. The Magnitudes are studied as hieroglyphics, particularly mathematical ones in the form of equations. For every directional form there is a corresponding magnitudinal form, if we can find it! They are the left and right hand of the same body of information that presents itself to us in the natural universe and the social universe that we create amongst ourselves.

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The Ways and the Truths

This book is the beginning. It is the shortest path I can find between the simple concepts at the beginning to the enormous representations at the end. It is with great difficulty that I have found a connective path at all amongst the voluminous notebook of this man. He is a copious writer and a gigantuan thinker. He applies himself to food,

health, medicine, astrology, engineering, designs, manufacturing, social studies, finance, people, future medicine, buildings, accounting, disease, music and so on. At sixty he climbs mountains following a heart attack a few years before, mixes with people half his age, teaches, talks and dances his way through life with peace of mind and lightness of heart. He has opened a Pandora’s box of work to be done that will take more years to complete than universities have staff to cope. The other books, which relate to the Ostberger notes, are not about mathematics or physics. They are about human activity, a subject which intensely interested this man to the exclusion of his further researches into the mathematics and other technical studies. We have lost a valuable contributor to this new science simply because we could not hear. There are many who can reason the future Ways of physics and chemistry and accounting and engineering, for that is where this road is leading. They will be able to predict the properties of composite materials, make models of atoms and  walk inside. They will be able to travel the universe for years just as Columbus travelled the ocean. They will be able to make products the size of the smallest dot and the largest city. They will be able to treat illness and learn longevity with the skills of the great medical masters who have been buried. But they will not even enter the gate of this beginning unless they come to the humility of loving each other and the nature that is around them.

I cannot think of a better reason to write a book. I hope you can enjoy its harvest. R.P.B

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I. The Chapters

 

Chapte Cha pterr 1. Einste Einstein in and all all that that 1.1. Introducing the reasons for this book Books that relate mathematics to physics and physics to our universe usually centre their attention on that part of  physics which deals with Einstein’s Relativity. Yet the reality is that the world of Relativity is not more than one third of the studies of physics and perhaps even one half of that third. Another world of physics exists in the study of Thermodynamics. This is a world in which we see the volumes and pressures of substances raised through temperatures by stuff called heat to produce a thing called work. The state of organisation of the working device or substance is negative entropy and a thing called the absolute or Kelvin temperature measures the level or degree to which it can achieve organisational efficiency. There are two cases of absolute temperature measurement. The first comes in working devices that man has created creat ed such as a jet engine, engine, a heat exchanger exchanger or a household boiler1. The second comes from the internal affairs of substances such as gases, liquids and solids2. In the former we find that the higher the temperature the greater can be the organisational efficiency. In the latter the reverse is the case, the lower the temperature the greater is the efficiency of the substance. In reality the two processes actually work in opposition to each other. Thus in a car engine the gases are made to disorganise themselves by exploding which in turn conveys a certain amount of organisation to the engine in producing work. Such devices are degenerate because they disorganise nature in exchange for something which a human being eventually discards. Thus a car which is manufactured with a great deal of human ingenuity, materials and fossil energy disorganises not only man himself but also nature. Nature inevitably retaliates by reorganising the waste into some new form which is foreign to the environment in which man first created the car. Thus we may consume all the oxygen by burning it in engines which drive machinery to cut down the trees which produce the oxygen. It is only in the twentieth century that some human beings are waking up to this doomsday scenario, and they have a remarkable ability to sacrifice their life for the ecology. We cannot be degenerate for ever. Yet like all animals before us we will, no doubt, wait for the storm to be upon us before we consider that we ought to have made a shelter. There is another great World of physics which is not pop chart topping. It is the world of Fluidics. In this world we have the great laws of Stress and Strain and their ratios, Young’s Modulus of Elasticity, The Modulus of Rigidity and Poison’s Ratio for materials and the laws of structural mechanics described by Mohr’s circle and all the resulting equations. There is the compressibility of fluids, their motion, their viscosity and all the stream lines and potentials that gather in the equations of Bernoulli. There are the great tensors of stress and strain and vectors of  potential. This is a world describing the ways in which our universe flows. It is the creations brought about in the universe through the combinative building up of the two worlds of Thermodynamics and Grav-electromagnetics 3 which we see in the great study of physics.

1

 

Chapter 1. Einstein and all that 

A sketch of a very dense geometry which Ostberger says encapsulates the laws of Physics. It’s presented in more detail in Chapter in Chapter 10. 10. This book is about the work of Telle Ostberger who sees the world, not in terms of the magnitudes of things, but instead in terms of their directions. He has discovered that the directions of space are just as much a reasoning tool as are the magnitudes. He has cemented his ideas into the language of mathematics and begun to show that perhaps all mathematics can be approached in this way as well. He argues that it is possible to see the universe in two ways, not one. We are able to examine the Ways of the universe as well as the Truths 4. The ways are the Directional5 studies that Ostberger has begun and the Truths are the Magnitudinal 5 studies that have been the subject of mathematics since the passing of Euclid. Ostberger takes the principles of Euclid expands them into a whole new paradigm of mathematical thinking, about the Directional relationships of things and not just their size.

1.2. Descartes Ostberger

Space, matter, energy; it is all fully occupied. There are no spaces into which we can insert our notion of Cartesian coordinates. We may use them to construct our local material environment with vehicles, buildings, possessions and  the like but we may not apply them to nature. She has her own language of curvatures. There are no straight lines and  she does not recognise the concept of distance measured in metres. The curvatures fold over and over transforming themselves into ever more dense forms of our universe. Only the Laws of nature are straight, as we perceive them, and 

2

 

Chapter 1. Einstein and all that  they do not exist. The Laws are our way of seeing very special exceptions in the world around us. They are an invariant   part of natur naturee that we cannot quite touch in Magnitude but can imagine in Direction! The yang of Magnitude and the yin of Direction vie with one and other for a place in our imagination. Depending on the subject one or the other never quite makes it. The synaptic gap is either joined by a yang magnitude or left open by a yin direction.

In a 3-dimensional world Rene Descartes decided that we should adopt a system of coordinate measurement consisting of three axes  x,y,  x,y, z  at right angles to each other. This became the standard for the next four centuries. We still teach it in schools to the exclusion of all other possibilities. The human mind has become so entrenched in its use that change is tantamount to the same heresy that befell Bruno and Galileo.

Every direction in space is occupied by some phenomena which we measure. These phenomena have Directions which belong to them. They are a part of their make-up and they have relationships as Ostberger shows. When we observe light we also observe with the Directional aspects which belong to light. It is a curvature as Einstein Light travels along geodesic curves to our eye. of as sight from to Earth is ashowed. curve belonging to the electromagnetic phenomena thatThe weline know light. Its the rateintergalactic of curvaturebody is given by the Lorentz Transformation. But change did take place through the work of Bernard Riemann in the middle of the 19th century. He constructed a mental image of a geometry of four dimensions which he described through the hieroglyphics of mathematics. His work formed the Tensor Calculus from which Einstein took his Relativity. Mathematicians like Minkowski, Maxwell, Lorentz and Planck made their name by applying the Riemann work to real living physics. Einstein then came to his peers and said look I can calculate the motions of the universe from Riemann’s work. He produced from the curvatures of the calculus three effects:

•

  The advance advance of the perihelion of Mercury Mercury

•

  The bending of light light around a massive massive body, body, our Sun.

•

  The red shift of light received received from interstellar space.

In the first he calculated the fact that mercury’s elliptical orbit does not just go around the Sun in a flat plane but that the plane of the ellipse rotates as well. His calculation fitted precisely the known facts at the time. In the second he calculated the bending of light from a star, which had its image line of sight in close proximity to the Sun. When the Sun was eclipsed by the moon the image light could be observed. He predicted that the star would be seen just a few minutes earlier than the straight line of sight would predict. His calculation fitted the observation.

3

 

Chapter 1. Einstein and all that 

In the third he calculated that light from bodies far away in the heavens would be shifted towards the red end of  the electromagnetic spectrum as it travelled towards our planet. What Ostberger observed was that all these phenomena are Directional effects. His calculations from fr om the curvature tensors of Riemann’s geometry produced directional results. The Mercury orbit changes its direction in a given direction according to some rate. The light changes its direction around an object with a dense gravitational field. Radiation changes its directional frequency through space. However Ostberger saw the third one slightly differently. The current wisdom says that the red shift is like a Doppler effect. When a train travels towards us standing at the level crossing we hear a high pitch. As it passes the pitch drops and as it goes into the horizon it drops further. The frequency shifts down as the direction of the velocity of the train changes with us as witness. This leads to an interpretation of space looking like the train. The farther away an object is perceived to be the greater the red shift and so the greater the velocity of the object away from our witness. This in turn leads to the Hubble constant which is central to the Big Bang and all that. Ostberger says, this is not true. It is an interpretation of what we believe to be true. It is like the moon which goes round the Earth. It rotates rotates once6 for each complete sidereal cycle around the Earth. What he says is that the geometries in his notebook, which are introduced in this book, show that the red shift is due to the curvature of  the electromagnetic tensor with which we view the intergalactic object.

1.3. Imaginary Time Steven Hawkins, A Brief History of Time. Only if we could picture the universe in terms of imaginary time would there be no singularities.

Steven Hawkins, A Brief History of Time. So may be what we call imaginary time is really more basic, and what we call real is just an idea to help up describe what we think the universe is like.

Is time real or is it imaginary? That is the question. Whether it is nobler in the mind to be a finite player living in real time or an infinite player7 living in imaginary time is a question which nature will not answer for us. We must find the answer for ourselves. Or perhaps it needs no answer and we will learn to live as both by alternating from one to the other as circumstances demand.

What Einstein did was to say that time was real. He placed the real time into an algebraic equation of his time which represented four dimensions. The time component (ict) was made imaginary by attaching the imaginary number  i  as a coefficient. The time itself was scaled by the magnitude of the velocity of light 8. The concept of time was now connected to that of  motion. (Ostberger connects time to Direction). All this made the time component negative. If we follow this idea we see that if time were imaginary the last component in the equation would become real 9

10

and go positive . Steven Hawkins says, When I tried to unify gravity with Quantum Mechanics, one had to introduce the idea of ’imaginary’ time. Imaginary time is indistinguishable from directions in space.

This is also what Ostberger says.  “Time”, he says,  “does not exist. It is a Direction in space. It is a figment of our  imagination to help us to relate to each other. It does not relate us to the universe by a number.”

Our clocks are a good approximation to the rotation of the Earth, the planetary body that we know the best and one that we can measure the easiest. What we have done is to take one rotation of the Earth, divide it into 24 bits, then divide it into 60 smaller bits, then 60 smaller bits for a second time to obtain a unit of measure we call the second.

4

 

Chapter 1. Einstein and all that 

We have measured a Direction in space. The accuracy of this always needs correcting because the rotations of the Earth are difficult to measure exactly. The Earth not only rotates on its axis but also rotates around the Sun. Its elliptical orbit moves to and fro and the Sun is moving and so on. An exact  2π  2π  revolution is practically impossible to measure but fortunately the remarkably exact laws of nature permit us to measure the rotations mathematically using the heavenly bodies. It is the fact that the heavenly bodies, and the rest of the universe, follows these laws so precisely that titillates our imagination and drives us to scientific exploration. What drove Ostberger to further explore the Directions of nature is the realisation that they too are precise in their relative juxtaposition as we shall see in this book. These Directions are the curvature of space which we can model in the laboratory. Even more than this we can model them on the computer. They speak to us as rotations. Rotations of all kinds and in all planes of geometry. They curve and rotate as they pass our eye and on into space. We can measure them by holding them still for an instant or creating a representation of stillness like a vector, as we shall see. In this way we can watch the curvatures roll over and over by rotating in the spaces of many dimensions yet controlled by the four dimensions of Einstein and Riemann. There are no singularities in Ostberger’s geometry. A singularity in one continuum is found joined by the next. The process generates dimensions from continuum to dis-continuum and back to continuum again alternating the processes of continuousness with those of discontinuousness. The key to all this is a rotation of one quarter of a circle. This is what Ostberger searched for in the study of  Direction, and his notebook is filled with examples of this11. Orthogonality is the quantum of Direction.

Notes 1. Syste Systems ms mad madee by man; man; the machine, machine, as Carse ([Carse87]) [Carse87]) calls it. Historically, so far, these are degenerate. 2. Syste Systems ms made by na nature; ture; the garden garden,, as Carse calls calls it. Th These ese are Agenerate. 3. Ostbe Ostberger rger use used d three new words words and they are all in this first first chapter. chapter.  Intravariant ,  Extravariant   and  Gravelectromagnetic. See the glossary of terms for full definitions.

4. New testament testament John John Ch. Ch. 14 v 6. 5. Ostbe Ostberger rger uses the words Direction and  Magnitude with a capital letter to talk specifically about magnitudinal and directional phenomena. Both terms are defined in the glossary. 6. The rreader eader is left left to think about this. 7.   [Carse87] 8. Our connec connection tion to nature is through our senses. senses. One of these is sight which uses the elect electromagn romagnetic etic phenomena in a very narrow band of frequencies. This mathematical phenomena scales everything we observe through the eye, even if it is transformed by other devices, with the large number  3  x  10 10 , the velocity of  light in meters per second. 9. In the equati equation on

ict becomes

−

10. 10. [Hawkin88]  [Hawkin88],, chapter 9.

ic ic((it it)) which is  +ict  +ict.

−

11. In Ostberger’s [note1166] Ostberger’s [note1166] for  for example, he shows that the Kronecker Delta in Tensor Calculus is no more than a metric of Direction. It is a quarter of a complete cycle.

5

 

Chapte Cha pterr 2. The eleme elements nts of new new geome geometry try 2.1. Rotations One example of rotations in space is the measurement of time. In the last chapter I mentioned the two methods of  visualising time, one real and one imaginary. The former describes time as a magnitude, a number that somehow exists in the real world. The latter describes time as a Direction in space and this accords with Steven Hawkin’s description of it later in this chapter. It is a complex rotation in space. The atomic clock is another example of rotations. This time the rotations are electromagnetic frequencies which operate at the speed of light. The Caesium clock will produce  299  299,, 979 979,, 200  vibrational rotations whilst the radiation travels one metre. That is because the speed of light  c  is  299  299,, 979 979,, 200  metres per second. One metre is therefore   0.000000003335640952  seconds. This accords with Einstein who used the scale factor   c   in his four dimensional equation. The directional rotation of the Caesium atom and the direction rotation of the Earth are connected by the laws of physics through the measurement we call the second. The atomic clock is ticking away at the rate of the reciprocal of the speed of light ( 1/c). We will see later that Ostberger used reciprocal velocity and velocity as contra parts of one of the laws in the field of Newtonian mechanics. From it he deduces that  “every momentum has a contra momentum”  and other Newtonian laws (Chapter (Chapter 8). 8).

2.2. Three Dimensions At school we learned about three dimensions through the process originally devised by Descartes in the 17th century century. . Three straight line were at mechan righthanica angles andineeri labelled . We them all g, our x,y,z They The y are ext extens ensiv ively ely used use d inaxes str struct uctura urallset and mec icall engine eng ering, ng, phy physic sics, s, chemis cheused mistry try, , surve surfor veyin ying, navig namathematics. vigati ation, on, and for all our local creative activity on Earth.

Figure 2-1. Cartesian frame of referen reference ce

6 

 

Chapter 2. The elements of new geometry

The essence of the Cartesian frame of reference is that it is taken to be numerically zero at the origin no matter which direction direction we approach approach it from. The negative negative direction is a translated translated axis of  x,   x, y,z . There is no reason to assume that this is right or even useful. We have simply not been given reason to use anything else. But Ostberger gives us a reason to approach a point in space with values other than zero. After all we are well aware that some sets of ideas and objects do not contain the number zero. Velocity is one such example. We cannot find a place in the universe and say this is absolutely stationary. Velocity and reciprocal velocity have the property that they meet at the number one. A place at which we must identify a normalising number, usually  1. Another example is temperature and yet another is the set of reciprocal real numbers.

Figure 2-2. The origin is numerically zer zero o in every direction

The scales of the Cartesian frame are generally taken to be equi-spaced and linear. But again we have no reason to stay with this concept, particularly in the age of computers.

Figure 2-3. Polar Cartesian

7 

 

Chapter 2. The elements of new geometry

The boxes that describe the Cartesian space are not always convenient. In navigation, for example, we need a radar to see objects in a polar frame of reference. But this too is only for our local creative activities. It still has all zeros at the origin. Mathematics provides for many different frames of reference for our local use. Spherical, cylindrical, conical, annular are all different ways of describing the space of our ordinary thinking. It is called Euclidean space. It is in the Euclidean space that Ostberger begins his journey of examining the directions of the universe. The theorems of Euclid form a basis on which to create geometries which describe the mathematics which is hidden in its hieroglyphic language. First, a simple demonstration. The page opposite shows a pen which is held in a stand by a magnetic ball. The pen is used as the radius vector representing the polar description of a 3-dimensional Euclidean space 1. At (a) the pen acts as the position vector  r 1  of any chosen chosen length in the z-x plane and together with the angle θ 1  describes a set

of points within the first right angle. At (b) the pen rises in the x-y plane to produce a set of points  (r  (r2 , θ2 )  within the next right angle. At (c) the pen falls forward a right angle to produce a set  (r  ( r3 , θ3 ). The pen now rests at (d). Seemingly the same position as at (a). But this is a different pen to the one we started with at (a). The pen clip has rotated by exactly a right angle. In mathematics we would refer to this as a new position vector because it is in a space which is at right angles to the original vector (pen). We can now begin the whole process all over again repeating the four motions of the pen from (d) but with the clip rotated ninety degrees. We can do this four times, each time rotating the pen ninety degrees. In the quadrant of the space2 that is before us in the picture there is a four fold set of points. Figure Figu re 2-4. The pen st stand and1

What is happening here? The angle  θ 4  is creating a new vector which allows us to rotate the pen with its new

8 

 

Chapter 2. The elements of new geometry

posi positi tion on an anot othe herr four four ti time mess in th thee sp spac ace. e. Bu Butt θ4  is an int intern ernal al rot rotati ation on wherea whereass the other other thre threee are ex exter ternal nal rotati rotations ons.. Yet they are all rotations and Ostberger’s work is about rotations in space, particularly ones that seem to go to infinity, so we cannot ignore it. But there is another question. Where is the  r4  that belongs to  θ 4 ? The pen stand is not the easiest method of displaying the four components of the space. There are clearly some problems in measurement. Not the least of which is the fact that we cannot see  r4 . We can see  θ 4  and we can ask 

the question  “Is  θ 4  orthogonal to the other three  θ ’s?”  We need also to ask what we mean by  “orthogonal”  in respect of the measurements we are making. The appearance of Rubic’s cube in the seventies was a great help to Ostberger since he could measure orthogonal rotations using the cube as a measuring instrument. It was clear at the outset that two intersecting lines could be orthogonal at one point in space but perhaps no other. If a curve had a series of points which were orthogonal to a series of points on another curve then there was something special about the two curves. But how would they be measured? One method was to associate the known mathematics with the curves and prove orthogonality by trigonometric functions. Another method was to use Euclid’s geometric theorems as proof.

Figure 2-5. An example curved g geometry eometry

One thing was clear Cartesian-ty Cartesian-type pe straight lines had infinit infinities ies of points in a one to many corres corresponde pondence nce when they were at right angles. There were so many points a quarter of a rotation apart that they represented something very special indeed. Ostberger eventually concluded that straight lines which were at right angles were so special that they did not exist! We will look at this in Chapter in  Chapter 4. 4. What did exist were the curvatures which had several points in one to one correspondence. The circular geometries shown here are particularly tractable and have many qualities that make them a sensible starting point for the study. These are both geometries in the plane and so measuremen measu rementt of orthogonalit orthogonality y does not present present a problem. problem. The Figure The Figure 2-5 2-5 is  is very close to the shapes produced     by Radical Circles except that the line AA is not a straight Cartesian axis and that makes the shape algebraically complicated. However an algebraic description of a shape is very different from using the geometry to represent a vector. The same shape of a geometry may have many uses in representations (see example in  Figure 5-6). 5-6). The lower geometry (which Ostberger calls  “of the first kind” ) is perhaps the most simple form to use but first we must rid ourselves of our Cartesian frame of thinking. If there are axes then they will form themselves in the course of constructing the representation. Let’s look at a couple of examples from the Ostberger notebook.

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Chapter 2. The elements of new geometry

Figure 2-6. An important basic geometry

The Figure 2-6 provides The Figure 2-6 provides a geometry which can have values of the curvatures from some arbitrary small value close to zero to some arbitrary large value close to infinity. There are no zero curvatures in this geometry. The note in Appendix Appen dix A3 says that we cannot have an infinite radial measure or zero curvature. The straight line is proven not to exist here and it is clear that Ostberger regards the simple theorem of  Appendix of  Appendix A as A  as extremely important. It is, he says,  “a geometric incompleteness theorem similar to Godel’s except that it can be seen plainly by the earliest student.” We see here that an infinity can be represented geometrically by a straight line given that proper attention is paid to the analysis of the curvatures.

Figure 2-7. Multiples of trigon trigonometric ometric functions

Some of the trigonometric functions are shown in the geometry of  Figure  Figure 2-7. 2-7. It uses the geometry of the first kind in in Figure  Figure 2-6. 2-6. We see here that the intervals between successive circles is constant with respect to the origin. This means that any scale between 1  and nearly zero is possible using the sine function here. This can form a basis for working with vectors. The amplitude of any sine or cosine function is clearly identified as associated with one of  the circles and can be related to a curvature. All amplitudes are possible 4. This becomes a library standard form together with all the other trigonometric functions 5. It can be used to construct further geometries because the space becomes unique if the nomenclature is followed.

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Chapter 2. The elements of new geometry

Figure 2-8. The first sets of inte integer ger surds6

Figure 2-8 is 2-8  is an example of the mixed use of ordinal and cardinal number. Here the line lengths in the geometry measure the roots of all integers. If the circles are of a continuous nature then it is apparent 6 that all real number number roots lie on the horizontal line at unity. By picking out groups of lines from point to point one may find several kinds of root functions expressed as the length of a line element. In this case a straight line element as this is in Euclidean 2-space. Figure Figu re 2-9. Elem Elements ents of the form6

 a(a + 1)

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Chapter 2. The elements of new geometry

Figure 2-9 is 2-9 is a brief study of the line elements in this geometry of the first kind which relate to a specific formula. In this case the study paid off because it is used later in modelling the hydrogen atom spin function. Many of  the studies do not pay off. At least for the present. But we never know when a piece of mathematics, in this case geometric mathematics, will be useful. So we need to build up a library of  Standard  Standard forms.

2.3. Standard Forms Within the library we need to define the nomenclatures that will be used. After all we are in the age of computing and we do need to have a common language to be able to communicate about these Directional Studies.

Figure 2-10. Two straight lines m meet eet at right angles

The earliest standard form is an examination of two straight lines meeting at right angles. We can imagine that each point amongst the infinity of points on the blue line can be measured as being orthogonal to every point on the red line at right angles to it.With an infinity of points on the blue line there is a square of infinite orthogonal connections in each quadrant. This representation leaves us with a many to one and a one to many relationship of  points. It simply wont do for a geometry in which we want to use orthogonality as a basis.

Figure 2-11. Two curves meet orthogonally

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Chapter 2. The elements of new geometry

On the other hand two curves can meet orthogonally at a point (Figure (Figure 2-11). 2-11). But we still need to discuss in what sense they meet orthogonally. Two curves of equal curvature meeting orthogonally will have a whole set of  orthogonal points in one to one correspondence. On the other hand if they are unequal curvature such as as Figure  Figure 2-5 then 2-5  then the meeting point is special. It may not be unique for there may be a pair of points as in in Figure  Figure 2-11 2-11..

Figure 2-12. We can apply directions along along line elements

So we can analyse the mathematics of the points in space. We can go further to ask ourselves in what part of the space the orthogonality manifests itself (Figure (Figure 2-12 2-12). ). We can apply directions along the line elements so that they become unique in the sense of having an inward or outward relationship.

Figure 2-13. The shortest path between tw two o orthogonal points is a quarter rotation

We can establish rules about of the lines in space say,the forshortest example, that the shortest path between two orthogonal pointsthe is juxtaposition a quarter of a rotation (Figure (Figure 2-13) 2-13)and or that path between two points which are three quarters of a rotation apart leave an orthogonal space. We may draw a simple line on the paper and posit that there are at least two directions in the line. If not then one of the ends must be at infinity. This is analogous to saying that we cannot form a number system with any less than two characters. The binary system is the minimum that will allow a complete set of magnitudes to be

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Chapter 2. The elements of new geometry

represented. But what do we mean by  “complete”7? The two magnitude characters  0  and  1  of the binary system are to be replaced by two Directions in a line element in the Ostberger geometry.

Figure 2-14. Different types of orth orthogonal ogonal elements

Two directions in a line element is the minimum. We can have more. But the more we have the greater the density of the representation as we shall see. If we leave out the directions altogether then we can choose any number of  directions to suit the problem.

2.4. More Than Two Dimensions Figure 2-15. There are eight possible orthogonal cur curvatures vatures of the single line element; four cylindrical and four annular

Just as we may embed a vector in mathematics into a one, two, three or n-dimensional space so we may embed a line likewise. But we may also apply a number of dimensions to the vector itself. Vectors may be curved and they must have width and thickness. We may compose theorems to these principles. In all cases a line element must have at least two dimension giving four possible curvatures ( Figure 2-15) 2-15) in the  x

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Chapter 2. The elements of new geometry

plane and four in the  y  plane. In each of the curvatures we have numerical scales in four senses. Two scales along the line element (forward and backward) and two scales orthogonal to it (in the radial and the normal). This amounts to saying that the Magnitudes can be associated with the Directions in four ways. In the first way the magnitude and direction are 0π -apart, in the second way they are π -apart; the orthogonal way is  π/  π/22-apart and in the case of the normal they are 2  2π π  apart and this is a theorem of a directional kind. It turns out that such a theorem is extremely powerful. So is the first rope trick theorem which we will look at in the next chapter.

Figure 2-16. The old ve vector ctor rules

In the existing rules the reversal of the direction indicates a contra- magnitude.

Figure 2-17. The new v vector ector rules

In the new rules the reversal of the direction is an accountable mathematical phenomena and it therefore can support both a magnitude and its contra-.

But what this means for mathematics is that Ostberger is saying that vectors may have reverse magnitudes. For example a positive one in the forward direction and a negative one in the reverse direction. Both the Direction and the Magnitude can be reversed. What Ostberger has discovered is really quite simple. Directions are just as much a tool of reasoning as are the magnitudes. Mother nature knows this and makes thorough use of the fact. She does not easily yield up her secrets but Ostberger has discovered that Magnitudes and Directions are separable entities. He has left us with a legacy which will serve us well in the next millennium. Our modelling techniques can now include pictures. But they can be pictures of a very precise and analysable nature. In the philosophies of the east Yin and Yang compose the Tao just as direction and magnitude compose our vector spaces. But the Tao is not there if the yin or yang is missing. Neither is it there if they are found because when they are found and placed together there comes another yin or yang to be found which is inside out to the previous and so the Tao never is. The student of Ostberger’s work will discover this for himself. The Tao is our attempt to represent and understand the world we have entered. It is how we visualise the world. Of itself it is not a reality only a model of reality, it is a void sequence of events, a null tensor in space, a vacuum. We may mark a post, scale a guideline or determine a metric in a tensor with our magnitudes but we are still missing the Directions that belong to nature.

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Chapter 2. The elements of new geometry

Mathematics is primarily a yang study. A study about the magnitudes of things. About trying to establish a number that we can to something. Buttapestry. Ostberger invites us to commence the yin study of Directions so that we may establish theattach directions in life’s rich To quote from his notebook,  “for what we have ignored is that the Laws of Nature follow very strict rules of   Direction, her yin, as well as of magnitudes, her yang. The idea that everything in nature is going to be described  by a number that can be extracted from a mathematical equation is as strange to me now as it was when I was a student. Although mathematical equations contain components of both magnitude and direction the later is easily lost in the craving for a number; for the Magnitude has components of both direction and number and so does the  Direction.”

Figure 2-18. The front page of a note on sine and cosine add addition ition and subtraction8

Figure 2-18 defines 2-18  defines the attributes of a vector of position like the pen. The length of the vector represents the magnitude. This has terminals (arrows) that yield its directional aspect. The Direction also has two aspects. One is the magnitude of the direction which we measure in degrees or radians but can be other measure such as the delta values in the atomic structure 9. The other is the direction of the direction which we measure with geometry such as Euclid and Ostberger. There are four aspects to the vector of  Figure   Figure 2-18,, not two as our prese 2-18 presence nce mathematics mathematics currently presumes. presumes. This results results in the possibiliti possibilities es shown in  Figure 2-17.. 2-17 The Ostberger work sets out to represent all four aspects. By selecting one of the aspects first we may bring our reasoning to bear on the problem that we wish to represent. We may select the Magnitudinal aspect first in which case we delve into our existing hieroglyphic mathematics to seek a magnitude solution. It is likely that we will travel via the eigenvector theory towards our solution. On the other hand we can select the Ostberger process by looking at our library of directional standard forms and seek a direction solution. We need to relate the former process to the latter and vice versa. In practice says Ostberger,  “We need to follow the processes which nature herself seems to use. She oscillates between a magnitudinal (yang) process and a directional (yin) process. First one and then the other. So, when we have come down the mathematical (yang)  path and arrived at impassable terrain we can swap vehicles and continue the journey with our new directional (yin) vehicle. We can change back again when we find the new terrain which suits our mathematical vehicle. The  process is endless. It seems that our right and left cerebr cerebral al hemispheres are a gift for this process.”

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Chapter 2. The elements of new geometry

However, it would be impossible to foresee all aspects of a representation before we start and so we have to creep up on it just as we do in mathematics by using the lesser calculus such as simple differential equations to build up into a denser one with operator nomenclature. The Ostberger process is broken into four main stages. 1. A yin representation of direction only. only. A generalised geometry which is much like a guess. 2. A representation of the stage 1 above, with the yang magnitudes magnitudes on it. The shape or form which is expected. 3. A representation which includes both the above and the yin of the magnitude. That is the directions that are attached to the line lengths which, at this stage are not defined in magnitude. 4. A complete representation which includes all the above sstages tages and the yang magnitudes which finalise it. A set of scale factors relate the whole to our measurement of reality.

Figure 2-19. A larger cu curved rved geometry geometry

Each stage is a slow process and none have been formalised in Ostberger’s work. He simply recognises the need for the stages. The ones that he took to arrive at solutions. He says,  “There is a symbiosis between all Directions and all Magnitudes which seems to say that the one set is mirrored in the other symmetrically. However, I suspect that the mirror is asymmetric just as we find in physics.”

Notes 1. The ssimple imple mathematics mathematics is given given in in [note103].  [note103]. 2. By usi using ng the full 360 360 de degree gree notatio notation n of  θ  θ  and the positive and negative values of  r  r  we can describe the whole spherical space in terms of  r  r  and  θ . 3. See  [note140]  [note140].. 4. Ostbe Ostberger rger m made ade not notes es on imaginary imaginary geometries. geometries. 5. See  Chapter 6 6 and  and the [note5xx] the [note5xx] series  series of notes.

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Chapter 2. The elements of new geometry

6. For th thee mathematic mathematicss of this this please please read read [note586].  [note586]. 7. The extre extremely mely powerful powerful Godel’s Godel’s Theorem Theorem (the incompleten incompleteness ess theorem) theorem) will lead us, in later chapters, to a system of geometric representation which has no singularities, but  “time must be a Direction in space”   in such a system. 8. See  [note117]  [note117].. 9. Delta values values are are explai explained ned in in Chapter 12. 12.

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Chapte Cha pterr 3. A story story and and a theore theorem m 3.1. A Story. This is a story about the poly-men. Its the kind of story that physicists have heard before, so they needn’t listen. In the land of one-dimension there was a fly fish called a wun-man. He lived in a lily pond with all the other wun-men. They were all confined to one dimension because they had two heads and because the top half of them breathed air with lungs and the bottom half breathed water with gills.

Now, all the little wun-men lived a lovely life floating amongst the lily leaves; and that is where they made their nests. Whenever they went walkabout they always travelled in straight lines on account of them being one dimensional and having two heads. They would go outward with one head thinking and come backward with the other head thinking. So when they were at home they simple changed their thinking head and went that way. Whichever way that was. As you can imagine the only track of a wun-man that could ever be seen was a straight line emanating from his home and every body accepted that that was the way it was.

Figure 3-1. The W Wun-man’ un-man’ss universe

Also, as you can imagine, the habits of their predators was made very easy. All they had to do was to home in on the straight lines of the wun-men and consume them to extinction. And we know that that is true because there are no more wun-men in the world today. But in their struggle to survive they learned a trick or two. They would leave one of their number at home as a sentry to measure the directions of the fly fish as they went walkabout. If the wun-men did not return then it was theorised that there was a hole in that direction or that the poor little chap had dropped off the edge. No wunman would go that way in the future. The sentry was know as the fly fish angler. But alas the angler was to be their demise for there came a time when no wun-men would venture out for fear of holes and edges. But, one day a baby wunman was born with a genetic defect. He only had one head. He was rejected by all the other wun-men.  “How can he survive with only one head?”, they would ask,  “he will never be able to get back  home”, they said.  “he will always need some-wun with him to show him the way home”, they said.

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Chapter 3. A story and a theorem

So, as he grew up he was never jolly because no wun would join with him and he was jibed and jeered as juveniles do. But he was a determined little fighter and whatever the other wuns would do he would do too. He soon got the nick-name of the Wun-too-man. But he never let them get him down and alway kept his spirits up. As an invalid he had to learn a new way of getting about. He was learning turning. He would turn his head just a little so that he could see where he had been. He would then do a thing he called  “backwards”. It was so exciting

that he developed a burning yearning for learning turning because this was such a great discovery; backwards. But as he grew older and wiser he discovered a new way of getting about he called  “turning”. He didn’t have to go “backwards” any more he went forwards all the time “turning”; and this was his first discovery. One day Wun-too went walkabout and was not reported back by the angler. A search party was sent out to try to find him. But they found nothing and so it was assumed that he was lost forever. A long time passed and one of the anglers on the other side of the lily leaf reported Wun-too as arriving home. “That’s impossible”  said all the members of the meeting of anglers, if you go out that way you must come back  that way too. So they called upon the young Wun-too to explain how he had managed such a trick. The timid voice of the little invalid reigned in the silence of the anglers.  “Well”, he began,  “What I did was to go out that way and come back this way.” “What?”, shouted an observing professor in a voice which sounded as if he had a plumb in his mouth,  “That’s impossible; and with only one head its doubly impossible. If you go out that way you must come back that way and if you go out this way you must come in this way. Otherwise you drop off the edge. That is an established  scientific fact. We have known it for years and no one headed, half-a-wunman is going to change it.” “But, sir”, began Wun-too like Oliver about to ask for another bowl of soup,  “I did it by turning.” “Turning! What the hell is turning?” , said the professor in his broadcasting voice. “Erm..., well sir, its like going out in two ways at the same time.”

There was roars of laughter as the little wun-men tried desperately to explain about another dimension. But he knew. He knew it was true for this was his second discovery. Wun could go out in one direction and arrive back in another direction and wun could go out in two directions at the same time. There really was a second dimension in the wun World. Wun-too was not one to boast but he knew one more than those professors. The little wunman became known as as a two-man, and he survived. And the end of this story is well known by all the humans in the world today. You see, the only survivor of the wun-men was the two-man; and the only survivor of the two-men was the three-man; and the only survivor of the three-man was the four-man and so on until we human are the last survivors of the hu-minus one-men. And the one-man was where we began. It was the wun-men who discovered, through a genetic mutation, that in order to survive an  n-dimensional world we need to know about an  n + 1-dimensional world. The moral of this story is that our survival in this 3-dimensional world depends upon our ability to understand a 4-dimensional world. That is what Ostberger is giving us.

3.2. And now for a theorem To lighten up the studying of these new ideas Ostberger invented a rope-maker who was all too often involving himself in problems with ropes which he didn’t understand and so tried to find the answer in mathematics. He usually failed and so created his own solutions. I rather think that Ostberger and the rope-maker were more than casual acquaintances. He begins,  “At the outset  it is difficult to imagine how we could construct a process of reasoning with geometrical forms which has a sound 

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Chapter 3. A story and a theorem

and logical basis and which passes beyond the work of Euclid. What kinds of axioms could we have? How would  they connect together? Here are two fundamental axioms upon which a great deal of subsequent directional study depends. They are my notes 140 and others; The Rope-maker.”

3.3. The first rope trick A rope-maker is contracted to make a rope to circumvent the Earth. The mean radius of the Earth is 6.37 million metres making the circumference some 40 million metres. Figure 3-2. A rope arou around nd the Earth

The rope-maker was overjoyed with his contract. He contrived a machine to make the rope at great speed and called upon his friend the ship-maker to freight it round the world. When the rope was finished and the beginning was returned to the rope-maker’s workshop he butted the two ends together and celebrated his success. But no sooner had he finished celebrating than he received irate telephone calls from around the world saying that the rope was far too long. The half rope-maker brain not wave. He drew how the ends thepossible, rope taught andwent measured spare rope. Itfor measured  just a metre.had Heacould understand this of was so he to his the local university help in calculating the problem.  “It’s far too trivial for us”, they said,  “we don’t get involved in trivia like that” . So the rope-maker did his own calculations. What he discovered was quite remarkable. The half metre overlap (Figure ( Figure 3-4) 3-4) that laid on his workshop floor translated into 15 cm of slack in the rope all the way round the Earth! The phone calls were right. There was a lot of slack in the rope. He could not believe that the half metre of rope that lay on his workshop floor could produce 15 cm of slack everywhere around the world. The rope-maker was so astounded by his discovery that he contacted his local University again to confirm his calculations but they just told him to stop bothering them with his trivialities. The rope-maker wondered if he could make a rope to go round Jupiter.  “What a contract”, he thought. “I wonder  if it will have to be as accurate?” accurate?”

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Chapter 3. A story and a theorem

Figure 3-3. A rope arou around nd Jupiter

He did some more calculations. To his surprise he came up with the same answer as for the Earth rope. The distance that the rope would stand off from the surface of Jupiter would be 15 cm for each 1 metre of error in the length of the rope.  “How can that be?”  he exclaimed.  “Does this mean that the stand-off height is a constant no matter how big the planet! Is it always 15 cm per metre?”

Figure Figu re 3-4. one uni unitt of overlap

The rope-maker considered a rope around the Universe.  “No matter who puts the extra metre into the rope there will always be a 15 cm movement at right angles to the rope” , he thought. )  “But suppose the rope were a part of  nature, the rope.” ropethen .” what? We would be part of the rope as if we we standing on it. So we might never see the motion of 

With nobody nobody to tell tell about about his dis discov covery ery the rop rope-m e-mak aker er sli slippe pped d his not notes es into into the wor workbe kbench nch dra drawer wer and pro procee ceeded ded home to bed. The rope-maker had indeed discovered something quite useful and this is given in  Appendix A. A. You do not have to be a mathematician to understand the arguments, although you will need some skill to extend the arguments to cases other than circles. But that is another story. The first Rope trick is a Directional Theorem, an idea that does not require specific magnitudes to elucidate the arguments and produce the answers. There is also another rope trick in the appendix and more in the notes.

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Chapter Chapt er 4. Orthogonalit Orthogonality y In this chapter I want to look at the way in which Ostberger slowly arrives at the idea of orthogonality in physics and then transcribes that idea to the mathematics.

4.1. The rules of Fleming Clench the fist of your right hand and point your thumb upwards. Now point your first finger away from you and bring your index finger out so that all three fingers are at right angles. This is your very own personal orthogonal direction indicator. You were given it at birth and you will now have to use it. In the study of magnetic fields there is a law called the Biot-Savart law. It was arrived at through meticulous experimentation and is essential to the calculation of such fields at a point in space. It puts into a formula the directional relationship between the current in a wire and the field outside it. However, it is not easy to see in the formula because the vectors involved are often separated. Fleming showed this rule on the hands. The left hand rule is applied to the motor; a device that uses electric current as an input to extract motion as an output. If you take the motor apart (one with coils on the rotor) and place your index finger (representing the current) in line with one of the copper wires and then rotate your wrist so that your thumb (representing the motion) into the line of motion of the motor then your forefinger will tell you the direction of the magnetic flux at that point in the motor. Of course, its a silly thing to do, so we do it with our heads instead.

Figure Figu re 4-1. Left hand ru rule le for the motor

So there it is. Fleming’s Left hand Rule for motors. And there it remains. And what is more it will remain for ever and ever, and ever. It will simply never go away. It is as fixed as the stars or the laws of the heavens that position the planets to the last decimal part of a millimetre. So Ostberger thought that he had better see if there were any more of these rules that are fixed, or invariant as the mathematician says. Fleming also had a Right Hand Rule. This time for generators.

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Chapter 4. Orthogonality

Figure 4-2. Right hand rule ffor or the generator

What Ostberger did was to say that these properties, represented on different hands, must in someway belong to each other. After all, a motion is a motion wherever it is. It is part of an idea that we call velocity. If we place our hands back to back with the first knuckle touching we can align these three properties into the set of axes in Figure 4-3. 4-3. His interpretation of this was of three fluxes flowing in an energy field1.

Figure 4-3. The unidir unidirectional ectional repr representation esentation

Ostberger goes a step further. He uses a bidirectional representation as in Figure in  Figure 4-4 4-4.. In this we see that the motor is represented by a net vector pointing outward in the diagonal of the box shown and the generator is represented by a net vector pointing inward in the box2. The two processes are: Inwa Inward rd

A pro proce cess ss of putti putting ng moti motion on in to get get cu curr rren entt out out usin using g a magn magnet etic ic field field as a co con nvert verter er is a

generator .

Outwa Out ward rd

A pro proces cess puttin ting g curren currentt in to get motion motion out usi using ng a mag magnet netic ic field field as a con conve verte rterr is a motor  . s of put

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Chapter 4. Orthogonality

Figure 4-4. The bidirec bidirectional tional repre representation sentation

Both processes are degenerate in that they consume energy. So the direction of the energy flow does not, here, tell us about any process that might generate energy as nature does. This concerned Ostberger and he made notes which suggest that he despaired that he might not be able to see the agenerate processes of nature in the new process. But we will see later that he succeeded in showing the remarkable processes which are agenerate in nature and ourselves.

4.2. Deflection of an electron beam In a cathode ray or television tube electrons are produced in the gun at high voltage and they motion towards the phosphorescent screen. They are deflected on their way by two fields. One is electric and the other is magnetic. These three vectors are mutually orthogonal just like in Fleming’s rules. What is important is that they reliably follow these rules. But what are the rules and how do they manifest themselves in geometry?

4.3. Vectors of light The vectors of light are a little more difficult to imagine. We have special equipment to see them. Light is seen as a wave travelling through space and yet the space that it travels through must be occupied. Ostberger treats the light as one of the occupants along with all the other orthogonal, phenomena that can be identified at every point of the space.

The wave travels in a sinusoidal manner, but what is it that we are measuring that oscillates so? Ostberger concludes that the directions belonging to the vectors of light are the surface vectors which are wrapped around the magnitude vectors show in Figure in  Figure 4-5 4-5..

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Chapter 4. Orthogonality

Figure 4-5. The orthogonal vectors of light

We can identify three vectors  E  the Electric intensity (force),  B  the Magnetic Flux density and  v  the velocity of  propagation. These are vectors of a magnitude character and we can ascribe a number to them according to the particular manifestation of the light. They are related precisely by Figure by  Figure 4-5. 4-5. The three vectors are orthogonal in space and we can identify them with a right hand rule just as we do Flemings rules. Again the preciseness of this relationship relationship is never never in question. question. Our experiment experimental al measurement measurementss indicate indicate that they remai remain n consi consistent stently ly orthogonal, in the manner of  Figure   Figure 4-5 throughout 4-5 throughout the universe. Ostberger asked himself,  “Is this velocity the same kind of velocity that we have seen in the electric motor and  generator?” He also asked himself how these vectors could remain orthogonal in a space of curvatures. It took a

 journey of many years to find the answers. It was on this journey that he identified the surface vector of  Figure of  Figure 4-5 as 4-5  as belonging to a group of tensors which are purely directional in character. He later named them  Electric Direction and Magnetic Direction. They were the Directional component parts of the vectors  E   and B . These Directional vector components manifest themselves as  Potentials  in physics. The connection between the two is by far the most remarkable discovery that Ostberger made. Two great four dimensional geometries containing the same complete set of vector elements appear to be inside out to each other. In fact they are orthogonal as we will see in later chapters. Ostberger called them intravariant  and  and  extravariant .

4.4. The vectors of the gyro This example of orthogonality is very much down to earth. The Earth is a gyro body and it obeys the mathematics of the universe. Or is it that the mathematics obey the rules of the universe 3? The Tao says that both coalesce. Get yourself a bicycle wheel. Hold the spindle on both sides with clenched fists. Use two thumbs on the spokes to spin the wheel and there you can feel the forces of nature in the gyro.

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Chapter 4. Orthogonality

Figure Figu re 4-6. Usin Using g a bicycle bicycle wheel as a gyro

With the wheel spinning at speed let go of your right fist and allow the spindle to rest on the fingers of your left hand. The wheel does not fall to the floor. What does it do? It rotates. Looking down in the Direction of the gravitational Looking gravitational acceleration acceleration it rotates rotates clockwise. clockwise. It will rotate clockwise clockwise for everybody in the world. This is called the precession and it follows a very strict set of orthogonal rules. To work this out you need your best brains in gear. So those readers who have gear box trouble should look at the pictures and pass on to the next chapter.

Figure 4-7. For the left hand holdin holding g the spindle; the magnitudinal par partt of the spin.

It is easiest to look at the spin of the wheel first. Hold your right hand out with your orthogonal indicating digits in position. Point your forefinger along the spindle of the wheel and follow the rotation of the spin with a right handed screw motion. Construct an axial vector with a ring around it in the direction of the screw motion (Figure (Figure 4-7). 4-7 ). This is the spin component with its velocity around the ring.

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Chapter 4. Orthogonality

Figure 4-8. Gravity pulling do down wn creates a torque

Next we find the torque component with its force around the ring. Look at the wheel end on ( Figure 4-8 4-8). ). With only one fist holding the wheel the acceleration due to gravity is trying to turn the wheel downward. Use your right hand to work out which way gravity is trying to turn the wheel. If your hand is the same as mine you will find Figure find  Figure 4-7 4-7 and  and Figure  Figure 4-9 to 4-9 to be correct.

Figure 4-9. For the right hand hold holding ing the spindle; the direc directional tional part of the vectors.

Finally we use our right hand to determine the rotation of the precession. To get the same answer as me you will have to point your forefinger up in the air. The right handed screw turns the way of the procession as we have observed (Figure (Figure 4-9). 4-9).

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Chapter 4. Orthogonality

Figure 4-10. The orthogonal su surface rface vectors of the gyr gyro o

We can now assemble these findings on to a single surface, the Figure the  Figure 4-10 4-10.. Here we can identify three axial vectors in the representation each with its orthogonal phenomena connected thus, 1. The axis of the Precession with its associated associated surface Direction. 2. The Spin with its associated surface Velocity Velocity.. 3. The Torque Torque with its associated surface Force. This is a second stage diagram of the type mentioned at the end of  Chapter of  Chapter 2 2.. There are no magnitudes and so further work must take place to make a stage 3 representation. But the directions will remain in their relative  juxtaposition. In fact, we will see later that the three surface vectors form what Ostberger called a   Law field  (Figure 4-10). 4-10). The reason why this is special will be discussed in in Chapter  Chapter 8 8.. All these vectors are well known. But, perhaps it is not so well understood that the Precession produces a Direction surface vector. What kind of phenomena is this and to what group of phenomena does it belong?

Figure 4-11. The gyro sur surface face produce producess a Law Field

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Chapter 4. Orthogonality

The Law Field helps us to group the phenomena as we will see in  Chapter 8. 8. Thus all forces of a Newtonian kind can be aggregated on to the Law Field by mappings. So they are a group of phenomena. So also are the group of Newtonian velocities4. The third group is a group of Potentials which is here associated with the direction component of the whole field. It is called Direction as a general term because, as we shall see, there are many of  these. It is a class with the term  flux  perhaps. The precession is associated with this Gravitic Potential. What Ostberger showed was that this Potential belongs to the group of Newtonian Potentials. And that is the reason the bicycle wheel did not fall off our finger. That the gyro produces its own Direction field is a strange concept at first. But we can soon become accustomed

to the idea as we work out our misconceptions in terms of these directional phenomena.

4.5. Other examples There are other examples both in thermodynamics and fluid studies where orthogonality is the key to separating the phenomena that we understand. The flow and stream functions in fluids, the gradients and contours on maps, the isobars and pressure gradients in weather charts, the lines and forces around a magnet, the flux and potentials around an electric charge, the vectors in a transformer, the force and potential in a Schwartzchild solution, the velocity, direction and force on a satellite and so on. There are even similar groupings in the mathematics itself. And, if that were not sufficient there are similar groups to our social activity too, such as accounting.

Notes 1. In  Chapter 10 10 these  these three fluxes will appear on the surface of a World geometry. 2. Now w wee see the the relationshi relationship p to the cl clip ip on the pen pen in in Figure  Figure 2-4. 2-4. 3. This is an exa example mple of intraintra- and and extrav extravarianc ariance. e. 4. See  Figure 8-2 8-2..

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Chapte Cha pterr 5. More More than than geometry geometry

5.1. Studying direction In studying directions I have come to realise that they are very real and tangible entities that enter into the structure of our world. Nature is possessed of directional elements everywhere. We have now a great need to study them. They are the yin counterpart of our yang studies which seek to deliver a number as a solution. In directional studies we might have expected that the only solutions that we can deliver are directional ones. This is not so. Numerical answers can be found amongst the geometry of curvatures that are just as precise as the ones we find by formula, but these two concepts are not in competition. They need to be in cooperation, for sometimes it is better to use the Direction process and other times it is easier to use the hieroglyphic process. They are both a part of Mathematics. In the final result we are looking for the truth and a way of expressing the truth. Both processes speak in their special way to the truth of the final result. The approach of the two processes is opposite in character. In our mathematics we seek a numerical solution and often try a directional sketch to achieve the result. The mathematics often hunts the directional eigen-vector as a path to the solution. In the Ostberger process we seek a directional solution and then apply the numerical scales to achieve a result. The two processes together must inevitably bring the best result. In most circumstances we begin the directional process with a generalisation. But we need not fear that we cannot whittle it down to a specific entity if we have already used the generalisation successfully. If one generalisation works then we may test it again to see how many times it does work. After a while we gain sufficient confidence in the process to be able to accept the generalisation.The trick is to get the generalisation right first time to avoid extensive researching of mistaken direction. The examples of orthogonal phenomena in the last chapter were intended to show this process at work. We can go further. If we can relate the generalisation to a known piece of  mathematics such as tensor calculus and see that it fits then we can gain confidence at the start.

Figure 5-1. A line minim minimum um element element

How many dimensions must a line have for us to be able to see it? Let’s look at a one dimensional line A  in Figure  in  Figure 5-1.. It is not there because we need to have another dimension to see it. The line must have a small part of a 5-1 second dimension if we are to see it at all. So  Figure 5-1  B  is a 2-dimensional line displayed for us to discuss one of the dimensions. If we now curve the line we are producing effects in 2-dimensions. This is the minimum display that we may expect nature to show us. If we take away one of the dimensions we cannot observe anything. We need two dimensions to see one. The line element is the basis of string theory. Indeed Ostberger makes notes about string theory and comments thus, “If string theory were to take account of both of the two most important theorems in Quantum Mechanics it  would accord with my own discoveries precisely. String theory has ignored the orthogonality theorem.”

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Chapter 5. More than geometry

So Ostberger built up geometries on the basis of curvatures starting with examples like those in  Chapter 2  2   and moving on to more condensed forms of geometry which we will see in Chapter in  Chapter 10. 10. It took thirty years in all. The most important geometry is the World geometry and this I will look at here very briefly. By the time Ostberger had reached the stage of drawing these pictures he had realised that the affixes and suffixes which are conventionally attached to the character of tensor calculus were, in fact, the Directions i, j  and k , of this geometry. In a note on the subject 1 he identified all the permutations of tensor calculus as products of this space. The picture at the top in Figure in  Figure 5-2 5-2 is  is one of the many diagrams he drew for that note. Interestingly he identified the Einstein tensor as a triangle moving with the transforming geometry. What is particularly interesting is that he identifies the Ricci and Weyl tensors as associated with Magnitudes and Directions respectively and hangs his potentials on the Weyl tensors.

Figure 5-2. A page from Ostberge Ostberger’s r’s notes on tensor calculus calculus

Look at Figure at  Figure 5-3. 5-3. It is a world geometry with co- and contravariant regions. They are identified by the nomenclature of tensor calculus2. Visualise the geometry as a pair of balloons with a hole interfacing the pair. There is a

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Chapter 5. More than geometry

whole set of variations of the space. The two balloons are identified as the covariant and contravariant components in the space.

Figure 5-3. Permutation of a w world orld geometry in tens tensor or calculus

Note that b and c are vectorial rotations. A Euclidean rotation would retain the same colours at b and c. Imagine that we have selected a metric (scale) for the space by determining the size of the hole forming the interface3. The two balloons are identified as the covariant and contravariant components in the space. Firstly, imagine that the covariant balloon is fixed in size and the contravariant balloon can expand into the contravariant space whilst keeping the  ij  junctions and all the others orthogonal. That is one possibility. Now suppose we have a whole set of the  j  circles spreading over the covariant surface. (See Ostberger’s sketch in the bottom left hand corner of  Figure  Figure 5-2). 5-2). That is another possibility. Each of the circles can range over the surface as did the first one. Now suppose we change the metric to a new hole size. The whole process can be repeated. Now suppose we reverse all the permutation of the process and range the covariant circles over the contravariant ones. We can further repeat the whole again game whilst keeping one of the  ij  junctions stationary and then the other. Then we can include the direction k as part of the geometry or we can leave it off. The permutations are enormous since every element can range from unity to near infinity in each of the directions. Then there are the permutations that derive from rotating the whole system and forming the metric in the  i  and  j directions in turn (the small pictures in Figure in  Figure 5-3). 5-3). The picture gets bigger because we can hang all three permutations in the  Figure 5-3 5-3 on  on to a single  World  geom  geometry. A sketched example is shown in Figure in  Figure 5-4. 5-4. And the picture gets bigger still because we can use each of the orthogonal states of this geometry to represent a group of phenomena. This is what Ostberger did with his process of Law Fields which we will look at in subsequent chapters. The result was geometric representations so large that it becomes difficult to handle the solutions they contain, in much the same way as in the Schrodinger equation.

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Chapter 5. More than geometry

Figure 5-4. Orthogonal curv curvatures atures in a W World orld geometry

But the picture did not stop there. An even bigger picture was obtained by Ostberger. He looked into the geometry of   Figure Figure 5-4 5-4 and  and imagined that he could stand inside the covariant components which encircled him, the ones that are outside outside the World. He imagined imagined that all the surface surface components components of the Worl World d (and there are two furthe furtherr sets that I have not mentioned here) were expanded to infinity and collapsed around him. The result was a new World geometry containing the same complete set of phenomenological elements. One can only see this when all the line elements are occupied on the World geometry. This is in Chapter in  Chapter 10 10.. The new World is clearly different. The first he called  intravariant  and   and the second he called  extravariant . The former addressed the problems of classical physics and the latter those of Quantum Mechanics. The extravariant World was of discontinuous character and the intravariant World of continuous character. They required different mathematical treatment. We will look at this later.

5.2. Growing geometry Every geometry can be used many times for different kinds of representation. Take as an example the simplest kind of geometry in Figure in  Figure 5-5. 5-5. It is Euclidean and undirected. It is the simplest form of one of the theorems we learn at school.

Figure 5-5. The interior Euclid Euclidean ean chord theorem theorem

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Chapter 5. More than geometry

No Now w ap appl ply y this this to some some geom geomet etrie riess wh whic ich h ar aree co cons nstru truct cted ed in Figure Figure 5-6 5-6.. The These se geomet geometrie riess are als also o Euclid Euclidian ian.. The They y are not even bidirectional, yet to describe them with algebra would require a considerable degree of mathematical skill for there are many circles in them arranged in a special displaced order. So, algebra is not the first choice for this process. Let’s look at the geometry. Figure 5-6. The magnitud magnitudee powers of  p  p

The two sets of circles are to show how different magnitudes are obtained from exactly the same geometric form simply by changing the choice of scalar in each set. How did he arrive at these magnitudes? Consider the Euclid’s theorem of  Figure of  Figure 5-5 5-5,, the internal chord theorem. It simply says that the opposing product of two crossing chords inside a circle are equal. If two of the chords are equal then the square root of the other chord product results in one of the equal chords. By applying that to the sets of circles you will see that the magnitudes are as shown. But, what is interesting is the effect obtained by exchanging the values of  p  and unity. Each value of  p  p n in the geometry of   Figure Figure 5-6 5-6’s ’s right 0 hand side is measured from the centre line and clearly  p is unity. Comparing the two geometries, one is for the values of   p p  less than one and the other side for values of  p  p  greater than one. The same shape suffices for both. That is the demonstration.

Note that these line lengths cannot be summed without separating them because the lengths are superimposed. This is a geometry of the second stage as mentioned at the end of  Chapter   Chapter 2. 2.

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Chapter 5. More than geometry

Figure Figu re 5-7. The field of an elect electric ric dipole dipole

There is another kind of geometry which is well known, the field an electric dipole (Figure ( Figure 5-7). 5-7). This time the geometry is of a vector character. The dotted lines are equipotential lines and the others are the strength of the electric field. the former is the direction component of some kind of vector which represents the electric field. (What kind of vector is this?) If the dashed lines are interchanged with the plain lines we have another vector of  this kind representing the field of a pair of parallel wires with currents flowing (Figure ( Figure 5-8). 5-8). This is the kind of  geometry that has its Directions orthogonal to its Magnitudes. Not forgetting that the Magnitudes have directions (here a set of circles with constant angular arcs) and the Directions have magnitudes (here a set of circular shapes with the pole at the limit point).

Figure Figu re 5-8. The field of a pair of paralle parallell wires

There is another kind of geometry in which the magnitudes are elements that oppose the directions. In Quantum Mechanical language they anti-commute. But there are no examples of these that can easily be given here. Suffice

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Chapter 5. More than geometry

to say that the study of the hydrogen atom displays such a geometry. In  Chapter 12 12 the  the geometry which produces Ostberger’s delta values is also one such geometry. In  Appendix B   the structure of geometric analysis is given; for geometry is as much subject to the rules of  Ostberger’s work as is any other subject. It is self analytic. In  Chapter 10 the 10  the structure of subjects is based on the principle of the Law Field which brings together the biggest possible concepts into a geometric form. The straight line becomes so special that it represents the Laws. The point is no longer; instead the micro-world takes its place. There are no points in a space of curvatures, only worlds of the smallest order and size. Our study of Directions began with Euclid’s geometries. There the lines are not directed. They have no arrows which point a way and there is no difference between forward and backward. The theorems of our school days describe these undirected line elements. From here we move on into the study of position vectors. These are directed line elements. These position vectors have transformations which operate on the vectors, making them enlarge, rotate, translate reflect and distort. These form a group as do the vectors themselves. “The mathematician Klein described Geometry as the complete set of all transformations over all vectors. I now add another set to this, the Interpretive set 4. In all I would describe these directed elements as "of the first kind"  making Euclid’s undirected geometry elements of the zeroth kind. The analogy with the magnitudinal study of  vectors would appear to be complete. The "scalars" are vectors of a zeroth kind. The vectors of force, motion and  momentum are those of the second kind 5.”

Curved vectors of the second kind are not commonly used in mathematics. Indeed their existence is scarcely admitted yet they are quite valuable tools. Vectors having multiple  heads   and   tails  are certainly new and not yet related to tensors as Ostberger has done. These are the vectors of the third kind. These four kinds form a mathematical group. It is interesting that Ostberger has produced a subject of study that is self analysing. He says, TAO This, in brief, is some of the work of my notebook. I have: Extende nded d Klein Klein’s ’s con concept cept of ge geome ometry try as con consisti sisting ng of the group group of all ttran ransfor sformatio mations ns ove overr the group group of all g geome eometries tries •   Exte to include a further group, the group of all Interpretations. The Interpretive group is over the Transformation group

and the Transformation group is over the Definition or Formative group •   Exchan Exchanged ged planes and straig straight ht lines for curves and curved surfaces surfaces making the former a very special case of the

latter. 3-dimension -dimensional al geometry to 44-dimensiona dimensional. l. •   Extended 3 •   Incorpo Incorporated rated the principle of ortho orthogonality gonality to all direc directional tional study study.. •   Added the principle of conve conversion rsion from intra intravariance variance to extra extravariance variance..

Notes 1.   [Note1254] 2.   [Note1250] 3. For the mathem mathematica aticall the these se are are g ij , gij   and gji  and their inverses. 4. See  Chapter 9 9 and  and [note220]  [note220].. 5. See  Appendix B B..

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Chapter Chapt er 6. The develo development pment of of standar standard d forms (rank one vectors) 6.1. Standard forms One of the tools that is particularly useful is the library of Standards Forms 1. Many equations can be represented in geometric form, many solutions can be geometricized, so it is helpful to create a library into which all the useful and sometimes useless geometries can be organised for future use. In engineering every process has a set of tools which can be called upon when a new job comes along. There are tools at every level of the company. The international corporation regards countries and governments as tools, the conglomerate regard sites and buildings as tools, the company regards computers and milling machines as tools and the operator regards keyboards and setting clamps as tools. Tools operate at every level. TAO  In the process process of building up these geometries I have established a small library of geometric standard forms. I would  like to show you some of these to give you some idea of how they assemble together into ever denser representations. These standard forms did not appear miraculously. They are the result of many painstaking weeks and sometimes months on a mathematical subject far removed from that which was relevant to the standard form. I have done no more than scratch the surface of a very large and unending subject.

To begin let us take one example and expose it as far as space and simplicity will allow.

6.2. The trigonometric forms There are many equations that result in trigonometric functions as solutions. We need to represent sine, cosine, tangent, cotangent, secant and cosecant in a geometric form which will be tractable for representing solutions of  equations. Also we need to be able to show in the geometry for each function,

•

  the amplitude

•

  the frequency

•

  the power

•

  the angle

•

  the multiple angle

We need to be able to add, subtract, multiple and divide all these aspects of the representations. Now, the set of Ostberger notes that describe all of these 2 occupies a space about the size of this book. Yet I am going to try to describe part of all these in a few pages. When I have finished there will be the cry.  “Well, that’s obvious. That doesn’t take much doing,”  and that is the point of the whole process. It contracts our learning methods into simple forms. Yet the time spent discovering these simple forms is immense and will only be understood by the student when he tries for himself. Building up the library of standard forms in Directional Study is the left-handed process which mirrors the process of establishing formula and theorems in mathematics today. The Directional process has the advantage that it

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Chapter 6. The development of standard forms (rank one vectors)

lends itself easily to presentation on the computer screen and can be used by a greater number of students for their applications.

6.3. The trigonometric standard forms We teach that the ratio of the sides of a right triangle are constant for each given angle and that these ratios have names like sine, cosine, tangent and so on (Figure (Figure 6-1 6-1). ). This is what Ostberger calls the  Directional form.

Figure 6-1. The directional fform; orm; the functions arise as direction directional al ratios

We may also establish these same ideas in a Magnitudinal form. Consider  Figure 6-2. 6-2. Here a triangle is bound by a circle of unit diameter. The angle at the circumference is always a right angle and the sine and cosine are always as shown. All the principle values of these two elementary functions are contained in the motion of the first half circumference. (But note that the angle θ  has only rotated a quarter of a cycle.) Here I will only show the first quadrant of the functions but in the notes are the complete set together with a computer animation of all the 3

functions . Figure 6-2. The magnitudinal fform; orm; the functions arise as line length

These two repres These represent entati ations ons are yin and yang. yang. Neithe Neitherr can cla claim im exclus exclusiv ivity ity.. The They y are com compli plimen ments ts of each each oth other er and serve useful purpose for different applications. The fact seems to be that this property of pairing Directional forms with Magnitudinal ones continues indefinitely. It has to be researched on every occasion. When a representation is discovered there will be the yin or yang counterpart somewhere. It is not always obvious or easy to find.

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Chapter 6. The development of standard forms (rank one vectors)

Figure 6-3. The magnitude form of tangent

In Figure In Figure 6-3 6-3 the  the tangent functions are added. In Figure In  Figure 6-4 6-4 the  the secant is added. The line lengths shown are always the magnitude of these two function when measuring the angle θ  as shown. Figure 6-4. The magnitude form of secant

We may continue in this way until we have produced the combined geometry shown in Figure in  Figure 6-5 6-5.. Here a rotating half vector is to be seen carrying with it all the elements in the picture as the point  P  traverses through the first half circumference. The angle  θ  traverses 90 degrees. Figure 6-5. Magnitudes of cir circular cular functions of the outer angle

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Chapter 6. The development of standard forms (rank one vectors)

What is to be understood here is that each line element representing a function is accurately the correct length. The limitation is the accuracy of the process used to construct the representation. The Direction of the function is  in  Figure 6-6 6-6.. The nomenclature must be followed  in Figure  Figure 6-5 or 6-5  or  O0  in Figure also correct with respect to the angle at  O1  in with precision.

Figure 6-6. Magnitudes of circular fun functions ctions of the the inner angle

What is the difference difference between the Figure the  Figure 6-5 and 6-5  and  Figure 6-6 6-6?? At first sight there seems little differenc differencee exce except pt that the angle of measurement has been changed from  θ 0   at  O 0   to  θ 1   at  O 1  (see  (see Figure  Figure 6-7). 6-7). What the student will find, upon careful study, is that the Figure the  Figure 6-5 is 6-5  is a discontinuous geometry whilst Figure whilst  Figure 6-6 is 6-6  is a continuous geometry. They can be assembled together in many different ways.

Figure 6-7. Directions of the magnitude magnitudess for the outer angle in the first quadran quadrantt

Figure 6-6 is 6-6 is mathematically wrong. It is not easy to spot, but it is. It is wrong to the extent that one might change the   x’s for   y ’s in an algebraic equation and expect to see the same graphical shape and solutions. This is an example of the preciseness with which we must view this process. It cannot be assumed that a geometry can exist

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Chapter 6. The development of standard forms (rank one vectors)

simply because we have drawn it. It may not be consistent or necessary or sufficient. Only when the geometry has all its stages (Appendix ( Appendix B) B) defined can we accept a given geometric configuration. In this case we need to define

the final stage, the stage of applying Directions to the magnitudes i.e. the line lengths. Figure lengths. Figure 6-7 6-7 is  is the final stage of production for the Trigonometric geometry. It makes it mathematically unique because all four aspects shown in in Figure  Figure 2-18 2-18 have  have been defined. The convention here is that one unit of amplitude is defined upward which means that in the bidirectional space of the representation minus one unit is downward in the same space.

6.4. Applying Euclid over Ostberger We may increase the depth of our understanding of the trignometric standard form (TSF) by applying Euclidean theorems to it. This is an example of using a lower density 4 geometry and applying it to an higher one. The lower density, the theorems, are undirected and unspecified in magnitude. The higher density, our TSF, is directed and specified. We can apply one over the other. Look at the figures in Figure in  Figure 6-8. 6-8. Here are just five examples of the use of Euclidean theorems to extract more information from the TSF.

Figure 6-8. Applying Euc Euclidean lidean theorems to the trigonom trigonometric etric standard forms

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Chapter 6. The development of standard forms (rank one vectors)

We have: Figure 6-8a 6-8a.

An amplitude  A  applied to the TSF of the outer angle. The student would see that this has the effect of scaling all functions relating to the initial scalar. This represents a big mathematical move. Were the amplitude applied to the TSF of the inner angle then this is simply a variable amplitude in a geometry of continuity. It is the fact the the outer angle TSF is a discontinuous geometry that makes  A  have such a big effect. In Figure In  Figure 2-7 of  2-7 of   Chapter Chapter 2 2,, for example, A would have the effect of changing every amplitude in the set. It is like the difference between a multiplier and an operator in mathematics.

Figure 6-8b. 6-8b.

Pythag Pythagora oras’ s’ theore theorem m applie applied d gi give vess a well well known known relati relation. on. Com Compar paree wit with h Figure 6-8e 6-8e where the external chord theorem is applied to give another relation.

Figure 6-8c. 6-8c.

Applying Applying Pythagoras Pythagoras yield yieldss this common common relationshi relationship p which can be seen to be satis satisfied fied for all angles including those greater than π  giving the negative magnitudes in Figure in  Figure 6-8d. 6-8d. Simply reversing the triangle in Figure in  Figure 6-8c 6-8c does not satisfy the conditions set. It would have the effect of changing the convention used. That is unacceptable. The formula at  Figure 6-8 6-8d d is currently considered to be trivial. Here it is shown to impart different directions to the representation. It belongs to a different region of the circle.

Figure 6-9. A proof of the reciproc reciprocal al relationship between tan tangent gent and cotangent

The interior Euclidean chord theorem shows that,   cot t. tan t   = 12 , which proves the reciprocal relationship between them. Two more examples before leaving this chapter. In  Figure 6-9 6-9 I  I have assembled two copies of the unit TSF, one above the other. The cot function belongs to the upper one and the tan function the lower one for a given angle. Because the large triangle containing sec and cosec is a right triangle the tan and cotan functions are contained by the large circle. We may apply the internal chord theorem to yield the fact that there is a reciprocity between tan and cot. An obvious fact in a new context.

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Chapter 6. The development of standard forms (rank one vectors)

Figure 6-10. A simple proof of the recipr reciprocal ocal relationship between between cosine and secant

The Euclidean chord theorem shows that,  a cos t.a sec t  =  a 2 , which says that there is a reciprocal relatio relationship nship   1 between cosine and secant, cos t  = sec t , which provides a simple proof of a well known relationship from earlier study of Euclid. 2

In Figure In Figure 6-10 6-10 the  the external chord theorem is used to find a similar relationship between cos and sec. Here we can apply the external chord theorem twice, once in each of the circles. These examples are mentioned here because some geometry requires a representation in which the forward and backward directions are the reciprocal of each other. Here are two reciprocal representations which may be useful in the future. It is currently common practice in mathematics to assume that the forward direction is positive and the backward is negative and that they do not overlap. This was Descartes I now see possibilities that this has,exist, unwittingly, been is a great hindrance to mathematics. In this chapter I have seededcreation. the idea that other not the least the idea that positive and negative may be represented as opposing and overlapping.

6.5. Amplitudes It should be quite evident that combining Figure combining  Figure 2-7 with 2-7  with the TSF of the outer angle will provide representations of all discrete increases in amplitude. The same with the inner angle will provide all the continuous increases in amplitude5. The geometric standard forms for frequency6 and multiple angles7 are particularly interesting because the geometry creates a unit circle rolling through the space of the multiple angle. Its meaning is elusive.

6.6. Powers of trigonometric functions

It is essential to realise that any power of sine or cosine is less than unity and that each successive power is less, in magnitude, than its predecessor. I am not going to examine the whole of these notes here nor parade through

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Chapter 6. The development of standard forms (rank one vectors)

the discovery process. I reproduce one of the resultant library forms for the powers of sine and cosine in in Figure  Figure 6-11.. This is reproduced from the Ostberger notebook. 6-11

Figure 6-11. The inside powers of cir circular cular functions belonging belonging to the outer angle

By extending the process of discovery here to the outside region of the bounding circle we find the secant and cosecant powers in Figure in  Figure 6-128.

Figure 6-12. The outside powers of cir circular cular functions belongin belonging g to the inner angle

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Chapter 6. The development of standard forms (rank one vectors)

6.7. Half angle tangent relations These relations are well known in A-level mathematics but Ostberger extends them. The page in  Figure 6-13 is 6-13  is 9 a copy from Ostberger’s notes which geometrise the half angle tangent relations and extends them to interesting and potentially useful geometries. The geometry shows that there are directional and magnitudinal cases of these relations and connects relations connects a unique and infinite series of them. The mathemati mathematics cs and geometries geometries are some 40 pages in all.

Figure 6-13. A copy of the half tangent study from the Ostber Ostberger ger notebook

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Chapter 6. The development of standard forms (rank one vectors)

6.8. Algebraic equations From the earliest days of algebra processes had to be developed for finding the roots of equations. The methods of factoring, finding solution formulae and trial and error are all a part of the history. That it is possible to use a geometric method may not be new but such a process has not been accepted as enhancing the understanding of  the algebra process. What does seem to be useful is that the geometric process using lines in space can lead more easily to bigger concepts. Thus, if we solve quadratics10 and cubics11 with a Directional process then it might be possible to solve quadrics and other higher powered polynomials 12.

6.9. Parametric equations The representation of parametric equations lends itself easily to the direction process.  Figure 6-14 is 6-14  is the normal form   x cos α  + y  +  y sin α   =   p  for the Cartesian equation   3x  + 4y   = 7. It is a specific case of the more general form described in [note90] in  [note90] (14  (14 pages). It is particularly useful in the learning process because it relates, in a proper pictorial theinrelationship a Cartesian space to its outer linear therefore of littleway, value handling theofcurvatures of inner later chapters here. Thespace. circle Itisis, thehowever same circle of and the unit TSF. To relate this mathematically the student will see that the parametric equation must be the negative case −x cos α − y sin α   =   − p. The reason for this is the labelling of the point  O 0 , which is essential for the measurement of the angle α  as shown, makes the sine and cosine negative and in the second quadrant. Figure 6-14. The normal normal form

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Chapter 6. The development of standard forms (rank one vectors)

6.10. Imaginary geometry Because the square of the imaginary quantity √ −1   produces a negative magnitude we need to examine just how geometries with imaginary values actually work. The Pythagorean hypotenuse is not necessarily the longest length. Ostberger has made a study of these 13. They are essential to the study of hydrogen. Next we need to look at the much bigger ideas which lead Ostberger to use reasoning as the basis for constructing pictures which are mathematical representations. Eventually we will arrive at two demonstrations of the process and theory. One is the calculation of the Sommerfeld fine-structure constant and an associated set of constants. The other is the modelling of the hydrogen atom14.

Notes 1. We need to bear in mind that this book has been created because because there has been no way of expres expressing sing Ostberger’s work through the conventional journals. Geometric operating is not acceptable in these journals. Indeed it would not even be possible to get the pictures published because they are not conventional mathematics. The resistance to new work is prevalent in all professions during the twentieth century and no mechanism has been set up to allow new work to entry the wisdom of mankind. Such wisdom is still the domain and privilege of the high priests. 2. The trigo trigonometr nometric ic notes are found found in notes 110-120, 110-120, 501-540 501-540 and 1200-1221. 1200-1221. 3. This anima animation tion is interesting interesting because because the vectors cover cover the region region enclosed enclosed by the circle twice twice,, yet the angle theta is covered only once through 360 degrees. It is a Riemann two-sheet. 4. See  Appendix B B.. 5.   [Note504] 6.   [Note112]. [Note112].

7. See  [note512]  [note512],, [note513] and  [note513]  and [note514]  [note514].. 8. These ttwo wo geom geometries etries are are the culminati culmination on of five months months work. work. 9. See  [note1206]  [note1206],, [note1207] and  [note1207]  and [note1210]  [note1210].. 10. See [note98] See  [note98].. 11. See [note96] See  [note96].. 12. Ostberger Ostberger predict predicted ed that all polynomial polynomial solutions solutions lie in a single single bidire bidirection ctional al plane although although there is no evidence that he proved this. 13. 13. [Note1230]  [Note1230] 14. The Hydrogen atom is presented in a separate book. It examines the process of arriving at the model through the Schroedinger equation and compares an almost identical model arrived at through Dirac’s equations of  vectors.

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Chapte Cha pterr 7. Four Four dimensi dimensions ons are are here If it were not for the fact that a four dimensional geometric representation was possible I do not believe that Ostberger would have continued his studies. This is now introduced so that the student may also be motivated.

7.1. Four dimensions There are several uses of the word  “ dimension”  in mathematics and so it is essential to iron out their meaning. Appendix D does D does this. In this chapter we are looking at the dimensions of the Einsteinian kind. Of course the wun-man was right, unless we can discover and understand the four dimensions of our universe we will always live in disharmony. It seems that we must understand a fourth dimension to be able to live stably in a third dimension. It is fortunate that we have a geometric picture that allows us to do this and that Ostberger describes it and applies it as proof of its properties. Mathematics is a specialist’s task and I am not going to enter into the detail here. I explain the directional aspect with just sufficient mathematics so as not to discourage a wider audience 1. Let’s look at the surface only of a spherical geometry like the one shown in  Figure 5-4. 5-4. We may colour the elements red, green and blue.

Figure 7-1. A W World orld point

The line elements in this sketch have no thickness and so cannot easily be said to be orthogonal. To describe this surface we will need three coordinate numbers. It cannot be done with two as can a spherical surface. The sphere as a volumetric object can be described with three coordinate numbers. The Figure The  Figure 7-1 7-1 needs  needs four coordinate numbers to describe, uniquely all the possible points. The question is  “are they orthogonal coordinates?”.

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Figure 7-2. Cylindrical elements

Look more closely at the junction of the line elements, Figure elements,  Figure 7-2. 7-2. The three coloured elements meet at a point 2 on the surface. surface. If they are cylindrica cylindricall elements elements they look like Figure like Figure 7-2. 7-2. If they are annular elements they look  like Figure like  Figure 7-3. 7-3. In each case there are two small sets of orthogonal coordinates formed by the meeting. One set is

formed by the  normals  to the elements and the other by the  bi normals.

Figure 7-3. Annular elements

If we were to remove the elements with their thicknesses we would not be able to identify the difference between the annular and cylindrical sets. But with the coloured elements present we can see that in Figure in  Figure 7-2 the 7-2 the thickness of the elements are in the direction of the bi-normal whereas in Figure in  Figure 7-3 it 7-3  it is in the direction of the normal.

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Chapter 7. Four dimensions are here

Figure 7-4. Measuring the d directions irections of the su surface rface

Let us take a trip over the surface with an orthogonal indicator, Figure indicator,  Figure 7-4. 7-4. A little practice and the student soon discovers that the surface is rather strange. for one might expect that the indicator would rotate about its own axis as it is cast around the surface. In fact, what happens is that the red, green and blue directions of the indicator remain in the same direction all the time. Wherever we go over the surface the indicator faces the same way. Try it and see. So what are we measuring and how do we know where we are on the surface? Let’s go back to the pen stand note of  Chapter   Chapter 2. 2. There we saw that there were four possible spaces in the one quadrant. Each plane was defined by a 90 degree orientation of the pen until the pen was back home after twelve moves through twelve planes. But actually there are 24 moves because the pen can have an opposite direction i.e. the clip could be pointing in or out. So there are eight 3-dimensional spaces in the quadrant, four inward and four outward. These spaces are the quadrants of the world. There are eight quadrants on the world which correspond to the eight spaces of the pen stand. The permutation of the directions is  4  x  3  x  2 = 24  which is 3 ways on each of  8 quadrants. The three ways are toward each pole of the world from the  W 0  point. The W 0  point is that one which subtends equi-angular measure from each of the great circle planes i.e. in the centroid of the forward quadrant. But how do we know that we have moved around the world surface? Our coordinate indicator hasn’t moved at all. It still lies in a fixed position as if constrained in a field like a compass 3. The answer lies in supplying the surface surfa ce elements of the world with some directional directional information. information. This will then produce produce a result result that resembles resembles the parametric representation of  Figure   Figure 6-14 but 6-14 but in three lots of two dimensions known as the Hessian form 4. The way of identifying each of the eight quadrants is by rotating the world surface and observing the indicator change its spacial orientation. The juxtaposition of the three colours red, blue and green are different in each quadrant when the rotation is made (for example Figure example  Figure 7-5). 7-5). These eight quadrants quadrants correspo correspond nd to the eight quadrants of  the pen stand. Four with the outward pointing pen and four with the inward pointing pen.

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Chapter 7. Four dimensions are here

Figure 7-5. Numbered octants

There are several ways of defining the magnitudes and directions belonging to the world surface. Each has its uses but some are special5. The curvatures here are described by the Gaussian curvatures 6. The three coloured elements have curvature which are tangible and which would have centres of radius somewhere along the straight line pole elements. However, according to the rope-maker, we are not allowed straight lines in the geometry, so the pole elements must be curved. These will turn out to represent the geodesic nulls of tensor calculus and the null identities associated with vectors7. But, where is the fourth dimension? The surface of the world includes the representation of the fourth component which appeared in the pen stand, the rotation of the pen. In the world surface we can also see a magnitudinal component of the fourth element. Look at the Figure the  Figure 7-4 7-4.. Consider the motion of the world as it enlarges; and there is only one, non trivial, transformation of this geometry, enlargement. As the enlargement takes place the orthogonal indicator at the  W 0  point would remain in the same juxtaposition if the pole elements were straight (inset in  Figure 7-4). 7-4). However they are not. They are curved. The orthogonal indicator rotates with the curvature of these polar elements. It is an  internal rotation of the orthogonal indicator at the  W 0  point. There is another rotation taking place as the enlargement proceeds. This is to be seen in the Ostberger sketch in the lower left of  Figure   Figure 5-2. 5-2. As the world expands the tangent and the normal to the surface rotates albeit that the plane of rotation is a geodesic plane8 and not one of the planes of the coloured elements. This is an external rotation. So what is the total rotation of the world geometry in this fourth dimension? Ostberger says that in proceeding from the first (unit) circular element to the last curved element before infinity the rotation is almost π/2 π/2. It consists of almost π/  π/44 for the internal rotation and almost  π/  π/44 for the external rotation. Is it orthogonal?

7.2. Observational platform The reality of our universe is that we see three dimensions in the locality in which we are. So we see the three axes of the orthogonal indicator in Figure in  Figure 7-4 as 7-4  as our measuring tool. We easily miss the fact that the other rotations are taking place at all. The practical, mathematical method of making these observations is to imagine that we are standing on the line elements and make the calculations from our external view of ourselves standing on the lines.

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Chapter 7. Four dimensions are here

7.3. Quarter points There are certain places in the geometries that seem to give better answers than others. The quarter points are those that are 45 degrees from the measu measuring ring elements. elements. The most importan importantt quart quarter er point is the  W 0  point as this 9 underlies the basis of the measurement of the whole space . Consider Figure Consider  Figure 7-6 7-6.. If this is representing a vector space then the upper geometry relates to a stationary observer watching the quarter points moving out into space. The two observers are standing, one at the centre of his own little world, the limit point labelled  “our observer  view-point B”, and the other at the infinite element on the same axis, generally called the  z  axis. If this is representing a vector space then the upper geometry relates to a stationary observer watching the quarter points moving out into space as we move at constant speed. They track a straight line as in Figure in  Figure 7-6 7-6.. If, however, the observer-point is moving (which is an accelerating observer) the quarter points rotate and the track is a curved line. Asymptotically they rotate almost  π/  π/44 throughout their journey. It is for this reason that Ostberger says, The Lorentz transformations need to be reinterpreted. There are several effects of the Lorentz transformations10. One is the Fitzgerald contraction. When an object, which includes a human being, travels into space at ever greater speeds his physical character changes. The old interpretation says that he contracts in the direction of his travel.

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Chapter 7. Four dimensions are here

Figure 7-6. Rotations associated with th thee Lorentz tranformation tranformation

The new interpretation says that he does not; his physicality does not change. What does change is the measurement  of our observation. We record correctly that we see a contraction of the parallel length because we see through our  eyeball into our head that he is moving out of our local observational region where everything is measured equally in all directions. What the journeying human being is doing is to join in with one of nature’s laws, the law of velocities 11 , for velocity is a  part of nature. We We have no copyright on it. It belongs to nature. If we join in with her we must accept her rules and travel on a curve into space. Her rules say that the traveller appears to contract. That is only an observation not a reality. It is

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Chapter 7. Four dimensions are here an observation and therefore imaginary, not real. The journeying human being observes us the same. He sees a contraction because velocities are relative and he does not  know which of us is moving or stationary. Neither do we.  Another effect of the Lorentz transformation transformationss is the slowing of clocks. The new interpretatio interpretation n says that two clocks separated by a velocity do not change their time, but we must take account of the fact that we will observe and measure the other clock as being slower by the amount of the Lorentz ratio.  Another effect is the chang changee of waveleng wavelength th of light that appr approaches oaches us from the universe universe.. The lengthening of the wavelength means that we observe light to be shifted towards the red end of the spectrum. The old interpretation of this is that  a Doppler effect is taking place and that the further away we observe the more red shift there is and therefore the faster  the objects are travelling. This leads to the Hubble constant which says that our universe is expanding at a phenomenal rate. That we are on the surface of a spotted balloon and the farther out we see the bigger the balloon. The bigger the balloon the faster the separation of the spots on its surface. This interpretation would be fine if we observe the universe in straight lines from our earthly home and ignore relativity.  But relativity formerly formerly identifies the Lor Lorentz entz Tr Transforma ansformation tion with a rotation in Mink Minkowski owski space i.e. the space of my world  geometry. The form of representation is independent of the particular directions in which the chosen set of orthogonal axess hap axe happens pens to poin point, t, so the Mink Minkowsk owskii repr represen esentatio tation n ensu ensures res that the equ equatio ations ns of rel relativ ativistic istic mec mechani hanics cs is independ independent  ent  of those in which the Lorentz observer is involved. All equations moving with constant relative velocity thus use equations of the same form to describe the optical and mechanical phenomena that they measure. We certainly observe the correct results in our universe. However we have omitted to take account of the four dimensional 12

 Minkowski world rotation that I have described.  Minkowski described. This rotation is real.  It is a part of natur naturee and we cannot observe our  universe as if we belonged to a world outside of it. We belong to this universe and so the rotation must be taken into account. The red shift is a measure of this rotation. As we view our universe we are looking round a corner. The greater  the red shift the more we see round the corner. At the speed of light we see almost 90 degrees round 13.  Looking out on our universe in radially straight lines is analogo analogous us to observing the Earth as if it were flat. Those days soon passed once the new worlds were discovered. Of course, there is more than one corner to peer round. I am wondering whether the types of constellations that we observe are related to the different curvature curvaturess of our observatio observations? ns?14

7.4. Standing in another place Perhaps the only really important discovery that Ostberger made was that there are a pair of four dimensional (Minkowski) worlds.

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Chapter 7. Four dimensions are here

Figure 7-7. Rotating the W World orld

We need to study carefully how the directions of the world behave. Look at the surface in  Figure 7-7. 7-7. As we move over the surface in a counter clockwise direction (a) at the N-pole becomes a clockwise direction (b) at the back-N pole. On the other hand a rotation of the world  π  radians leaves the directions counter clockwise (c). Identifying the types of rotations is important. All these rotations have no affect upon the World. The N and back-N identifiers determine the direction of the surface and not the Cartesian rotation that we have done.

Figure Figu re 7-8. Curv Curves es in the planes of  “straight”   “straight”  nulls  nulls

To understand the transformation that takes us from intravariant to extravariant (which we will see again later in the Law Worlds) we need to stand at a position somewhere outside the world surface. What we see is the external

elements which are orthogonal to the surface extending out into the space around the World (Figure ( Figure 7-8 7-8). ). But do these curves belong to a common surface? If they are the curves belonging to  Figure 7-9 7-9 then  then they are not in common because because they are parallel to the sames planes from which they eminate. However However,, if they are curves belonging to Figure to Figure 7-10 they 7-10 they are part of a common surface because they are orthogonal to the planes from which they eminate. The circles in the illustrations attempt to show this. These curves come from a geometry in which the pole elements are curved. The three external curved elements now rotate as the World enlarges (see Figure (see  Figure 5-2 showing a section through the enlarging World.)

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Chapter 7. Four dimensions are here

Figure 7-9. Curves not in orthogonal planes

Figure 7-10. Curves in the movin moving g and rotating orthogonal planes

It was this new surface that surprised Ostberger. He notes,  “I must be standing in a new world. A world in which all of the imaginary region that envelopes the outside of the one I am looking at 15 will pass over my head as the

world enlarges. enlarges. If I can find a way to stand on the outside of this new world I will see a world with an imaginar imaginaryy 16  region inside which is enveloped by a real region outside. ”

He did not work out that these two world were orthogonal for many years. He simply thought that they were inside out to each other. It is difficult to show this transformation on a single piece of paper. There are six pages in the notes.

Notes 1.   [Note1120], [Note1120], [note1121] and  [note1121] and [note1122]  [note1122],, amongst others, contain the mathematics of the geometry. 2. Wit With h the discovery discovery of the extravariant extravariant geometry geometry that assembles assembles back on to its counterpart counterpart the intrav intravarian ariantt geometry (in Chapter (in Chapter 10) 10) the need for  points in a curvature space completely disappears.

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Chapter 7. Four dimensions are here

3. I have written it this way to show show this as an example example of extravariance. extravariance. We We do not see the world world existing becaus becausee our indicator shows no change at all when passed over the surface, yet we may sense its presence. But when shown in this form it is clear that the world exists and possesses the 3 dimensional properties mentioned. 4. The Hessia Hessian n fform orm is is [note409]  [note409].. 5.   [Note250] 6.   [Note1021] 7. For exampl examplee Div Div u Div  u  is identically zero. 8. The geod geodesic esic plane plane is the one that w would ould pass pass through the the centre of the world world if it had one. It does not  have   have a centre in the conventional sense because the pole elements are curves. In fact it will pass through a limit point. On the other hand when the pole elements are straight there are great circles passing through a centre. The World then has six conic geometries. 9. This accor accords ds with mathe mathematic matics. s. 10. 10. [Note1110]  [Note1110] and  and [note1111]  [note1111].. 11. Velocity is one of the elements of a Law Field. See Chapter See  Chapter 10 10.. 12. It is real in an intravariant space and imaginary in an extravariant space. 13. See Appendix See  Appendix E. E. 14. Ostberger makes notes of this. He reverses the roles of the intra and extra worlds in a geometry of the second kind with a view to investigating its relation to this statement. 15. He later called this intravariant. 16. He later called this extravariant.

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Chapter 8. The develo Chapter development pment of of Law Field Fields s (rank two vectors) 8.1. Law Fields TAO  I did not invent the Law Field. It grew out of the studies I was making in 1969.

It is difficult to know how to introduce a subject which is as new as this one. I have decided to present the Law Fields just as Ostberger constructed them and then show how they work. There areField two ways of seeing One up-view, which bythat mathematical the Law diagram worksthe andprocess. the other is is thethe down-view, whichisistotoshow show the results reasoning produced that are successful. In the sensible space available I will do some of both. Let’s look at two Law Fields. The first the Newton Law Field (LF) in  Figure 8-2. 8-2. The second the general case in Figure 8-1 which 8-1 which will serve as the pattern for further examples.

Figure 8-1. The generalise generalised d Law Field

You need to recall that straight lines were extremely special. They are so special that they do not exist in the representation of magnitudes. Only the directional component of a vector can be represented by the straight line 1. Even then, the straight line is so special that it represents the exception to the rule. It turns out that the exception is the rule!

8.2. The Newton Law Field Consider Newton’s first Law,  “Every body continues in its state of rest or motion unless a force is applied to change it.”

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Chapter 8. The development of Law Fields (rank two vectors)

So, a body continues in a straight line when there is no force field, it says. That is the law; a straight line 2. And that is how Ostberger represented it. It later turned out that this same straight line was in fact the gravitational potential 3 of physics. He labelled it Direction  Z g  because that is what the line representing gravitational potential really is. It is the part of the LF which is most Directional in character. In order to obtain a magnitude one must measure the radius or the curvature4 (which is the reciprocal of the radius). But measuring along the line of the potential yields nothing. It is this nothingness that makes the phenomena a candidate for being described by the idea of  absorption (Figure 8-2). 8-2). Absorption takes the total representation of both the positive and negative directions and says that they must always be zero. This is described later in Chapter in  Chapter 9 in 9 in several different ways including a matrix method.

Figure Figu re 8-2. The Newto Newton n Law Field

It remains useful to refer to this law element as either the Direction belonging to the Newton or gravitational Field, or the Potential of it. The next element has the property of  reciprocity  or Inversion. It took Ostberger many years to realise that reciprocal velocity was an admissible idea. He admits that he included it in the first instance because all the other Law Fields contained the same pattern and not because he deduced it. But later he was able to show that it is the correct deduction. We must bear in mind that there seems to be two kinds of velocity. One belonging to the World of Fluidics and this one. Here the velocity creates a thing we call momentum and that gives us the clue to the reciprocal nature of  velocity. In the World of Fluidics the velocity is internal and may have a different property 5 to this one. This one implies that, “to every momentum there is a contra-momentum 6 ”. This is really a re-statement of the law of conservation. In this case a conservation of momentum. The real evidence for a reciprocal velocity lies in the depths of the atomic structure where it begins to appear as a contra for the energy in the Neutron. If engineers and designers really understood this law we would not have the wasteful engine designs of today7.

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Chapter 8. The development of Law Fields (rank two vectors)

The third element of the Field is evident from the Newton’s third law,   “to every force there is an equal and  opposite reaction”. Newton forces are quite real. They can be measured and quantified by the real effects upon our bodies and our instruments. But what about the reactions? A man stands on the floor. His reaction is in the floor but what he feels is the force in his body, it presses hard upon the soles of his feet. We cannot get at the reaction and say  “here it is”. Its measurem measurement ent is an assumptio assumption n from the measurement of the force. The reaction is imaginary. The reaction is imaginary and so is the whole top half of the Law Field.

8.3. The origin Unlike a Cartesian representation, this one has different numbers approaching the origin from different directions.

1. The Directions arrive at the origin in the two numbers zero plus and zero minus. 2. The Velocity Velocity arrives at the numbers one and its inverse inverse.. 3. The Force arrives at the origin in a singularity. A real singularity and an imaginary one each the reflection of  the other. The Law Field has the special characteristic that the numbers  0, 1  and a singularity (near infinity) arrive orthogonally at the origin. And that is very special indeed.

8.4. The relations At the simplest level we may observe that, if there were a large (compared to the electron) mass in the field then, 1. A rate of change of a direction with respect to time le leads ads to a velocity. velocity. 2. A rate of change of a velocity velocity with respect respect to time leads to a force. It is as if the first is necessary to generate the second and the second is necessary to generate the third. They seem to be contained in each other in a generic order like the parts of a car that build into sub-assemblies. There also seems to be two cases of each of the laws. One magnitudinal in character and the other Directional (see Appendix (see  Appendix F). F). I need to remind the reader that there are no Magnitudes in the Law Field. It has no size. Neither will there be magnitudes in he Law Worlds later. It is not until we begin to use the process that we need to apply the magnitudes in a formal way. There are also no rates of change. Even enlargements are not easily depicted, at present. Let’s look at two more Law Fields belonging to the World of Grav-electromagnetics (GEM).

8.5. The magnetic Law Field There is a general pattern to these Law Fields which will become self evident as we progress. In the magnetic LF8 there is an intensity of magnetic origin which we designate  H  which   which exhibits a force F m  (Figure  ( Figure 8-3 8-3). ). There is also a magnetic potential which has the same absorption properties as the potential in the gravitic LF. There is also a flux, we designate B , which has properties which are isomorphic to velocity in the gravitic LF. There is also a potential, one of a magnetic kind.

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-3. The magnetic Law Field

This Field is on a different level to the gravitic one. It is a field of  firstness  firstness9. The Newton Field is one of  thirdness  thirdness.

8.6. The electric Law Field The field of secondness is the electric Law Field. It is shown in Figure in  Figure 8-4 8-4.. All its properties are well established in the literature.

Figure 8-4. The electric Law Field

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Chapter 8. The development of Law Fields (rank two vectors)

This is a field of action. We cannot grasp the secondness fields. They are the operators; the players in the game. The transformers, relators, movers, actors; they are the doers in the set. The magnetic LF is the script of the play,

the formulation or specification, that sets the scenes and derives the plots within which the drama will take place. The Newton field is like the audience who interpret the play and upon whom the judgement of the performance will rest. Like all the other fields they have rates of change and enlargements which are not shown in the LF diagram but can be worked in the form of the Law World where the curvatures become evident. We will assemble these Law Fields into Law Worlds in  Chapter 10 10.. In the meantime let’s look at the general Law Field of  Figure   Figure 8-5. 8-5.

8.7. The general Law Field Remember that in these studies the Directions of the things that we study are just as important as the size or Magnitude. This is the case in reality. For example, suppose my friend and I were going to London and we want to chance a bet on who would get there first. We might assess each other’s transport and consider its performance. We might also consider the chance of failure or the skill in handling the transport. The one thing we are unlikely to do though, in our present way of living, is to consider the map reading skills of each other. We would assume that we both knew the directions. But what if we did not know the direction or we did not have a map or a map did not exist? Would our assessments of each other’s transport be of any value? If one of us did not know the direction to London then the bet has a high probability of being won by the other person. The direction becomes the the salient parameter in our assessment of each other. Now, in nature there are directions, albeit of a different kind the to the London one, but we do not know where they all are. Indeed it is doubtful whether any of them are recognised as directional phenomena at all, and yet it is clear from our betting game that the Directions of nature may be more important than her magnitudes. So too for the directions of our social activity, perhaps. The Law Field acts as a guiding tool. Ostberger demonstrates its use in many examples, the most significant of  which are the representations representations of the Hydrogen10 and Helium atoms. It brings together in a single diagram three essential principles. Each principle is represented by a line element orthogonally. Each has a contra-principle represented bi-orthogonally (opposite).

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-5. The general pr properties operties of a Law Field

The origin of the LF has six magnitudes depending upon the direction of approach. Along the Absorption11 axis the origin is approached in 0+ or 0- (Figure ( Figure 8-5c). 8-5c). Along the Reciprocity or Inversion axis the origin is approached in 1 or 1/1 (=1) and along the Conversion or Real/Imaginary axis the origin is approached in a real or imaginary singular point12. I will explain these in Chapter in  Chapter 9. 9. It is to be understood that these  axis principles  are very much the starters, the prologue to the drama. We must find a more exact basis on which to tease out a solution to a problem or to make a representation. Thus in Quantum Mechanics we find the Unitary transformations  13 which fit the LF and which can then form a basis for a geometric representation. But we still have a long way to go, even from there. The Nomenclature X, Y, Z in Figure in  Figure 8-5a 8-5a is not to be confused with the Cartesian representation which does not

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Chapter 8. The development of Law Fields (rank two vectors)

appear in the geometries at all except to make the connection between the current mathematics and the Ostberger process. When mathematics, in the sense of present-day equational mathematics, is used with the geometries then the Law Field diagram has to be transformed (rotated for example) in the space of the room to permit the ordinary Euclidian space to be related to the Law Space. In doing so, it becomes apparent that some of the Law space cannot be measured (represented) because we have run out of dimension (Einsteinian). Another way to see this is to understand that the Law World fully occupies the space of 4 dimensions. There is no room for any more line elements. Any attempt to occupy the space with any other dimension such as a Euclidean line element extinguishes the availability of some other element in the geometry. This makes it unavailable to sight or size. Measurement cannot be made. This is Heisenberg’s principle.

8.8. Continuity and discontinuity Many of the phenomena we see around us are divided into Continuity and Discontinuity. We humans are examples of this because internally we are continuous whereas externally we are discontinuous. Our organ, cell, lymphatic, skeletal, muscular, motor, nerve and mental systems work together as a whole. They are part of an holistic process; all work together for the well-being of the organism, none are privileged. Externally, on the other hand we are separate bodies, we live separate lives with laws of discreteness14. Continuity and Discontinuity vie with one and the other in life 15. So also in the geometry as we shall see in the later chapters of this book where extravariant Worlds anti-commute with intravariant ones. Here in the LF (Figure 8-5b) 8-5b) one axis is always continuous and another discontinuous. There is a third which is the real and the imaginary of which we are all consciously aware. There is another which is seen in the 4th dimension: the process of Condensation which converts the imaginary from outside to inside and begins a new process of discovery.

8.9. Matrices With the discovery With discovery and use of the Absor Absorbing bing matrices Ostberger Ostberger puts the remaining kinds of matrix into the LF form. In mathematics the most commonly used matrix is the reciprocal or inversion kind. These are used for solving multiples of linear equations, for example. The absorbing matrices are new and are explained in  Chapter 9 and 9  and [note100].. The Conversion matrices are few; they contain both real and imaginary elements in the same matrix. [note100] On Condensation Matrices Ostberger writes, “I cannot see how these will work particularly as we are not at ease with the singularities of mathematics. However I am hopeful that the geometry will help.”

Let us look at a few more Law Fields.

8.10. The Law Fields of number These are the Numbers that make the things we call Magnitudes, the sizes of things, the bigness or smallness, the amplitude or the length. These are the concepts with which we measure our universe, that lend uniqueness to the curvatures of space whether that be a Gibb’s ensemble in thermodynamics or a Loan space in Finance. This kind of Number is called  Ordinal. They describe the magnitudes 16. There is another kind of number called Cardinal . They tell us, not about the bigness of things, but about their state of order. They are labels.

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-6. The number Law Field (firstness) (firstness)

1. Integ Integer er number are wholesom wholesome. e. They cannot be broken down into parts. In a binary system system there are only two. Integers have no reciprocals. 2. The rational number set has a pair of zero at its origin. One belonging to the positive rationals and the other to the negative rationals. 3. The point point at the origin is no point at all. It turns out out to be a world of number number.. 4. The set of irrational numbers is spe special cial because there is a contra sset et which are reciprocal. They They are continuous in their nature. Our decimal point expresses the nature of this reciprocity. 5. Imagi Imaginary nary numbers are all prefixed with the square root of minus unity unity.. Although they are  imaginary  they are none the less essential in solving real problems. Cardinal numbers describe the possibility of a face turning up on a dice (1 in 6 or 1/6th) or of picking a spade from a pack of cards (13/52nds). They label the cars on a race track and specify their winning order but they say nothing about their size. These cardinal numbers belong to the extravariant number World 17. I am not going to describe these here. The notes give a description and definition of the fields. No doubt they will cause discussion and maybe dissension but that is what Ostberger is about; causing debate and discourse. What we have here are the sets of number that are well recorded in the literature. •

  N, the set positive positive integers integers

•

  Z, the set of integers

•

  Q-, the set set of negative negative rational numbers

•

  Q+, the set set of positive positive rational numbers

•

  R , the set of real numbers numbers

•

  C, the set of complex numbers numbers

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Chapter 8. The development of Law Fields (rank two vectors) •

  I, the identity 0, 1 and a singularity

•

  S, the set of irrational numbers

If this Law Field is correct then the sets Z, N and S have no zero and a singularity is a form of Identity. The LF of secondness is about the operations that we can do with numbers. Although it has the same isomorphic patterns as the first Field it is very different in so far as it cannot be contained, just like the Electric Law Field.

Figure 8-7. The number oper operations ations Law Field (secondness) (secondness)

In one of his early notes Ostberger exposes his concern at educating children into believing that all operations commute, that A   B  =  B   A except in higher mathematics; when in fact the case is completely the reverse. Only when we are using pure numbers does the commutative law work. In all other cases   A   B  is not the same as B   A. Thus 3 bags of four apples is not 4 bags of three apples. The bags may be as important as the apples. Yet we are teaching children to think solely of the magnitudes and launching them into a yang world without a yin perspective. There will come a generation that will despise their parents foe hiding such facts.  

 

 

 

The fact is that the purity of number is so special that it is common to all homosapiens on the planet.

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-8. The interpre interpretation tation of number Law Field (thirdnes (thirdness) s)

The third LF is Figure is  Figure 8-8. 8-8. This is the field with which we interpret number. Thus we lay them out in indicial order using the addition operator in a base of 10, 2,345 is  2   103 + 3   102 + 4   101 + 5   100 .  

 

 

 

We could use any base. For computing we us a binary system with base 2. At school we learn about bases, number rings and clock numbers. These appear on the Law World which we will see in  Chapter 10. 10. The Imaginary halves of these three Law Fields are worth mentioning. In the first field there is a whole region of the complex numbers. In the second field the imaginary region belongs to the operators which are themselves imaginary such as is used in the study of electric currents where  j  is the usual symbol. In the third Field there is a question mark. What are the imaginary bases of number?

8.11. The Law Fields of thermodynamics I can remember spending many hours as a student plotting engine diagrams. I later saw these in practice in the aircraft industry where the performance of turbines and jets were displayed on three dimensional diagrams. I wondered what the connection was between the Pressure-Volume diagram and the Temperature-Entropy diagram. I was delighted to find the connection in the Law World of Thermodynamics. The fact that the P-V and S-T diagrams lived in orthogonal planes, were on different levels 18 and had curvatures helped me to understand why so many thermodynamic devices deviate from the theory as the parameters of  volume, pressure, frequency and temperature become extreme. The distortions are an inevitable part of the natural curvatures belonging to this World. I am now going to show in a couple of pages the essence of Ostberger’s work in Thermodynamics which took  him ten years and more to discover. Discovery that is far from finished. Indeed it has only just begun. TAO  At first I thought I had discovered discovered something very final, something that would remain at the foundation foundationss of future education and science. But I soon realised that I had only scratched the surface, that there was so much to do and build that 

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Chapter 8. The development of Law Fields (rank two vectors)  I despaired despaired that I would ever get it down on paper paper.. Only with the advent of the computer computerss did my happiest realisations come to fruition. At last I could manipulate the geometries on a computer to match the visions that were in my head.

This is the order of the Thermodynamic World. It is a differential World.

Figure 8-9. The work law field (thermodynamic firstness)

•

  The absorption principle for volumes I explained explained earlier in this book. V Volumes olumes do not move discretely; the they y move in a differential manner. These are sometimes referred to as internal and external volumes.

•

  I do not understand the three kinds of temperature. They appear on the Law Law World World as fluxes belonging to three different generations.

•

  Dalton talked of pressure probabilities. The The react to the real pressures in a sstatistical tatistical way. way.

•

  The fourth field her is Enthalpy Enthalpy..

Table 8-1. The Work Law Field (firstness)

Volumes

remaining and consumed

Pressures Temperature 1

real and probable of the first kind and its reciprocal

 

Absorption

   

Conversion Inversion

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-10. The organisation law field (therm (thermodynamic odynamic secondness) secondness)

Table 8-2. The Organisation Field (secondness)

Entropy

internal and external

Internal energy

U and its statistical counterpart rt??

Absolute (Kelvin) Temperature of the second kind and its reciprocal

 

Absorption  

Conversion  

Inversion

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Chapter 8. The development of Law Fields (rank two vectors)

Figure 8-11. The reation Law Field (therm (thermodynamic odynamic thirdnes thirdness) s)

Table 8-3. The Reaction Field (thirdness)

Mass fraction

participating and remaining

Chemical Pot Potential

and its likely reaction

Temperature

of the third kind and its reciprocal

 

Absorption

 

Conversion  

Inversion

TAO There were several aspects of the World that surprised me and it therefore took several years for me to settle on the facts as I had found them. Not the least was the fact that there were three different temperatures on three generic levels. They are each orthogonal and have a reciprocal which accords with Onsager’s “Reciprocity Theorem”. The pressure element  accords with Dalton’s principles of probability pressures in so far as there is seemingly always a real pressure which reacts with an imaginary one in any containing volume. What I find interesting is that we may be able to refer to temperature as Thermodynamic Direction and pressure as the  potential to work.

There are many questions to ask about these Law Fields and that is Ostberger’s intention; that question should be asked and the subjects discussed. He saw his work as a foundation for discussion in the search for order, not as a fete a compli. The ultimate test is the application of the process. For that reason he chose to represent the Hydrogen atom. But even in that he says,  “... this is not proof that the Law Worlds are all in order. Only time and  more applications will cement the foundations of this building.”

Here in the Fields of Thermodynamics we see that the elements which possess the property of Conversion are perhaps all of the character of potentials. That is if we regard Pressure and Internal Energy as a potential. This suggests that the World of Fluidics may have elements of a potential character as possessing the property of  Inversion; since in the GEM World potential had the property of Absorption. Here we see three elements of temperature which are Directional in character and possess the property of Inver-

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sion. In the GEM World we saw three elements of Potential which were Absorptive in character. This suggests that the World of Fluidics may have three elements of a directional kind which have Conversion as a property. And that seems to be so19. Once having seen so many corresponding patterns in these geometries one begins to say that perhaps we should try to use them as a basis for our mental state of order regardless of whether they are perfect or not. Indeed we need to ask even more serious questions about our approach to the universe. The end of the twentieth century has seen the meeting of ideas about life from the East and the West. This seems to be characterised by either an intravariant or extravariant view of life. In the intravariant view we see our ourselves as the masters and owners of nature, controlling new developments and forging a new kind of nature. It is a predominantly yang view.   “We control nature for societal reasons”, says Carse,  “The control of nature advances our ability to predict the outcome of natural processes... Indeed,  prediction is the most highly developed skill of the Master Player, Player, for without it control of an opponent is all

the more difficult. It follows that our domination of nature is meant to achieve not certain natural outcomes, but  certain societal outcomes.”

In the extravariant view we see ourselves as a part of the universe and belonging to it; as servants of the universe helping to improve its lot as well as our own. This view is predominantly yin and aims to decrease the entropy (disorganisation) of the planet. Infinite players understand that the vigour of a culture has to do with the variety of its sources, the differences within itself. The unique and surprising are not suppressed in some persons for the strength of others. The genius in you stimulates the genius in me20.

Notes 1. See  Appendix A A.. 2. There are no str straight aight lines lines in space because because there there are always always forces. forces. 3. We unders understan tand d  Potential  in terms of the effect it has on the Newton Force. The gradient of the gravitational potential is the gravitation force (see Appendix (see Appendix F). F). 4. The measurement of this element is not simply th through rough its single curvature curvature as a law line. The law line appears to us in all three dimensions. The proper measure is vector function and Gaussian curvature by the vector gradient (see Appendix (see Appendix F). F). 5. The notes relating to the World World of Fluidics are not well well developed, but it is clear that a velocity associated associated with fluids exists and that it has an independent nature to the gravitational kind. The former is internal whilst the latter is external. That is to say that the velocity that we associate with gravity makes the whole mass move and thereby imparts momentum, but the fluid velocity makes the particles move, or perhaps interchange, and thereby imparts no additional momentum to the mass as a whole. In a flowing river both velocities are evident. (Could it be that we can account for the slowing of e-m radiation by the existence of these two kinds of velocity?) 6. There are copiou copiouss notes about the application application of this Law Law.. The engineering engineering designer will think very diff differerently when this Law is borne in mind. It makes the current reciprocating engine into an antique device. It also makes the current current rail transportation transportation seem primitive primitive.. In the the [note9xx]  [note9xx] series  series notes several Carnot cycle engines and Sterling engines are suggested which use this principle. The Sterling engines also make use the Thermodynamic ’lasing’ principle which appear in the theory. 7. The [note9xx] series  [note9xx]  series notes describe designs which make use of the contra-momentum principle. 8.   [Note1702] 9. Other than the T Taoist aoistss I found only one philosophe philosopherr whose work accords accords with these conc concepts. epts. Charles Charles Saunders Pierce (1939-1914) was a mathematician. He wrote an essay entitled  “The Architecture of Theories” .

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It was published in  “The Monist”  in January January 1891. Pierce introduced introduced the ideas of firstne firstness, ss, secondness secondness and thirdness and described them as   “principles of logic”. From this he developed a metaphysics of evolution. Today, there are plenty of sites on the Web discussing his ideas. He defines the terms as follows: “First is the conception of being or existing independent of anything else. Second is the conception of being relative to, the conception of reaction with, something else. Third is the conception of mediation whereby a first and second  are brought into mediation.”

10. The representation of the hydrogen atom is explained in a separate document of some 60 pages. The Som-

merfeld fine-structure constant is derived from a special geometry in [note665] in  [note665]   and and [note666]  [note666] and  and shown, summarised, in Chapter in  Chapter 12. 12. The Delta values which seem to be useful in scaling the atomic elements are given in Appendix in Appendix J. J. 11. These principles are discussed and quantified in  Chapter 9. 9. 12. Shown in the Figure the  Figure 8-5 as 8-5  as + or - infinity, the different types of singularity are the subject of a note XXXX which one? XXX. 13. For example, A = A-1, where A is a squar squaree matrix. 14. Accounting is a process of discreteness. The law endeavours to resolve civil dispute with discrete Acts and Rules of procedure. The [note20xx] The  [note20xx] series  series of notes are about the use of the Law Fields in social activities. It has become evident to many people that we are not an animal that lives easily alone. Like necrosis in the body cells something of us dies when we live alone. When we live together we feel a greater warmth and security which leads us to greater harmony and creativity. 15. XXX cite these properly XXX  “To have or to Be” , Eric Fromm.  “Finite and Infinite games”, James P. Carse. 16. Intravariant World of Number. note nos XXX 233-238. 17. See Chapter See  Chapter 10, 10, also [note257] also [note257] and  and [note258]  [note258].. 18. Generic levels, meaning firstness, secondness and thirdness. 19. The notes on Fluidics are [note1750] are  [note1750],, etc. The World of Fluidics is given in Chapter in  Chapter 10. 10. 20. 20. [Carse87]  [Carse87]

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Chapte Cha pterr 9. The omnip omnipote otent nt laws laws

In this chapter I am going to briefly examine the magnitudinal aspects of the Law Fields. We will look at their relationship to well established mathematical procedures. The first example is not well established. It is new. TAO  It comes from my personal experience in accounting. accounting. I realis realised ed that I was able to express the whole edifice of finance in terms of a new and simple kind of matrix which I called “Absorption matrices”. They are easily missed by academics who have not spent years applying their subject in a real commercial environment.

9.1. Opposites There were certain geometries which Ostberger studied that disappeared when assembled. They could disappear by virtue of having having opposing directions directions or by there opposite opposite magnitudes, magnitudes, but the fact remaine remained d that they mathematically disappeared. This was primarily due to the fact that the representations had both positive and negative attributes in the same space. They overlapped in opposite directions. A conventional Cartesian graphical representation would not disappear. So what does one do with a geometry that has disappeared? Or, at least parts of it? I have not the space here to demonstrate this process geometrically 1 but it can be understood from what follows. It gives us a special insight in the the workings of nature. We wonder how it is that we can take some energy out of space and leave even more behind. Well, that is exactly what the geometry does. When a part is taken away the part that was previously  absorbed  re-appears.  re-appears.  I doubt that there is a philosopher philosopher who has not made good use of the concept of opposites; sag sages es and prophets too. There There are so many examples and so many philosophers, sages and prophets that I would scarcely have the pages to write them.  In Taoism opposites merg mergee to form the Tao. In Buddhism too the Koan Koanss are given to the students by the masters to help them understand the two opposing concepts thoroughly. The more the student uses his experience to merge the two concepts the more he understands a world in which opposites play a common part. Opposite ideas can be merged into nothing forming the structur structuree of nothingne nothingness. ss.

This is my experience too. Not only does the physics of nothingness (the void or vacuum) have a structure which is not yet revealed but the reverse process applies to our social activities and we can form structure out of nothingness. A simple example is the rise of Local Exchange Trading Systems 2 in which money is created from nothing. Before the system begins there are no Brights in Brighton or Trugs in Lewes. But a year later there are thousands of monetary units being used by the people.

9.2. Absorption What Ostberger did was to incorporate the principle of opposites into the geometry. But first he expressed it in the form of new matrices3. He called these  Absorption matrices and the spaces absorption spaces. It was these spaces that he found were disappearing. Let us look briefly at these matrices.

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Chapter 9. The omnipotent laws

Figure 9-1. An absor absorption ption matrix

Figure 9-1 is 9-1 is a simple absorption matrix. It has the property that every line horizontally and vertically aggregates to zero. It is a zero matrix. Its Identity is zero. We do not have to add or subtract the numbers, we merely have to take them as a complete set which is empty. These could be entries in an account. So that the top line could read, Debit 4

to insurance account

Debit 6

to stock account

Debit 2

to freight charges account

Credit 12

supplier suspense account

which is precisely the accounting process. If accountants were to use these matrices they would have to handle ones with hundreds or thousands of rows and columns. But that does not make the process any the less understandable.

Figure 9-2. An associative matrix with missin missing g elements is still completable

What is interesting is the fact that a large number of the elements (numbers) of the matrix can be left out without destroying its structure. Figure structure.  Figure 9-2 is 9-2  is the matrix of  Figure  Figure 9-1 yet 9-1  yet it can be determined completely. Notice also that the magnitude in each quadrant is the  determinant  in   in the corner, 38. This is true of any absorption matrix. Even if the elements of the matrix are mixed signs and the two lines are drawn orthogonally at any point, the four quadrant magnitudes are equal. Figure equal.  Figure 9-3 is 9-3  is an example in which the quadrant magnitude or determinant is 10. But the elements of the matrix are the same. The commercial applications of this are explained elsewhere 4.

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Chapter 9. The omnipotent laws

Figure 9-3. An associative matrix with a quadran quadrantt magnitude (determinan (determinant) t) of 10

These seemingly trivial matrices are extremely useful. An academic person who has not lived in the world of  commerce would have missed the point of them entirely. Indeed because they are empty sets they do not appear on the mathematical scene at all. Ostberger called these  “Matrices of the first kind”. The matrices of the second kind are the reciprocal matrices5 which we use to solve groups of simultaneous equations. The Absorption matrix could also be used to express Archimedian volumes in which the first line of   Figure Figure 9-1 means could express the bathing of babies, 4 volumes

is the first baby

6 volumes 2 volumes

is the second baby is the third baby

12 volumes

is the displacement from the bath

We may have thrown the baby out with the bath water. Figure 9-4. The absor absorption ption bottle5

The Archimedian principle is quite fundamental to our learning process yet we are stumbling over this truth and passing on. The displacement of water in a jar (Figure ( Figure 9-4 9-4)) is the exact counterpart of double entry accounting. If  the air is treated as credits and the water as debits then the movement of the surface gives the balance. For each volume of air that is changed so the same volume of water is changed in reverse. Except for compression (which is the next level system up) the air and water contra-exchange. We cannot get water in without taking air out. There is nothing more mystical to double entry accounting than this.  It is my deepest concern that we should teach our children the truth as we find it and to explain to them that it is “as we  find it” and no more. If, however, we find a truth and do not teach it then woe will betide that generation.

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Chapter 9. The omnipotent laws

In accounting the basic tenet is that,  “No credit exists without its corresponding debit” , and this is something that the physicist must learn for it applies equally to the directions of space and all of those phenomena that are Directional in character. Again,  “when a man walks from the East he also walks to the West” . This is the basis character of physical direction. Each metre West is the same metre East. They overlap, as the rope-maker found. When the student taoist was ask to consider the sound of one hand clapping he was asked to discover these simple facts so that they may become the foundation of his understanding. We do well to heed his experience.

Figure 9-5. The absorp absorption tion identity m matrix atrix

The identity for the absorption matrix is given in Figure in  Figure 9-5. 9-5. It is understandable that we have not found it before. It simply isn’t there. What is there is the realisation that there are two zeros, not one. Zero plus and zero minus are very real entities. They appear in directional studies quite frequently. They also appear in computer systems and accounting. What happens is this. As a magnitude dies to  0 + its Direction changes from plus to minus and the magnitude is born again at  0 − . The pole or zero point of a magnitude is an indication that the direction is changing abruptly. Zero plus and zero minus are oppositely directed. Absorption is a very special property and occurs in many different guises. One interesting guise is Entropy. Entro tropy py appear appearss in the World of thermo thermodyn dynami amics cs in Chapte Chapterr 10 10.. It is the the st stat atee if diso disorg rgan anis isat atio ion n of the the ther thermo mody dyna nami micc − system. Its counterpart is negative Entropy,  S i  , which is the state of organisation of the system. It is clear that in any organisable system the boundary between organisation and disorganisation moves in an absorptive way. A change towards organisation is the same change away from disorganisation, it  double enters. But it does so in a differential manner and so such a matrix needs to be expressed as  dS e+  with dS i−  . Another guise is the volume  V  , with which we measure space in thermodynamics. We may divide any thermodynamic system into two parts. The Volume Occupied and the Volume Remaining. These are absorptive and are +

−

expressed in terms of  dV   dV  with dV  . In physics the Ostberger Worlds show three kinds of Direction; Magnetic Direction, Electric Direction and Gravitational Direction. All possess the property of absorption, yet they are all of a different character because they are on different levels. It turns out that they are the  potentials  of our mathematics. In accounting the two zeros that occur in the system are of paramount importance. In  Appendix C C is  is an actual experience in which the outcome of a discordant exchange between two companies over a tool that was missing was finally resolved by the signs of the zeros in the accounts. In Taoism opposite concepts merge to form one of the states of Tao. In Buddhism the Koans are given by the masters to help the student understand the two concepts that are born out of nothing.  “Listen to the sound of one hand clapping”, says the Master, “Feel the space in the pool when the foot is withdrawn”. These are the teachings of wisdom. Teaching the student to see two sides to every event, just as the accountant must and, in the future, the physicist must too.

Absorption does not just apply to the nuts and bolts of a system, it also applies to the operators as well. It applies to our actions, our organisation, the transformations in geometry, the operators in Quantum Mechanics, the transactions in accounting, the operations in number theory; it applies to all operations of secondness.

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Figure Figu re 9-6. The Law Fiel Fields ds of Matrices

There may be four Kinds of matrices. 1. Absor Absorption ption Matrices. Matrices. New but simple and useful matrices matrices with meaning meaning to their rows and columns as de3 scribed here and further in the notes . 2. Reciprocal matrices. The ones we use to solve solve linear simultaneous equations and which form groups theory at university. They are well established in the literature. 3. Conversion Conversion matrices. The upper triangle is imaginary whilst the lower is real. The Pauli Pauli matrix is one example. 4. Condensati Condensation on matrices matrices.. A new kind of matrix which encompasses encompasses normalisations normalisations.. There are no notes of  these. They are speculative. If this is the case then they form a Law Field. There are other matrices of a lesser kind which are disordered. Then these are the groups of Ordered matrices.

9.3. Reciprocity

The second Direction of the Law Field is divided into two parts by the number one. In fields of greater complexity this can be divided again by the process of renormalisation. In primary school I learned about the two ways of using numbers. I could not possibly understand such a philosophically deep idea as I had no experience on which to hang my understanding. There are cardinal numbers and ordinal numbers. Cardinal numbers label things. They provide a means of putting things into a state of order. This leads to the ideas of probability, chance and statistics.

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Figure 9-7. Cardinal and O Ordinal rdinal numbe numbers rs

Ordinal numbers, on the other hand are those that give magnitude or size to things. These two aspects of number are not to be found in a single World of number. The cardinal numbers belong to an extravariant World whereas the ordinal numbers belong to an intravariant World. Such a big idea could not possibly be conceived by my inexperienced mind at primary school. Yet it is not long before we can appreciate the difference of these ideas and use them from the Law Worlds. In mathematics a single cardinal number is used to identify a group; the identity of the group. to find this identity we say that it must have a reciprocal. Even more than this we say that every element of the group must have the same identity. Thus the rational numbers   ab  such as   32 ,   34 ,   45   and   67  have reciprocals such as   23 ,   43 ,   54   and   76  and the identity to the group is the number  1. What we probably fail to observe is that the number  1 , in this instance, is not an ordinal number but a cardinal number. The cardinal number   1   identifies the group and allows it to be almost infinite in both the number of  members and their magnitude, both of which are ordinal numbers. This is explained by the application of a World geometry of the second kind6 to Number theory. A group of people also must have an identity. It is part of our nature to seek identity. People do not join the ranks of  the aimless. We are all different and we all seek different identities. We are members of the golf club, the physics Institute, the liberal party or Greenpeace. And each of these has aims and objectives and vector themselves toward them. It is these aims, which are incorporated into the constitution of the group, that determines the eigenvector of society7. That’s where we are going.

Table 9-1. From the notes on social activity

Every agreement requires two persons. The exception is that a person may choose to make an agreement with his god, or not. One of the fields of human activity that uses the reciprocal relation is that of Agreements. Every day of our life we make these things we call Agreements, but do we stop to think objectively about them? What are they? How do they occur? Do the different types of agreements fall into any kind of pattern? The more we ask, the more there is to know. The law deals with agreements and breaks them down into their natural groups. It deals with the making and breaking of agreements. It deals with how they should operate and how they should be interpreted. What we allow and what we forbid is a part of the eigen-vectored journey of the human race 8. The principle of Reciprocity litters the technical literature. There is   “Onsager’s reciprocity”  which describes the temperature process in Thermodynamics. In the Law Fields this is interpreted as the special property that we can attach to Thermodynamic Direction which we call Temperature. There are three different types of temperature at three generic levels9.

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There is als There also: o: Electr Electrica icall net networ work k recipr reciproci ocity ty,, va vario rious us mathem mathemati atical cal recipr reciproci ocity ty Theore Theorems, ms, the recipr reciproci ocity ty reg region ion for neutron energy, the law of reciprocity in photography, reciprocity in acoustics, the Maxwell and Betti reciprocity theorem, linear and rotational reciprocity in mechanical systems and so on. There is Reciprocity of all kinds. It is one of the mental processes that we use to lend order to the world around us. It must also be in our heads.

9.4. Conversion Without imagination neither we nor any other part of nature would exist. Even the smallest part of nature that we understand, the hydrogen atom, has its imaginary parts and the mathematics of Quantum Mechanics which describes it is itself imaginary. Indeed, in mathematics there is probably as many theorems about the imaginary as there are about the real. There are many amongst we humans for whom the idea of something imaginary has no meaning and does not form a part of life. Yet again there are more for whom the imaginary is a God outside of themselves and cannot be reached except in death. But there are a rising few who have a god inside of their being which forms an equal part of their lives in reality. They are the converted. There is, perhaps, nothing more real in life than a person who gather up his wares and travels to market. It is an activity that has gone on for thousands of years and looks set to continue for another thousand. Such people are the salt of the earth. Yet these people must live a part of their lives in the imaginary world of planning. They must plan what they will take to market. The greengrocer must assess what he thinks he can sell according to what the weather might be or what the people might want or what the telly might advertise. What is going on in his mind is a planning operation that is imaginary. There is nothing real until his stall is laid out and his sales begin. A larger company does the same planning operation. It is the annual Sales forecast and budget operation. The whole company is involved in planning the future. Everybody is guessing what might happen next year. It is all pie in the sky.

Every business spends a great deal of money preparing for the future and when it is all done the accountant will translate it into a Sales and Profit Forecast10. The accountant will not call the forecast money  imaginary; instead he will call it  notional  money and as the year passes he will offer a tracking system that compares the notional forecast with the actual performance in his  variance accounting. But it is all imaginary and the process is conversion. We have seen that one of the best known examples of conversion is in the third law of Newton,  “to every force there is an equal and opposite reaction”. The force is real and the reaction imaginary. Another example of conversion occurs in the field of Investment. The decision to invest in a new venture is accompanied by  risk . Risk is purely notional yet it is considered essential to spend considerable amounts of real money assessing the risk of a project. Risk analysis is a very serious study and demands a person of considerable experience, education and acumen. It is a part of our social fabric. On the Law Field risk is represented by a single line. This does not imply that the subject is simple but rather that the geometry is complex. The element of Risk is the  eigen-line  which draws together all the resultant risk vectors that could be assembled into the space which is the whole of the imaginary half of the diagram.

Notes 1. Ostbe Ostberger rger cons considere idered d that nature contracted contracted herself herself by this process. In the World World of the second kind this is exactly what happens. 2. LET LET-Systems, -Systems, new m money oney.. Michael Linton and Angus Soutar realised that money could exchange hands without the use of banks. banks. Individuals Individuals trade their skills by writing writing their own personal personal cheques in a local currency currency

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(Truggs in Lewes, England, and Brights in Brighton, England). An account is kept of the income and expenditure of individuals. The system has reached most parts of the planet. 3.   [Note100] describes [Note100]  describes these matrices and their properties. They are shown to be the basis of accounting and used for a complete accounting system in the book entitled XXX “The Information System belonging to the  Manufacturing and Distribution Sector of an Economy”. 4. See the book entitled entitled “The Information System belonging to the Manufacturing and Distribution Sector of an  Economy”. 5. See  [note124] 6. An exam example ple of a World World geometry geometry of the second second kind is given given in Chapter in Chapter 11 11.. The intravariant ordinal numbers are the structural framework for three orientations of extravariant cardinal number. This means that the group can be identified by any cardinal number, its statistical operator or interpretation. 7. The  [note20xx]  [note20xx] series  series notes are about Social order. The The [note23xx]  [note23xx] series  series notes are about order in Finance. They show that both of these subjects follow the patterns of the Law Worlds. 8. XXX 9. A comparison of the T Temperature emperature Law W World orld with the Agreement Law World World gives an excellent excellent insight into the understanding of these three generic levels. 10. The [note23xx] The [note23xx] series  series notes are about finance. The Law Fields of finance are isomorphic to those shown in this book. The book XXX  “Systems in the Manufacturing and Distribution Sector”  also shows the isomorphism between finance, internal data systems and mathematics.

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Chapter Chapt er 10. Law Worlds Worlds of the the first kind kind 10.1. Vector as pictures It is now time to look at the constructions that Ostberger called World geometries after Minkowski 1. But first a word from Ostberger.  I decided to give up my job and go write about these pictures. I sold the house, gave up my digs, travelled back to West  Sussex and began studying geometry from a caravan. I simply felt the need to spill out all the pictures that were inside of  me. And I did. There seemed to be endless amounts of geometric study to do and never enough time to do them. The ball really began to move fast when the ideas of curvatures took over my studies. I had suddenly abandoned the strictures of my imprisoned education and launched myself into a new world. From then on I could see that I would never have enough time or money to complete the gigantic task of studying all the Directional properties of all the mathematics that  had ever been done in history. Many tasks would have to be left undone, many gaps would have to be filled in by other  workers in the field. I began to aim for the mathematics of the Hydrogen atom because that was the best known and most  exact application of mathematics. Riemann and Einstein would be companions on the way and Dirac would be the man

 I had to emulate directionally. directionally. There is a poignant statement by P.A.M. Dirac 2 which captured my imagination: “In the case of atomic phenomena no  picturee can be expected to exist in the usual sense of the word picture, by which is meant a model functioning essentia  pictur essentially lly along classical lines. One may, however, extend the word picture to include any way of looking at the fundamental laws which makes their self-consiste self-consistency ncy obvious.”  A picture is exactly what I was aiming for for.. It would not be an ordinary picture picture,, it would be a vector pictur picture. e. I was going to prove Dirac wrong about the Directions (his picture) and right about his Magnitudes (his mathematics). The vector picture was likely to contain some new ways of looking at things which I had not been taught and which nobody else had either. For I had already learned that vectors can produce some funny looking effects on paper that do not always follow our instincts. For example it was not my instinct to believe that a single point on space could have three separate numbers attached to it or that the point would turn out to be no point at all but a kind of inside out World  that I was creating at the time. My Cartesian training had firmly implanted the idea that one point in space had one value. Yet there was something left from my A-level days that haunted me. One question that had never been answered. When we represent a vector in mathematics we draw a line, put a point for the beginning and an arrow for the ending and label the length as a representation of its magnitude. Then, when we want a negative vector we simply turn the vector round to  face the other way. My question was, “why was a nega negative tive vector always facing the other way?” Why could I not have a vector whose magnitude changed from positive to negative. Why was it always the direction of the vector that had to change?  I resolved resolved to accept that I was right and that vectors could have nega negative tive magnitud magnitudes es as well as positive ones. This  presented  prese nted me with a headac headache he that I would rather had left behind. The solutions of mathematic mathematicss were doubled. I was  further resolved to continue.  My headac headache he would be compoun compounded ded by the fact that I had accepted curvatures as the basis of my geometr geometryy giving an even greater scope for sailing into uncharted waters. The pain was even worse with the realisation that the pen stand  said that there were four times as many points in a Euclidean space as I had been led to believe. But my resolve was unattenuated.  Now in Quantum Mechanics Mechanics of the type that Dirac constructed the V Vectors ectors are not just the stage hands they are are the whole cast. There are a few hands off-stage that help with the scenery but otherwise the play is written for the vector. The cast are the magnitudes and what they say are the directions. Both are a party to the play. The importance of this is that vectors have both magnitude and directions. There are no vectors without both. Dirac calculated the probabilities,  produced  pro duced some excellen excellentt numerica numericall results and even showed that a nega negative tive electr electron on must exist. But, where where,, I ask, are the Directions? Dirac has a cast of players but no script. We have magnitudes but no directions. Where have they all gone? Do they just disappear leaving the cast frozen on the stage? Or is it that the magnitudes come out to see us, leaving the directions in some kind of hiding place?

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Chapter 10. Law Worlds of the first kind   If Quantum Mechanics Mechanics was a vector study then both the numbers which issued from the probability equations and their  respective Directions were available for me to find. So I set out to find them.

10.2. Separating direction from magnitude The whole theory says that we can separate Directions from Magnitudes. That we can study each separately before combining them in a final solution. The yin process is working with Directions. The yang process is working with the Magnitudes. Each contains a part of the other in the working process. Thus an equation will contain some elements which represent the directions and the Direction geometry will contain some scales and numbers representing the magnitude.The two will coalesce in the solution.

The basis of the Directional process is that we seek orthogonality 3 in space. In particular we seek orthogonality of  points in space in the dimensions defined by the pen stand and later the World geometry. They are not always easy to find when a geometry is compounded into a new space. We must get used to orthogonal conditions appearing unexpectedly. It becomes very easy to make assumptions about the amount of a rotation in space; as history has proven. Historically we have ignored the internal rotational of a line element and thereby passed over the opportunity that Ostberger has revealed.

10.3. Ordinary space The space of a Law World is not that of ordinary space. The Law World is embedded in ordinary space. So, the best way to visualise it, in the beginning, is to imagine that the World can be rotated in the room in which it sits as a model. The rotation in the room is then the ordinary Euclidian space. But, we need to remember that the room-space is also 4 dimensional in the same sense as the World model. That is to say that the pen stand applies to the room-space as well. Vectorially we would say the the World is a set of vector spaces over the Euclidean space. It is a Hilbert Space. The reader will have gathered that the word  “Direction”  is used when speaking about the Law World space and the word “direction”  when speaking about the Euclidean space. Consider an example of the mixture of these spaces. In the normal course of mathematics we would refer to the velocity of a car as, say, 17 Kph. We then represent this on paper to a scale of 0.5 cm per Kph. There are two magnitudes and two directions here: •

  The velocity of 17 is a Law Law W World orld Magnitude.

  The velocity is undefined in Direction. It simply is not pres present. ent.   The direction of the scale is relative relati ve and on the paper. paper . •

•

•

  The magnitude of the scale is 0.5.

We have not actually expressed a Direction to the velocity. When we do, we automatically relate it to the other Directions of the Universe which are part of the natural Law. It becomes a part of the physics around us. Its effects on other phenomena, such as the generation of a De Brogli wavelength, becomes evident.

10.4. The parallel principle When we represent a vector in a vector space any parallel vector having exactly the same magnitude is a representation of the same vector. The vectors must be unique to the representation.

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If we look at the world geometry we see that one set of curved vectors passing over the surface provides all the magnitudes from the greatest definable curvature at the pole to a fixed value at the great circle of the same colour in the same plane. In the enlargements there are an infinity (less the last) of possible fixed values belonging to the great circles. It is therefore possible to fit the curvature of this space with all the values of numbers from  0 through ∞ to ∞ − 1, but they need to be of the same class of continuous number. The other coloured surfaces are then available to define the other classes of number; the discontinuous and the complex. There is only one

World geometry in a purely directional space and all possible curvatures are present somewhere. This is the property of the Riemann unit Spherical surface. Enlargement is the only operation available to us. Rotation is not available because it must be along one of the curves already defined and therefore simply produces a parallel vector. Translation also produces a parallel vector because it too must be along one of the geodesic pole elements which is already defined. Thus it follows that if we can describe an atom using a geometric representation then any identical atom is described by the same representation.

Figure 10-1. Equal vectors are par parallel, allel, and of the same magnitude

10.5. Direction only In a social environment which teaches a predominantly yang view of life it will be difficult for most people to spring their magnitudinal jails and see the World geometry in terms of its Direction alone. Particularly in the Western hemisphere where we are taught to believe that size is what matters. It is not so. In what follows there are no magnitudes unless they specify themselves or we apply them in searching for solutions. They specify themselves by producing natural numbers. The relation of  2π  2 π  in a circle is one example. It is a purely geometric 4

constant and entirely universal. So is  e , the natural constant of growth. So too is the rope-maker’s constant , the reciprocal of  2π  2π  and the Sommerfeld fine-structure constant for Hydrogen  α, as we will see in Chapter in  Chapter 12. 12. The world geometries are constructions of Direction only. There are no magnitudes. Only when we begin to formulate a solution do we begin to see the numbers that specify the magnitudes appear in the shapes. Some elementary examples were given in earlier chapters, e.g. Figure e.g.  Figure 5-6 5-6.. The Law World geometries are no more an answer than the Schroedinger or Bernoulli equation. They provide the tool with which to approach an answer. A tool which lends guidance to an otherwise desperate situation. The true understanding comes from their application. It is hard work despite its visual presence.

10.6. Curvatures Nature is as much about shape and form as it is about number and size. This is particularly evident in biology where function and form are often visually separable. She does not produce a single tree that is so big that it over shadows all the other plants. She knows that her diversity would be compromised in this way. She allows the

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growth to be tapered by some rule which permits other plants to grow. If a flowering plant were too tall it would not be pollinated by the bee who could not reach its height. At the lowest level of our understanding of nature we measure the phenomena of our universe in terms of curvatures. These are her shapes and her shape is beautiful. The magnitudes of these curvatures are the essence of relativity which is born out of the Tensor mathematics created by Riemann.

10.7. Assembling law fields There is an excellent analogy of the way we currently use geometry in mathematics which relates it to the Ostberger process. Our current use of geometry is like laying out the components of a motor car in a long line. Each component is something different and not recognisable as belonging to any particular car. There are a few, expert individuals who can relate the component to the the Volkswagen that it really is. The students and the rest of us simply learn the bits as we pass along the education trail. The engineer, however, is the great organiser of society. He has constructed a generic system of the parts so that they can be easily related to the car as a whole or in parts. Starting with the saleable product he identifies precisely every generation of every family of parts which go to make up the car. With the advent of computers he now also reverses the process and constructs the  “Where used next level up”  data which every buyer of spare parts for his car can see on the stockist’s computer. Every change of component by date and model is now tracked and made available to the trade. The car, to the engineer, is a simple family of connected generations. Ostberger is creating a generic family of geometries for use in mathematics. The geometries are connected. They are just like the engineer’s  “Where used next level up”  generic map of the car.

Figure 10-2. Curving an an element

TAO  It became evident from the laws which associate electricity and magnetism that these Fields would assemble in some way. My early attempts were centred around a block-like construction that was Cartesian (see Figure (see  Figure 10-2). 10-2). But I later  realised that the elements of these constructions should be curved.

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Figure 10-3. Early sketch joinin joining g the Law Fields

Notes:

1. Lenz’s law says that the flux change opposes the electro-motive force. 2. Neuman’s law says that the magnitude of the flux change is proportional to B . The constant k  is the product of the number of turns in the circuit and its area. 3. The static terms  E.D  E .D  and  H.D  are zero. Energy is dynamic.

In Figure In  Figure 10-3 the 10-3  the key feature was that a changing magnetic flux  B   produced an electromotive force   e  which always opposed it. So, in some way the element  E  of electric force would oppose the flux  B . But  E  had to be an operator on  B  because E is in the secondness field. Which it is. This seemed to work and gave a space for the Poynting vector H   D  and a volume for the energy  dω . But the block-like construction did not measure every point in space of the pen stand.  

10.8. Minkowski’s worlds The next development was the Law World. This starts with a generalisation that we can apply a curvature to one

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or more of the elements of the Law Field ( Figure 10-2). 10-2). These are the worlds of which Minkowski 1, the great teacher at the turn of the century, was talking about in his lectures.

10.9. The grav-electromagnetic world Figure 10-4 is 10-4 is the Grav-electromagnetic Law World. The dotted lines show the imaginary regions and the surface of the world contains three elements of what Ostberger called  fluxes. In this way he referred to velocity as the flux of the gravitational field. These are all the elements that obey the general reciprocity principle. External to this are all the elements that obey the general absorption principle. The interior of the World is an all real region in which the Forces reign.

Figure 10-4. The intravariant Gra Gravelectromagne velectromagnetic tic World

•

  In the intravariant World World the whole of the exterior region is imaginary which is indicated by the use of broken lines throught the notes. The interior region is wholly real.

•

  The directions of the elements are crutial crutial to the correct relationship of all the elements.

•

  This diagram is illustrative illustrative only. only. The external elements are not not shown in their correct orthogonal orientation.

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This is an approximation since the external elements should themselves be curved, but it is probably as much as one could put on to a piece of paper without causing confusion. The notes are more definitive in places. At this point I remind the reader that there is no scale to this diagram. It could be the size of a galaxy or the size of an atom. We do not know. We have not specified. The World occupies the whole space. There is no room for any more elements. In particular there is no room for a Euclidean space. That can only be found by transforming the World in the space of the room in which it occupies. If we look at the Worlds in terms of their phenomenological form we see three Fluxes, three Directions and three Forces. That is about as simple as it comes. These are ranked by their generic level, The firstness Field

Magnetic Laws

The secondness Field

Electric Laws

The thirdness Field

Gravitic Laws

and each of these fields had Fir irsstness

a Dire recction possessing the property of Absorption

Seco Second ndne ness ss

a Fl Flux ux po poss sses essi sing ng th thee prop proper erty ty of Reci Recipr proc ocit ity y (In (Inversi ersion on))

Thirdness

a Force possessing the prop roperty of Conversion

and that is the general pattern for every Law Field and Law World. There is a fourth field in each of the spaces which, in general, follows the pattern of Condensation. The intravariant World is a World of continuity and therefore the line elements pass over the surfaces in the manner of spirals. The shape of the worlds may be of at least two types. In one type the scales change the shape of the surfaces; a yin process. In the other type the scales produce rings in the surface which belie the scale that is applied, the yang process. The two types seem to interplay. The exterior region is imaginary. This endows the World with instability. It is divergent and would, if left alone, expand into oblivion. But it seems that nature provides a stabilising influence on this World through the properties of the extravariant World. If all the elements of the intravariant world are transformed through an infinity (a singularity) the result is the extravariant World. The two worlds contain the same elements precisely. Compare Figure Compare  Figure 10-4 and 10-4  and Figure  Figure 10-5 10-5..

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Figure 10-5. The extravariant Gra Gravelectromagnetic velectromagnetic World World

•

  The exterior region is in  real  space.

•

  The interior region region is in imaginary  space.

•

  This diagram is illustrative only because the external external elements are not shown in their correct orthogonal orientation.

How much of a rotation in space is required to make this transformation? The answer is almost  π /2  of all the elements5, which is rather remarkable since the imaginary region has swapped from being external to being internal which we would normally consider to be a  π  transformation. Yet, looking at the two worlds we see that the surface elements of the intravariant World (Figure ( Figure 10-4) 10-4) have become the external elements of the extravariant one (Figure (Figure 10-5 10-5). ). What is important is that all the elements are in exactly the same juxtaposition with respect to each other.

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This apparent anomaly is similar to the apparent anomaly in the electron that says that a spin up electron is  2π  2 π apart from the spin down electron. Yet its spin vector direction suggests that it is only π apart. The exact geometric 6

measurement of this is given in the notes . In the World of the second kind the extravariant World is assembled back on to the intravariant World. The vectors of the surface of the intravariant World are then anti-commuting with those of the extravariant World. The results of this are explained in Chapter in  Chapter 12 12.. There are six ways of making this assembly. The extravariant World exhibits the properties of stability 7. It tends to condense its structure as it develops. What is surprising is that there are two worlds not one. We imagine that a four dimensional world exists as an expression of the omnipotent ending of our understanding. This is not so. Another World emanates from the first to start a new four dimensional process. Yet the new process is constructed of the old World elements. TAO  It was a great surprise to me that there was a four dimension geometry and it took me many years to over overcome come that  surprise and start to work in earnest as if it were so. But even more of a surprise came the fact that there was another   four dimensional geometry which could be construc constructed ted from exactly the same line elements as the first and yet be very different in character. But the really remarkable thing was that the latter was derived from the first because that meant  that such a process could be continued indefinitely. Four is suffice but it births itself twice.

10.10. The thermodynamic world There is another World that follows the same patterns as the Grav-electromagnetic World. The three Law Fields of the previous chapter come together to form this world. The first is the Work LF, the second is the Organisation LF and the third is the Reaction LF.

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Figure 10-6. The intravarian intravariantt Thermodynamic W World orld

1. In the intravariant World the whole of the exterior region is imaginary which is indicated by the use of  broken lines throughout the notes. The interior region is wholly real. 2. The directions of the elements are crucial to the correct relationship of all the elements. Looking into a different octant produces a different directional relationship. 3. This diagram is illustrative only because the external elements are not shown in their correct orthogonal orientation.

really interesting comparison is between and our own social activities. We setsame. to work a project or aThe business, we organise the business and then this we react with others who have done the Thisonforms bigger work groups which need organisational effort and then a bigger interaction takes place. The process goes on until we have large conglomerates that are irredeemably inefficient. That is the unstable way of the intravariant World.

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Figure 10-7. The extravarian extravariantt Thermodynamic W World orld

In nature, however there is an extravariant process taking place. This is the World of  Figure of  Figure 10-7 10-7.. Like the other extra-worlds it is a recipe for stability and when combined into a geometry of the second kind  8 it acts as nature’s stabiliser. Eventually the whole structure of thermodynamics becomes stable by virtue of it dominating the Laws. There are several interesting predictions in the field of Sterling engines and conductance which are extracted from the thermodynamic World.9 But this is not the place for them. TAO There were several aspects of this World that surprised me and it therefore took several years for me to settle on the  facts as I had found them. Not the least was the idea that there were three different generic levels of tempera temperature; ture; or  temperature-like phenomena. They are each orthogonal to each other and have a reciprocity in accord with Onsager’s  Reciprocity  Recipr ocity Theor Theorem. em. I cannot predict what this infers in reality but that is what the theory says! The pressure element  accords with Dalton’s principle of probability pressure in so far as there is seemingly always a real pressure which reacts with an imaginary one in any contained volume.

10.11. The World of number The Law Fields of  Chapter  Chapter 9 come 9 come together to form this World. The copy of the notebook, Figure notebook,  Figure 10-9 10-9,, shows the intravariant number World as Ostberger drew it in 1969. The orientation is described by numbering the octants of  the World as if it were an ordinary sphere. This is clearly important because every orientation of the World gives a new and different set of directions in the geometry.

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Figure 10-8. The Extravarian Extravariantt Number W World orld

There are no notes about the development of the world of number into a geometry of the second kind.

Figure 10-9. A copy of a page of the notebook showing a sketch of the W World orld of intravariant Num Number ber

10.12. Other Worlds There are other Worlds which are shown in the notebook. •

  [Note1750]: [Note1750]: The Fluid World

•

  [Note1001]: [Note1001]: The Fields of Engineering System Dynamics

•

  [Note23xx] series: [Note23xx] series: The World of Finance.

•

  [Note2333]: [Note2333]: The World of Commercial Operations.

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Chapter 10. Law Worlds of the first kind  •

  [Note1030]: [Note1030]: The mathematics in Minkowski’s World.

The Fluidic World is incomplete, but is important because it completes the set of three Worlds of physics which are depicted as a sketch in Figure in  Figure 10-12. 10-12.

Figure 10-10. The first issue of the three Law Fields Fields of Fluids

Although much of the fictional discussion about our physical universe is centred around the Grav-electromagnetic Worlds we see here that the other two Worlds are of equal stature in the Universe. The Fluidic World of the second kind, which stands in the place of thirdness in the physics Universe, which is shown at the top of this sketch gives rise to the three states of matter. It represents them in the microscopic form. The size is absent. Amongst this Fluidic World is Mohr’s Circles and its three dimensional extension. In those are the formula for the bending moments and shear stresses of physically loaded structures. The Mohr circle occupies just one of the eighteen planes in the Fluidic sketch.

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Figure 10-11. The first issue of the incomplete Fluidic Fluidic Law World World

Yes, there is a great deal more work to do in the detail structure of these geometries. But just how much work can one man do in one lifetime and still survive on the planet? I believe Ostberger has given us a very big leap forward and it is something to grasp and apply to our life on this planet. The fact that he has been able to show that our financial dealings have an isomorphic pattern to the world of GEM is of great value to the whole of humanity. TAO  I did not set out to discove discoverr the laws of physics. They arrived at my door like with the messenger messenger.. I had no reason not to receive them and display the process as I discovered it. The messenger was not announced, he did not say “I have come  from God” or “I have brought you the keys to the kingdom” kingdom”.. Nor did he say that he knew the contents of what he was carrying. He didn’t know and neither did I. There was, however something interior to my being that gave me impetus and caused me to believe that this was a job that I had to do. I was later to discover what that something was. I had no idea that it would take so long and cost so much. Yet I felt quite comfortable doing work which, maybe, no one else was doing or even would understand.

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Figure 10-12. Sketch of the Universe Universe of Ph Physics ysics

Each of the three worlds in the sketch are what Ostberger called  “of the second kind”. Each of these contains six worlds of the first kind. Each world of the first kind contains six law fields. Each Law Field has seven degrees of  freedom; the last of which is half in the next. Additional degrees of freedom appear as the geometries grow. A World of the first kind has twenty two degrees of freedom, the last two of which have one half of their freedom in the next world. In all, one World of the second kind has more than sixty degrees of freedom. This sketch would have more than one hundred and eighty degrees of freedom. Every degree of freedom is orthogonal to every other. A degree of freedom is one dimension of the Magnitudinal kind. Every group of line elements, Law Fields, World and 2nd World is in four dimensions of the Directional kind. The Worlds of the Second kind are assembled twice in different orders. The ones in the sketch, here, are extravariant worlds mounted on to intravariant backgrounds. They can equally be assembled in reverse. One of the notes suggests that the intravariant world when mounted on the extravariant world is representative of the macroscopic Universe. The one that we find invitingly curious.

Notes 1. Herma Hermann nn Minkowski Minkowski (1864-1909), (1864-1909), the Russian born Swiss-German Swiss-German number theorist, theorist, algebrais algebraist, t, analyst and geometer who developed the thory of four-dimensional space-time that laid the mathematical foundation for relativity theory. 2. See  [Dirac70]  [Dirac70].. 3. The half half quantu quantum m of Dire Directi ction on is   π/4 π/ 4. It would seem that, as with the Magnitude half quantum, it has a

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valuable place in the theory. 4. The consta constant nt 2  2/π /π  is a magnitudinal constant. It is the chance of a XXX? certain stick landing between two parallel lines. Also π/  π/22 is a directional constant. It is the quantum of direction. 5. See  [note1170]  [note1170],, [note1171],  [note1171],  [note1172] and  [note1172]  and [note1173]  [note1173].. 6. This This is expla explaine ined d in in [note1823]  [note1823] in  in terms of measuring the rotations that are needed across a 3-dimensional surface to get from an up-state to a down-state of the world. The up state electron lives on one side of a World geometry of the second kind and the down state electron lives on the other (Figure (Figure 11-5). 11-5). They are different electrons. 7. Stabil Stability ity is where Ostberge Ostbergerr began. He examined examined the Nyqui Nyquist st Diagrams of enginee engineering ring stability stability and was amazed that stability depended purely on the direction of rotation of a locus on a diagram. The reader should be aware that when the World pictures are applied to subjects which are not  “below”  us but  “above” us such as the Galactic Universe then the role of stability seems to reverse between intra and extra Worlds. The same seems to apply between subjects which are  “within us”  such as emotions and psychology and those outside of us such as Social Justice and Finance. This is a particularly big concept. 8. See  [note1713]  [note1713],, [note1714],  [note1714],  [note1715] and  [note1715]  and [note1716]  [note1716].. 9. Se Seee the the [note9xx] series  [note9xx]  series notes.

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Chapter Chapt er 11. Worlds of the the second second kind 11.1. Worlds of the second kind The Ostberger story is now becoming more difficult to write into a book that sufficient people will want to read that makes its progress self initiating. The difficulty lies in presenting images on paper that are multi-dimensional. Nevertheless, with a few modifications, I will write it in the way that Ostberger created it. The modifications are in the way the curvatures are shown on the page. They are not shown exterior to the World surface as curves but rather as straight lines with near zero curvature. This would be satisfactory for the applications to the first atomic elements where the curvatures are thus. But even Helium will require a small amount of curvature for its representation.

Figure 11-1. A minimum line element

The real difficulty lies in the fact that journals and publications of most kinds are not accustomed to printing the geometric forms that this process demands. Mathematical journals simply don’t do it. They certainly could not conceive of depicting the geometry to be the leading part of an article that contained a mathematical text. The text would be regarded as the leading part. Also, the line elements should show their thicknesses in each of two planes so that their orthogonality can be ascertained but it is unrealistic to believe that such a picture could be read from the page. Such detail must be left in the text of the notes. But a computer can display such a picture. I hope that there is sufficient here to convince the reader of the usefulness of the process and that one, at least, will want to examine the Hydrogen atom mathematics and its geometry.

11.2. Bosons and Fermions There are two kinds of statistics which are especially relevant to particle physics. One is attributed to Bose and Einstein and called the Bose Einstein statistics and the other to Enrico Fermi and Paul Dirac and called the FermiDirac statistics. They are shown in Figure in  Figure 11-2. 11-2. The only difference between the two is that one includes the first particle (Fermi-Dirac) and the other excludes the first particle.

Figure 11-2. Two kinds of particle statistics

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Chapter 11. Worlds of the second kind 

We may think that such a small consideration does not bear relevance to the kind of systems that the statistics describe, that the first particle is so small that it does not warrant our attention. This is not the case. In human systems we see the effects of inclusive rules and exclusive rules. When certain members of society are excluded from the consequences of their actions by virtue of their wealth or their legal standing or their club membership then the whole society becomes unstable. Just as it does in the physics systems of Bose Einstein. The Bosonic particles are unstable when not in the presence of the Fermi particles.

Figure 11-3. The four fold infinite transformation which births the extravariant World from the intravariant

On the other hand if all members of a society are included and bear the consequences of their actions, both legally and financially, then the society soon becomes stable. These are the Fermi people. 1

The reality of life is that both Boson people and Fermi people do exist. However if one understands the way in which Fermi particles coalesce in physics we may be able to stabilise society in the same way that nature does in the atomic structure. But that is another story. For now it is sufficient to understand that there are Boson particles which have whole number spins such as 1, 2, 3 and there are Fermi particles which have half number spins such as 1  3 ,  and   52 . These correlate precisely with the two Worlds geometries (Figure ( Figure 11-3). 11-3). The intravariant is boson-like 2 2 and the extravariant is Fermi-like. The transformation that takes the Boson World to the Fermi World is shown in Figure in  Figure 11-4 for 11-4  for the case of Gravelectromagnetic World. Or, at least the results are shown. The transformation itself is virtually impossible to show

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on paper. Each great circle element has to be expanded to its infinite case and then contracted into the extravariant geometry. It is suffice to say here that all the line elements in the intravariant World are the same as those in the extravariant World and they are all in the same orthogonal relationship. TAO One set is rotated with respect to the other. The rotation makes the imaginary region of the World geometry transfer from the outside to the inside.  It was a great surprise to me that a 4-dimensio 4-dimensional nal geometry could be rotate rotated d at all. It can be seen in in [note1127]  [note1127] that   that  in the passage of enlargement to infinity a rotation takes place during the first half of the transformation from  0 π  to the π/4. The continued passage from infinity to the smallest case of the rope trick is a further quarter rotation from π/ 4  to π/2. I call these two Worlds intravariant and extravariant because these would be the correct mathematical terms. Both geometries geometri es contain identically the same elements. They are orthogonal to each other other..  It turns out, from working with the two geometrie geometries, s, that the intravar intravariant iant one is continuous continuous and unstable whilst the extravariant one is discontinuous and stable in nature.  Most important, in my opinion, is the fact that these two are orthogonal for this enables us to continue the orthogonality  principle further into the Worlds Worlds of the second kind. Indeed, it sugge suggests sts that the intra to extra transformation transformation enables any representation to be continued indefinitely. And that is rather remarkable because it implies that there is no centre to the geometries either at the beginning or at the end. The Alpha and Omega of the universe is imbued in ourselves and the nature that surrounds us. Which way round we see it is instilled in the very nature of our soul.

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Figure 11-4. Assembly of the repr representations esentations of Ferm Fermions ions and Bosons.

Ostberger found this transformation in the seventies. It was many years before he realised that the two Worlds could assemble together under certain conditions. He had already worked on some geometry which, because of its absorptive properties, disappeared when assembled. He attempted several combinations of assembly that culminated in the  “World geometry of the second kind”.  Figure 11-4 11-4 shows  shows how he conceived the assembly. The extra-World was reduced to its minimum state and then assembled on to the intra-World so that the corresponding elements (of the same colour) were in opposition to each other. The external elements of the extra-World met the surface elements of the intra-World in opposing directions. This opposition or anti-commutation led to the reciprocal elements producing a unity at the surface2. Unlike the absorptive opposition of elements that produced annihilation. The result was Figure was  Figure 11-5 11-5,, the Grav-electromagnetic World of the second kind. There are six ways to assemble the two Worlds. These are labelled   ,   and   with their contras back-   , back-   and back-   3.  

 

 

 

 

 

This diagram is the last of a series of developments in the notes. As time passed and more work was done so the

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elements on this World were changed to correct errors that had inevitably been made from being in the forefront of the research.

11.3. GEM World of the second kind Figure 11-5. The anti-commuting assem assembly bly of the intra and extra Grav-electromagnetic Worlds Worlds that cre4 ates the representation of atomic elements

A description of the geometry in Figure in  Figure 11-5 11-5.. First the nomenclature. Ostberger often took plenty of time to develop a nomenclature that would be useful many years into the future. Here the subscripts are intravariant and the superscripts are extravariant. The zeros indicate a reference element which generally seems to be unmeasurable. The plus and minus signs are here the forward or backward directions and not the sign of the magnitude. The suffix   m  is magnetic,   e  is electric and   g   gravitic. The   Z  are the potentials. The   B ,   D   and   V   are the so called  fluxes  of the three Fields. The number affixes count the elements of the system. Thus  B +1 is the forward element of the extravariant magnetic field.  B+0  is the reference element of the intravariant surface. To generalise the surface elements the Greek  ξ   ξ ,  ψ   and  ζ  are used for the magnetic, electric and gravitic fields respectively in upper and lower case.

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11.4. The atomic elements This book is not the place to describe the attributes of this geometry that makes it wholly suitable for the description of atomic phenomena. The notes are for that. What follows must suffice for my reader. There are six positions of the extra-Worlds. As the nomenclature implies they are the representations of the lectron, the   roton and the   eutron. Remember this is a vector geometry and and it has no magnitude magnitudess only orthogonalities.  

 

 

1. Each Each   ,   and   is part of a pair which also includes an opposite spin partner. partner.  

 

 

2. The spin of the electron is the half-vector half-vector that sits diametrically diametrically across the gravitic extr extra-elem a-element ent (green) g labelled Z  . 3. The charge on the electron is ca carried rried by the half-vector which sits diametrically across the element element labelled e Z  . It has a spin direction opposite to the Proton but in the same plane. 4. Th Thee Z m in the electron is the element that creates the Zeeman effect. 5. Th Thee  Z g is the element that counts the electrons in terms of its gravitational effects. Its length, the circumferential length, determines the number of the electrons present. This is unlike our conventional view of the counting process. We are accustomed to counting things as separated parts but here we count both in series

and parallel the rings that are on the surface. Adding the rings in series makes a larger surface. Adding them in parallel creates rings providing that the conditions allow them to be accommodated on the extravariant 4-D surface. The pairing-particle must also be counted contemporaneously. In the case of electrons this means that the down spin electron which is at the  “back” location must be counted too. This process is shown in the notes to account precisely for the singlet and triplet cases in the H atom. It also has the correct attributes to account for the multiplets which arise in heavier atoms. 6. At   the e counts the protons in terms of its electrical (charge) effects. It is accompanied by an axial vector  Z  which measures the charge.  

7. Any two coincident line elements elements creates an observable observable  “particle”. 8. The plane plane of the first ele elemen ment, t, e.g.  D+1 , lies out of the plane of the zeroth element. This is because the orthogonality conditions demand it. We may speak of  “the first element plane”. This first element is also skew to the second plane. It passes over   , in inside side back-   , under back-   and outside   . The other two fi first rst plane elements take a similar path. In this way all three maintain their orthogonality to each other and to the rest of the elements. This is made more clear in the  Figure 11-6 where 11-6  where the three planes of the first tilting element are each clarified  

 

 

 

9. Ever Every y junction on the geometry geometry has three mutually orthogonal orthogonal elements and is orthogonal to every other  junction with the same. There are at least 36 such junctions making at least 108 orthogonal relations. 10. The intra- elements elements which face face on to the extra-surf extra-surface ace seem to be  “silent”. That is, they lend virtually no attribute to the particle. Thus the electron has   Z g silent. We may interpret this to be that the electron is virtually unaffected by gravitational potential. 11. The neutron neutron has Z e silent. It has no charge. It is unaffected by an electric field. In the proton  Z m is silent and so the proton should have no intrinsic spin magnetic moment. 12. The intrav intravariant ariant region is termed “orbital”  in the literature and the extravaraint region is termed  “spin”. 13. The Bosonic spins are of three types. The They y are axial vectors that measure across the intra intravariant variant interior interior.. The ISO spin, for example, is the intravariant axial vector of the   -   plane.  

 

14. The Helium nucleus or alpha particle turns out to be the comp complete lete set of four fermi particles in the   -   plane.  

 

correctly 15. The   -state electron is a 2π  2π rotation from the back-   -state electron when the rotations are measured correctly  

 

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Figure 11-6. The first set of orthogonal anticommutin anticommuting g elements on the intravariant intravariant surface

A vect vector or mode modell of the the Hy Hydr drog ogen en at atom om wh whic ich h co comp mpli lies es with with th thee dire direct ctio ions ns of this this geom geomet etry ry is the the su subj bjec ectt of an anot othe herr book on the subject (available from the Ostberger website). The magnitudes which are extracted are consistent with current theory. It may be possible to remove certain anomalies from current theory with the realisation that the vectors in the intravariant space are infinitely separated from those in the extravariant space. The Sommerfeld fine-structure constant has been derived and calculated from this vector model in  Chapter 12. 12. The table of Delta values in the Appendix the  Appendix J not J not only holds the key to the fine-structure constant but also seems to have important scales which can be applied to other atoms. There is much work to do. It will be difficult for the reader to grasp the idea that atomic phenomena can be represented in this way. After all, it is a far cry from the Bohr model of electrons whizzing round in orbits. Yet that model worked. At least sufficiently to get the results that we have arrive at today. What the picture says is this. The atoms have strict orthogonal relationships. These relationships are directional in character and they can be displayed in a geometry; the World geometry of the second kind. But this is not the end of the story. This is not the final solution. This is like a Schroedinger equation except that this equation is a directional one. It gives the general picture and provides the essential relationships but it needs a more detailed solution for each of the atoms. The solutions are not easy to find. The rules of operating must be followed rigorously. Yet sometimes new rules are found which may be sown into the fabric of the theory just as with any other theory. The essential attributes must be that the theory is consistent throughout; from beginning to end. Consistent that is in its formulation, operation and interpretation, which the reader will remember is the basis of the theory in the first place.

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The picture does show us that protons and neutrons are similar particles. But how do they work in the geometry? How can a proton turn into a neutron? How do para-hydrogen and ortho-hydrogen fit into the model? Can we explain the magic numbers and isotopic groupings? How does a singlet differ from a triplet? Can we do the mathematics, as well as see the picture, of these motions of the atoms? The answer must be a very likely yes  because Ostberger has already demonstrated some of these in one of the atoms, the Hydrogen atom. Nature is antisymmetric. The electrons are arranged as the antisymmetric eigen function determine. It is a matter of relating the eigenfunctions to the arrangement of circles on a surface and getting all the quantum numbers in

the right order. In the case of Helium5 this has been done. Not all the directional permutations will be allowed in nature and so a table of the permutations showing directional reasons for eliminating some has been made by Ostberger. The surprise is that there is such a great redundancy of permutation. Figure 11-7. Counting electr electrons, ons, the triplet states

To illustrate how the process works look at  Figure 11-7 11-7.. This is the representation of the triplet state for two up down wn spin electrons. This is a spin electrons. The two could equally well occur at the back-   location and be two do two electron state if the circumferential measure is a single Planck  h  of energy. The two electrons are in parallel and form a surface. The 4-D surface area is minimised, representing the net lowest energy both magnetic and electric in the structure. The result is that the two electrons can be assembled on the surface such that the greatest diameter (a great circle) is not more than the next quantum size up, in this case  2h  2 h. In fact, in this case (Figure (Figure √  11-7), 11-7 ), the diameter of the blue element is 2h or 1.  1.414 414h h. When we have a surface which measures  2h  2h across (the great circle) then we have a potential for four electrons (two spin up and two spin down) as well as all the lesser permutations.  

One must remember that the scales in one orthogonal plane are not the same as in another. Neither are the units of measure. So the blue (magnetic) plane scalar is different from the green (gravitic) plane scalar.

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Figure 11-8. Counting electr electrons, ons, the first multiplet multiplet state

Now look at Figure at  Figure 11-8. 11-8. This is the singlet solution. One spin up and one spin down electron. They are still in parallel but in separated locations. Which is one spin up and which one is spin down would not easily be recognised. What is more, since nature does not understand  “metre distance”  as a property, these two electrons could be any distance apart. And that seems to lead to Bell’s theorem. Figure 11-9. A sketch of a three electron state

Figure 11-9 is 11-9 is a sketch of a possible surface with perhaps three electrons on it. If such is possible and complies to the interval rules then this may be one state of Lithium. The theory says that there must be two separate ionic bond groups. One belonging to the   location and the other location. on. In the formation of simple diatomic diatomic molecules molecules such as   and  the theory belonging to the back-   locati 2 2  H   N  requires that the two atoms be back-to-back with each other thus forming a kind of pseudo antiparallel electron pair. The two electrons in the diatomic pair are of the same type. Either   electrons (spin up) or back-   electrons (spin down) but not mixed.  

 

 

 

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Chapter 11. Worlds of the second kind 

What the directional process gives us is a clue to the next solution. It is not a panacea for all solutions. We still have to find our way. We still have to understand how it works. There is a long journey ahead.

11.5. Computer Modelling The process lends itself easily to computer modelling. Work is underway to produce interactive models of the Law Worlds and tools to construct and manipulate the geometries. They are part of an exciting new journey too.

Notes 1. The Ferm Fermii people are are the infinite infinite players players of Prof. Carse’ Carse’ss  “Finite and Infinite Games” ([Carse87]  ( [Carse87]). ). 2. Equi Equivale valent nt to saying saying that th thee metric metric gij = g ji = g i j  = 1. 3. The term used to expres expresss the revers reversing ing of the character in the pairing locat location ion of the geometry. geometry. It make makess it easier to print in the absence of the new character. 4. See  [note1748]  [note1748].. 5.   [Note820] (Spin [Note820]  (Spin Eigenfunctions for Helium).

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Chapter 12. Using Chapter Using the proces process s for for Sommerfeld’s fine-structure constant 12.1. The first application When we are in the forefront of discovery there is no way of knowing what will come next, so it is prudent to allow our intuition do some of the guiding and take advantage of any new development that comes our way. Naturally, many new developments take us down a cul-de-sac, but that is all a part of the journey for we soon find out which kind of discovery has the cul-de-sacs and intuitively stay on the main highway. Ostberger did just this. He developed a flavour for the main highway towards his goal of modelling the Hydrogen atom. He gives his reasons for this:  I had to find some part of natur naturee that would lend itself to the theory in such a way as would give the opportun opportunity ity of   prediction.  pred iction. The macro macroscopic scopic universe did not do this and I could see no way of repeating repeating the kind of pred predictions ictions that   Einstein made. For one thing I had not read astro astronomy nomy and for another I did not have a theory of its content. Yes, I  could see that we were in error for looking through telescopes and mapping the heavenly bodies as if they are a radial cartology; yes, this would upset the Hubblist’s and stop the universe accelerating into oblivion, but this was not going to be a measurable observation in my lifetime. It is true that we may be able to send out high speed spacecraft and show that new objects appear around the entrance perimeter of black holes as the speed increases, but that will be a long way off too. So I chose to look at a white hole, the Hydrogen atom. The only atom that has no neutron and therefore has a hole inside. From this I could measure against the known mathematics. From this place I could see if there were any possibility of   predictive  pred ictive consequence consequences. s. This was my journe journeyy and anything that was not on the main highway would be spun aside.  I began began work to find a geometr geometryy that would fit the equations of the Hydr Hydrogen ogen atom. I had worked ordin ordinary ary polynomia polynomiall equations and found that the solutions lie in the coefficients; that one could construct simple shapes to solve equations without the use of the any numerical method. I had applied the same process to differential equations discovering that  the coefficients created whole geometric forms. In the most difficult equations and particularly those with complex coefficients the geometries representing the coefficients were moving in the space. They would form a step-wise series of  geometries moving synchronously with other similar sets. By the time I had got to the kind of differential equation that  Schroedinger produced I was well versed in the technique. I was able to follow the accepted solution which divides the equation three equations. Each pair of equations meet in a common 1 terms thisinto meant thatseparable there wasdifferential a common line of magnitudes between each pair of solutions . constant. In geometric  I knew that I would have to do this job myself. Nobody had been interested in the past and it would be a diversion trying to attract anybody to be interested in the future.  I knew that the Hydrogen Hydrogen atoms had to have Directions. Pa Particularly rticularly as I had established that a 4-dimensional world geometry was available for the modelling pro process. cess. If quantum mecha mechanics nics was a vector study then there had to be directions  for the vectors. The Hydr Hydrogen ogen atom was described by a quantum vector picture and so the atom had to have directions that I could find. I set about finding them. There was one strange coincidence in the physics of the atomic structure that caught my imagination. How did the  Differential  Differ ential Equation appr approach oach and the Quantum Mecha Mechanical nical appr approach oach arrive at the same answers to the Hydrogen Hydrogen atom? I discovered that there was a very small and subtle difference between the two.

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Figure 12-1. Finding the ne new w universe

At the outset, it seems that Ostberger had very few clues as to the way in which geometry could serve the purpose of representing the atomic structure. One of the clues that is marked in the margin of the notes forms a relationship between the Rope-maker’s first  “trick”  and Planck’s2 constant. Figure constant.  Figure 12-2 12-2 is  is the case of the rope trick in which one unit of rope is added around a body of zero size . This is compared with Planck’s    in Figure  in  Figure 12-3 12-3 which  which is identical in geometric form. There is a scribble in the margin. It says,  “The Planck trick... there can be only one shape that connects  h  with    and this is it”. The next sketch shows a shaded circle with the word  “energy” inside and the words  “Joules  per cycle per  second” are scribbled, with the emphasis.

Figure 12-2. The one u unit nit rope trick

The message is that a Joule-sec is actually a Joule per cycle per second if the cycles are ignored. But in the geometry the cycle is a line element, and this line element is the element of Gravitational Direction, the Direction that we call Potential in the intravariant case. In the extravariant case the same Direction is the one in which the frequency of light rotates. The area of the circle would be a representative of the energy if the circumference were a representative of the frequency. So Ostberger, relating the trignometric frequency to circumferential measure 3 had a starting point.

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Figure 12-3. The Planck T Trick rick

The first breakthrough for Ostberger seems to have come out of one of the many library Standard Forms that he had collected. In a series of notes4 he had found a set of geometries that seemed to allow the squaring of the circle. They did not really, but it seemed so at the 5start. In essence these geometries are very simple in so far as they are 6 well known shapes. They appear in algebra as Radical circles. They appear in system dynamics as  “M-circles” which relate open and closed loop responses. They appear in magnetic and electric fields and many other places in mathematics and physics, but here they appear in a new guise because these  red  geometries   geometries are quantised. XXXX below we say that the blue geometry is quantised. and above we say that the red geometry is quantised. ??? XXX

Figure 12-4. The complete set of Red geometries of the inner pr product oduct

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Chapter 12. Using the process for Sommerfeld’s fine-structure constant 

If we look at the two chords in the centre we see that an application of the inner chord theorem mentioned on page 4 XXX of  Chapter   Chapter 3 results 3  results in the square root formula7.  red 

Thismeasurements is easily referred tohorizontal. as the  geometry. All the elements arrive at a pole situated on the infinite element and the  p  are There is also a  blue  geometry. In the blue geometry of  Figure of  Figure 12-5 12-5 we  we may take the outer chord theorem and apply it to obtain the same formula! This time the measurements  p  are vertical. All this data must be taking into account. In mathematics we would not see these small, but important differences. The blue geometry shape is also well known in mathematics, but here it is quantised.

Figure 12-5. The complete set of Blue geometries of the outer product product

No Now w, suppos supposee we ass assemb emble le these these two geomet geometrie riess ort orthog hogona onally lly,, as in Figure Figure 1212-6 6. A we well ll know known n sh shap apee is prod produc uced ed,, except that it is quantised. The whole space is littered with quantised values and quantised (orthogonal) points.

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Chapter 12. Using the process for Sommerfeld’s fine-structure constant 

Figure 12-6. The red and blue geom geometry etry assembled orthogon orthogonally ally

The red8 geometry and blue9 geometry assembled orthogonally. The red elements can be used to represent the magnitudes of the vectors 10. The blue elements can be used

to represent the Directions of the vectors 11. There is a two-fold infinity of trigonometric functions which can be arrayed geometrically in this space. Functions of all amplitudes, powers and phases can be represented 12. Although these orthogonal geometries are well studied in terms of algebra, the trigonometric functions contained in them are not well studied in terms of geometry. They have a close relationship to Fourier Analysis. In this form the general picture of the application of this kind of geometry says that the blue elements are representative of the directions of the vectors and the red elements are representative of the magnitudes 13. By using the trigonometric geometry in Chapter in  Chapter 6 this 6 this geometry can be used to represent Fourier components. Suppose we assemble these two geometries in parallel! Or suppose we assemble them antiparallel!  Figure 12-7 12-7 is  is the antiparallel assembly.

Figure 12-7. A magnitude ess essential ential to Quantum Mechanics Mechanics

  p( p( p + 1)

Ostberger studied this assembly and produced the table of constants in Appendix in  Appendix J. J. Thes Thesee constants constants come from − q  + q inserting quantum values into the formula in Figure in Figure 12-7 12-7 for  for δ  p  and δ  p . These are his  delta values.

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Why are they significant? 1. These are natural natural constants. constants. 2. They have arrived out of a geometry that has has shape without reference to magnitudes. 3. They reflect the angle of tilt of the first eleme element nt of the W World orld of the second kind. 4. They contain the magnitudes essential essential to Quantum mechanics. The geometry in the Figure the  Figure 12-8 has 12-8  has some peculiar properties. As shown it represents all the geometries with integer values in the lower half plane.

 = 1)14 Figure 12-8. The delta geometry sets are part of the Hydr Hydrogen ogen atom solutions (q  =

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1. Despi Despite te the fact that it is circular circular it is none-the-less none-the-less quantised quantised.. The values of  p  p  and  q  are integer and half  integer. 2. There is a special conn connectio ection n between the areas of the green geometries geometries and those of the red and the blue geometries. The area formed by corresponding circles of equal values of  p  p  is always equal to the area within the green element circle regardless of the values of  p. 3. The whole geometr geometry y is scaled by  . 4. The m magnitu agnitude de of  δ   δ  is less then the magnitude of   ! The argument for  “hidden variables”  is here partially vindicated. I say partially because the values of  δ  are  are linear measure whereas the values of    are  are an extraction from the value   h  which is an area representing the energy. So the argument is not so much a question of  whether there are hidden variables but rather one of whether we can measure something less than energy. 5. There are two half states states of the green circle. circle. 6. To help our imagination imagination we may regard the green circle as a ball which when depres depressed sed opens the gate formed by X . The X R  is the close state and the  X B  is the open state. 7. Th Thee  red  and  and  blue  geometries are antiparallel. They have opposite directions. 8. This geom geometry etry was used by Ostberger Ostberger as the first part of the solution to the Hydrogen atom atom.. It fits into the

World model on page 9 XXX of   Chapter Chapter 11 11 in  in the  D  plane. The exact method of fitting it to the orthogonal conditions is given in the notes. 9. The values values of   q  q  are   are a part of the extravariant region which is shown green here. The values of  p  are a part of  the intravariant region in red and blue. The two parts anti-commute and cannot be joined in a classical sense. 10. In the Hydro Hydrogen gen atom the value value of  q   q  =  = 1. In this geometry when the integer values of  p  p  have reached ∞ − 1 then a new set is available with the value of   q   q   = 2, and so on. We can now extend the study of atomic structures to integer values  q >  1 ; the green circles.

12.2. Sommerfeld’s fine-structure constant Look Look at the the ta tabl blee of delt deltaa val alue uess in Append Appendix ix J. Lo Look ok at th thee fir first st val alue ue,, when when p =  p  = 1 and q  =  = 1. I t i s 0.08786437626. This is noticeably close to the square root of the fine-structure constant of  0.  0.085423666, but Ostberger ignored it for many years as pure coincidence simply because it did not fit the geometry. Some time around 1983 he realised that the element in the world geometry which must be used to measure the scale of the model, has a tilt and he had ignored it. He had been trying to fit the delta geometry into the  D0  plane. But when he applied it to the  D1  plane and calculated the projection of the delta geometry on to the  D0  plane he arrived at exactly the Sommerfeld constant.

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Somerfeld’s fine structure constant is calculated from the first value in the first table of delta values 15. The fundamental scalar of Hydrogen Hydrogen is derived from the fact that some natural length of magnitud magnitudee exists, − δ 11 , which is spacially spac ially rotate rotated d thr throug ough h an ang angle le whos whosee mag magnitu nitude de may also be me measur asured ed as − δ 11 . The direc direction tion − δ 11 and the ma magnitude gnitude − 1 δ 1  are orthogonal.

Figure 12-9. The calculation of Sommerfeld’ Sommerfeld’ss fine-structure fine-structure from the delta geometry

The Figure 12-9 The Figure 12-9 shows  shows the final stage of the calculation. The value of  − δ 11 which is shown in Figure in Figure 12-8 12-8 becomes  becomes 1 the sine of the tilt angle. It follows that the cosine is 1 − − δ 1 . This sine-cosine product is a chord (like  C   in Figure 5-5) 5-5) and the fine-structure constant is an orthogonal chord (like  A  in Figure  in  Figure 5-5 5-5))16.

 

One has to study the geometry very carefully to see how the two parts, the red and the blue, fit the world model. The atom seems to be partly in two states. They appear to be in two half states and so the construction of the

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Chapter 12. Using the process for Sommerfeld’s fine-structure constant 

geometric picture is particularly geometric particularly difficult. difficult. It eluded eluded Ostberger Ostberger for many years until he assu assumed med that the two half  states were Magnitude and Direction. The Magnitudinal representative, the blue geometry is the leader of the first half state and the Directional representative is the leader of the second half state; first Yang then Yin. However, there is nothing to say that the reverse process cannot be constructed to yield exact results also; first Yin then Yang. The one case has the potential to satisfy the differential equation (Schroedinger) approach and the other the Quantum Mechanical (Heisenberg) approach. Either case yields the fine-structure constant. So that students may

follow him Ostberger made notes on his intuitive assumptions, particularly the ones that worked.  I find, in genera general, l, that every yang solution or repr representation esentation is followed by a yin one. In the Hydrogen Hydrogen atom it would  seem that every successive energy level alternates as a yin and then yang solution. In this case the yin is the directional  part and the yang the magnitudinal part. That also seems to be the lesson in life.

It is because we need to learn how to find solutions with these Directions and at the same time seek to apply them to know known n physical results that could lead us into blind alleys. alleys. Thus, makin making g assu assumption mptionss about the way in which these processes work is essential to its development. However it will add nothing to our understanding or the future application of the process if we cannot claim consistency or reflect upon the accuracy. In 1930 P.A.M. Dirac wrote in his book on quantum mechanics17 the following:  In answer to the first criticism (the idea that a photon can be partly in each of the two states of polarisatio polarisation) n) it may be remarked that the main objective of physical science is not the provision of pictures, but is the formulation of laws governing phenomena and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better, but whether a picture exists or not is only of secondary importance. In the case of atomic phenomena no picture can be expected to exist in the usual sense of the word “picture”, by which is meant a model functioning essentially on classical lines. One may, however, extend the word “picture” to include any way of looking at the fundamental laws which makes their self consistently obvious.

The geometries of Ostberger are certainly not along classical lines. They are an extension of the word picture which includes the measurement of vector curvatures. They are a consistent  “way of looking at the fundamental laws” which exposes the truth of their nature. It is the truth that we seek in order that we may understand the path that leads us on to the next discovery. Without the truth, in whatever form, we are wandering in the wilderness. To avoid the mathematics entirely in this introductory book would be impossible. There are other writings from Ostberger which may ease the burden of understanding mathematics for those who are not conversant with its idiosyncratic language. I have minimised the mathematics by leaving a trail of references to the notes.

Notes 1. The g geomet eometries ries used used aare re gi given ven in the [note18xx] the [note18xx] note  note series. In particular the notes 1870-90 give the detailed mathematics and associated geometric constructions. This is being issued as a separate booklet of some 60 XXX pages. 2. See  Appendix A A.. 3.   [Note112] 4.   [Note6xx] series. [Note6xx]  series. 5. The algebr algebraic aic versions versions are graphical graphical shapes. It is useful and necessary necessary to know the relat relationsh ionship ip between the Ostberger geometry and the graphs because the equations are expressed in the algebra. In algebra we describe the shape but we can be easily blinded to the linear relationships of the shapes such as are shown here. 6. See the notes on M-circl M-circles. es. 7. All Eucli Euclidean dean Theorems Theorems can can be applied to a true vector vector geometry geometry to produce produce a true result. 8.   [note610]

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9.   [note612]

10. 10. [Note608]  [Note608] 11. 11. [Note609]  [Note609] 12. Refer to the [note5xx] the [note5xx] series  series notes. 13. This idea is too rich to express here but is clarified in the notes. 14. Notes 1850-65 1850-65 15. 15. [Note1850]  [Note1850] 16. The detail is given in the [note666] the  [note666] and  and [note667]  [note667].. 17. 17. [Dirac70]  [Dirac70] (page  (page 10)

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Chapte Cha pterr 13. New New wine into into new skins skins If I do not stop this book here it could become as voluminous as the notes themselves. The notes are numbered from 0 to 3000 although there are many spaces left vacant for future additions. In all there are probably half this number. The notes begin with the words,  “Some people write a diary which is a record of the past. I have written this notebook which is something for the future”  and that is how he sees the world for he says again,  “there is always some part of today that has some something thing for the futur future” e”. He sees today as a part of the future and lives to continue

it. Others see yesterday as part of today and wish to consume it. This is like the thermodynamic Volume that obeys the absorption rule (Figure (Figure 8-9 8-9). ). Ostberger views the  “Volume Remaining”  but some view the  “Volume 1 Consumed” or as professor Carse says,  “There are at least two kinds of games. One could be called finite, the other infinite. A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play”.

These two kinds of games are the ones that people play; and so does nature with her Bosons and Fermions. It is not that either is right or wrong in the universe but rather that one must have preference over the other if the flow of life is to be maintained. The intravariance and extravariance of the world determines its stability. That is the Ostberger discovery. discovery. There is a subtle difference between the two ways which cannot be observed except in action. It is the manner in which nature acts that tells us which is Boson or Fermion. So also for the finite or infinite player, it is the action that foretells his play not his words. The numbers in nature do not reveal its stability, only direction does. There is an interesting part of Roger Penrose’s book 2 in which he divides up the Riemann tensor into two components, “Riemann = Weyl + Ricci” . This is a division of the magnitudinal and directional aspects of the Riemann tensor. The Weyl part, with its ten components, are directional in character whereas the Ricci part, with its ten components, are the magnitudinal part. This is easily related to Figure to  Figure 2-18. 2-18. The Weyl describes the  “direction of the magnitude”  whereas the Ricci describes the “magnitude of the magnitude”. The other two parts of   Figure Figure 2-18 2-18 belong  belong to Ostberger. Forming the “direction of the Directions”  is making the geometric shapes and their orthogonalities. Finding the  “magnitudes of the Directions”  is seeking the answers such as the examples given of the trigonometric representation or the Delta geometry. All four attributes are essential to our understanding of the universe around us. Now that we have been presented with the remaining two we have one heck of a lot of work to do! This is in complete harmony with Steven Hawkin’s remarkable view of such a discovery: [Hawkin88] page [Hawkin88]  page 175  However, if we do discover a comple  However complete te theory theory,, it should in time be understandable in broad broad principle by everyone, everyone, not just  a few scientists. Then we shall all, philosophers, scientists and just ordinary people be able to take part in the question of why it is that we and the universe exist.

Steven Hawkin’s remarkable insight continues, [Hawkin88] page [Hawkin88]  page 168 Seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity. Nowadays tens of thousands of university graduates graduates do, and many millions of people are at least familiar with the idea. If a complete unified theory was discovered, it would only be a matter of time before it was digested and simplified in the same way and taught in schools, at least in outline.

and again his remarkable insight into the future yields, [Hawkin88] page [Hawkin88]  page 169  A complete, consistent, unified theory is only the first step; our goal is a complete understand understanding ing of the events ar around ound us, and of our own existence.

There clearly is a lot of work for tens of thousands of people to do.

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Chapter 13. New wine into new skins

This book has been an overview of a self-consistent theory about strings in space, albeit that those strings may have thickness in two dimensions and be of the simplest circular kind. But it differs from String Theory in two essential attributes. The first is that it is in the language of lines in space that can be drawn on to paper and computer. The second is that the Orthogonality Theorem of Quantum Mechanics has been placed foremost in the process in place of the Fundamental Theorem. Let me explain. There are two fundamental parts of Quantum Mechanics. The first appears at the top of page 32 of Dirac’s Book 3. THEOREM: Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal.

This principle is clear. If there are two eigenstates of an eigenvector and they belong to different values then they are orthogonal. This means that if there are two states for which a measurement of the vector is certain to give two different results then those two states are orthogonal. The insight to this is that each different state is orthogonal. Thus Orthogonality is as important as the number that we can attach to the eigenvalue of the state. In this theory the Orthogonality of the state is the first objective of the study and the numerical value the second. This is not the case in String Theory where the reverse process takes place. The second part is the fundamental equation and the fundamental conditions which are on page 87 of Dirac’s book 3 and are shown below. These are the primary objective in String Theory.

This equation also expresses one aspect of the Law Field. For example a rate of change with respect to the first element (Direction) produces the second element (velocity). Providing both Magnitudinal and Directional effects are considered separately. Eddington, back in 1928 foresaw that the fundamental quantum conditions were of greater significance than simply a formula. He predicted that we would be able to base some of our understandings on something other than number. He virtually predicted the Ostberger discovery. I have placed Ostberger’s scribbled notes along side the two key equations of Quantum Mechanics on the page below. He notes also that there will be two directional compliments to these. One is clearly the orthogonality conditions which comes from the Orthogonality Theorem of Quantum Mechanics4. The other, I do not know what he meant.

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Chapter 13. New wine into new skins

5

 “The Nature of the Physical World”

Sirvirtually Arthur Eddington in his of my Ostberger’s notes. It predicts thewrote the contents

 the following passage in 1928, page 209.

 I ventur venturee to think that there is an idea implied in Dirac’ Dirac’ss treatme treatment nt of [the above equation] which may have great   philosophical  philosoph ical significance indep independently endently of any success of this particular application. The idea is that in digging deepe deeper  r  and deeper in to that which lies at the base of physical phenomena we must be prepared to come to entities which, like many things in our conscious experience, are not measurable by numbers in any way; and further it suggests how exact  science, that is to say the science of phenomena correlated to measure numbers, can be founded on such a basis.

Notes 1.   [Carse87] 2. See  [Penrose]  [Penrose] page  page 271. 3.   [Dirac70] 4.   “Tw “Two o eigenvectors of a real dynamical variable belonging to different eigenvalues ar aree orthogonal.” [Dirac70] page 32. It is difficult to believe that every eigenvalue is going to be orthogonal to the next because that would mean that there are vast numbers of orthogonalities in the space. But Ostberger shows that such large numbers of orthogonalities can described and depicted for study. 5. A Frie Friend nd..

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II. The Appendices

 

Append App endix ix A. The first first rope rope tric trick k A.1. The note 104b In the following I consider the possible quantised extremities of a connected curvature. That is to say a continuous curvature that can change in integer intervals no matter how small the interval. I look at the largest possible curvature and then the smallest possible curvature. I conclude that a straight line cannot exist in a space which is to be used for the measurement of a magnitude. This infers that a magnitude space has no zero. A directional space has. With this conclusion results which accord with experimental reality can be obtained. By way of introducing the idea consider a rope around the Earth ( Figure A-1 A-1). ). Let us say it is 40 million metres round the Earth and the rope just touches the Earth around its entire circumference. Let us now add one metre of  arc into the length of the rope. How far from the Earth will this rope be if it stands away from the surface equally all round?

Figure A-1. A rope aro around und the world

Let the diameter of the original circle (the Earth) be   d1   and the diameter of the next up integer circle be   d2 . Likewise the circumferences are  c1  and  c2 . Then,

c2  =  c1  + 1

( 1)

The distance that the c2  circle stands away from the Earth  h , is given by:

d1  = 2h   (2)

d2

− Since c =  c  = π  πd d  in general we may express (2) as,

1/π /π((c2 − c1) = 2h 2h   (3) But from (1) the difference of the  c’s is a unit of arc so that,

1/π /π =  = 2h and

h  = 1/2π   (4) So we have the fact that if a rope were put around the Earth so that it just fitted snugly and we then stretch the rope by one metre it will stand away from the surface a distance of nearly 160mm all the way round! This is not our normal comprehension. There is something wrong with the proportions of our thinking and that is why the student must play with string and pencil to acquaint himself with the reality of an otherwise difficult magnitudinal concept.

Figure A-2. For each m meter eter of circumference circumference ...

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 Appendix A. The first rope trick 

Given that the rope is 40 million metres in length and that we have added only 1 metre to its length our sensibilities tells us that the stand-off is likely to be of the order of  1/  1 /40 40,, 000 000,, 000th It is not. It is  1/  1 /2π   times the distance added in.

Figure A-3. A rope aro around und Jupiter

But it is important to note that the one metre added is a length of arc whereas the distance of the stand-off  h  is a linear measure. Consider now a rope around Jupiter that is some 446 million metres round. Stretching the rope by one metre makes the stand-off 160mm. Again! The reason is because the same simple mathematics applies.

Figure A-4. The largest possible cur curvature vature of the rope

Consider now a rope around the Universe (Figure ( Figure A-4). A-4). We have established that for each unit of arc added into the circumference of a circle there is a  1/  1 /2π  displacement radially (and therefore orthogonally) to the circle. So that, if we were standing on the curvature that fits around the universe and someone adds a unit of length into the arc of that curvature at a point that we cannot easily observe then we will motion with that curvature a distance of   1/ 1 /2π  orthogonally to the curve. Providing, that is, there is continuity in the curve. Without any other line of  reference we could not detect the displacement because we are moving in conjunction with an infinity.

Now consider a rope at infinity. The  infinite element  is  is the straight line in Figure in Figure A-5. A-5. Whether this is by definition or by reason does not affect the argument. When the curvature is zero the radius is infinite; both are extremities.

Figure A-5. Curvatur Curvatures es at infinity

Consider the curvature that is last before the infinite element; the one in Figure in  Figure A-5 A-5 that  that has its centre of curvature somewhere below the infinite element. A unit of arc added into this will cause a displacement  1/  1/2π  upward. Next consider the first curvature after the infinite element. A unit of arc added in to this will cause a displacement  1/  1/2π downward. Where in the space are these displacements? Either the two curvatures are  1/  1/2π  apart or they are twice 1/2π  apart. Which is it?

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 Appendix A. The first rope trick 

Figure Figu re A-6. One un unit it

To answer this consider the smallest possible circular path in which  d 1  is zero. That is we put the rope around nothing (Figure (Figure A-6. A-6. We cannot have a smaller circle since there is only one unit around its circumference. To divide this unit into smaller parts is to divide the arc unit into the same smaller parts and we are doing no more than scaling the whole sheet of paper including the infinity. Whichever rod we use to measure the  c’s we use the same rod to measure the d’s. We cannot pretend to change the unit of measure of the smallest arc without changing the largest arc likewise. It would be cheating on the continuity principle of the circles.

Figure Figu re A-7. Half a step to infi infinity nity

So what happens to the maximum curvatures in Figure in  Figure A-5 A-5?? Do the displacements cross over in the distance h  making the space between them bi-directional as in Figure in  Figure A-7? A-7? Or is it like Figure like  Figure A-8 in A-8  in which the space remains uni-directional but the distance between the two maximum curvatures is  2h  2h?

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 Appendix A. The first rope trick 

Figure Figu re A-8. One step to infi infinity nity

If the Figure the Figure A-8 is A-8 is admissible then the infinite element exists and the two curvatures each have a step of one unit 1 to infinity. If the Figure the  Figure A-7 is A-7  is admissible then the two curvatures have a step of one unit of       to each other and the infinite element does not exist.

A.2. Concluding remarks 1. The paths shown here are circular and simply connected. What happens if the paths are not circular? In note 144 I deal with the non circular paths using line integrals and the Residue theorem. However I can leave my paths circular without loosing generality for several reasons. Firstly I can apply the Riemann mapping theorem and map all the non-circular paths on to my circles thereby allowing me to develop non-circular paths later as part of the solution finding process. Secondly I need to stay with these circular paths because I am going to seek orthogonal conditions which are based on establishing rotations in space. I must therefore find the rotations first and then apply any distorting mappings afterwards. Thirdly I need to be able apply the shortest distance axiom to the geometry and since the shortest distance between to orthogonal points is a quarter of a rotation I must keep these quarter rotations as simple paths i.e. circular. So, circular paths are first and other paths, whether simple or multiply connected, must follow. 2. There arises the question of how many directions there are in a line. Although I believe that I have dealt with this in my notes it is worthy of mention here. I argue as follows, ....if both ends of a line element are at infinity the line exists as a direction but has no measurable magnitude. (Analogous to a single zero character in number theory.) ....if one end of a line is at infinit infinity y then there is only one direction direction in the line. (Analo (Analogous gous to the chara character cter “1”  in number theory.) ....if neither end of a line is at infinity then there are two directions in the line. (Analogous to saying that we have two characters, 0 and 1, to create a system). Thus we must have two directions in a line element to create a system of Directed geometries. This is analogous to a binary system. 3. The principle of absorption applies to directions. “When a man takes a step to the East he contemporaneously takes a step to the West. 2”  For each action forward

we can measure a counter-action backward. This also demands bi-directional line elements. The question here is, which of the two geometries Figure geometries  Figure A-7 or A-7 or Figure  Figure A-8 A-8 is  is the one which exists at infinity?

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 Appendix A. The first rope trick 

I have considered the arguments, made numerous mistakes, and concluded that both are permissible. Both are necessary and then they are sufficient to make the representations. Indeed, it turns out that both are required. They are  “necessary”  and  “sufficient” to extract solutions to our universe. Everything that I see and understand comes in pairs. I have never seen an exception. Perhaps we should accept this principle and make progress whilst allowing somebody to prove it otherwise 3. What has brought me to this conclusion? My representation of the Hydrogen atom required the calculation of the Sommerfeld fine-structure constant from a geometry (Note 666). In this geometry the use of both the  Figure A-7 and A-7  and  Figure A-8 A-8 are  are necessary. Both are required in the atomic representation. The Figure The  Figure A-7 is A-7  is the magnitudinal vectors and the Figure the  Figure A-8 the A-8  the last of 

the directions belonging to them4. There are a host of other supporting arguments. ....Godel’s theorem supports my point of view. Any system of circles and connected mappings are incomplete. ....this argument supports Bells theorem. I therefore conclude that the infinite element cannot exist in a geometry of curvatures which represent Magnitudes. But it can exist if it represents the Directions. The first orthogonal condition. An orthogonal condition arises when the rotational measure between two line elements is   π/2 π/2  at the point of  intersection and nowhere else connected. In Figure In Figure A-8 A-8 the  the smallest circular path and the largest circular path have no intersections that can be orthogonal. The two meeting points either meet at zero or  π -apart depending upon the directions of the elements. However Figure A-7 does A-7  does have a possible orthogonal condition at the intersection of the last curvature before infinity and the circular path with radius  1/  1/2π . i.e. between the last curvature and the first curvature. But is it orthogonal? I have used Dirac’s     to   to label the distance h/2p here to emphasise the significance of dealing with concepts at infinity. But      is is used in Quantum Mechanics solely for the quantum of energy per cycle, 6.602 x 10-34 in the Grav-electromagnetic world. There is no reason why the same principle cannot be used in other representation in Thermodynamics, for example. The fact is,    is not local. It belongs belongs to measu measuremen rements ts near infinity infinity. Quant Quantum um Mechanics seems to be taking the relationship between h and    as indefinitely linear. It cannot be. As the geometries grow    is also subject to the the effects of the curvature space The relationship h = 2ph-bar can only be a circle. But the first circle in this theorem is the only one that has this relationship. The next circle contains a small element of curvature and so the relationship is an approximation.

A.3. The   1/4π  connection The area of  Figure   Figure A-6 is A-6  is  1/  1/4π . It is a constant and has no units of measure. It forms a constant relationship with h and with other well known constants of the grav-electromagnetic World. In the Note 1830 Ostberger makes it clear that he regards the Bohr radius as a connective constant between our world of distances and the gravelectromagnetic World where there are no distances. Distances are simply not understood by nature. In scaling hydrogen this idea is used to clarify the following expression:

a0 Rµ  =

 1 α 4π

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 Appendix A. The first rope trick 

in whi which ch a0 is Bohr’ Bohr’ss radius radius,, Rµ is Rydber Rydbergs gs con consta stant nt and α is Sommer Sommerfeld’ feld’ss constant. constant. Rydberg’ Rydberg’ss const constant ant belong belongss to Hydrogen. Bohr’s constant belongs to us and the product of the two belongs to Sommerfeld whose constant is the geometric constant derived from Ostberger’s  δ  value   value in Chapter in  Chapter 12. 12. The constant  1/  1 /4π  joins area with linear measure.

A.4. Another theorem Do as the rope-maker did in the second rope trick. He had a spare metre of rope left in his workshop. So he twisted it into a circle which was 1 metre in circumference. Figure A-9. One extra unit of circumfer circumference ence internally

This had the effect of taking up all the slack and bringing the rope into perfect contact with the Earth all the way round.

Figure A-10. One extra unit of circumfer circumference ence externally

He wondered what the relationship was between the areas of the rope with a twist and the one without. He had another surprise for he found that area produced by un-twisting the unit of rope was proportional to the original radius of curvature and that he thought was impossible since it is known that the area is proportional to the square of the radius. Something funny was going on. Somehow a two dimensional area had got mixed up with a one dimensional radius. But when he spoke to his friend, who was an engineer, he was told about something called Green’s Theorems which do just that. They relate an  n  dimensional domain to an  n − 1 dimensional domain5. The rope-maker remembered in his first rope trick the constant  1/  1/4π was a multiplicative constant. He was curious to find that the constant  1/  1/4π  had become an additive constant in this rope trick.

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 Appendix A. The first rope trick 

The rope-maker decided to leave that to the mathematicians. But he was curious enough to want to discover more about this Green guy and so he continued his rope tricks. Telle A Ostberger

Notes 1. I hav havee deliberately deliberately labell labelled ed the distance distance h/2  h/ 2π   as    . In Quantum Mechanics the Dirac     is   is used solely for the quantum of energy per cycle  1.  1 .055   x  10 34  joules per cycle per second. But there is no reason why this principle should not be generalised so long as its special nature is preserved. It is special for at least the following reasons:

1.     is   is linear and belongs to measurements near infinity. It is non-local. 2. It is ima imagin ginary ary.. I can cannot not see that it can be divorce divorced d fro from m its partner partner  i   because it is derived from an extravariant extrav ariant geometry. geometry. 3. The re relatio lationship nship h  = 2π    has has a unique geometric shape. The connection is circular. However we must be vigilant because the linearity of      is is only guaranteed for the first quantum. It too lives in a curved space.

2. Notes 4 and 124 and Not Notee 2014 et.se et.seq. q. These become become axioms axioms of the process. process. 3. One example of this problem is the teaching of multiplication. We We tell the children children the A   B  =  B   A and they are left with the impression that this is the rule. In fact we know that the truth is completely the reverse.  A   B is never  B   A  except in the one special case; where B  and  A  are pure numbers. I fear for the future if we do not teach our children the truths we already know.  

 

 

 

4. In mathemat mathematics ics we have not passed passed the stage where we believ believee that a vect vector or can have its directions directions separa separate te from its magnitudes. We teach only vectors whose magnitudes are parallel to their directions e.g. a car velocity. But in my notes I have shown otherwise. Vectors can be found that have their direction orthogonal and antiparallel to their associated magnitudes. Indeed the vectors come in sets of four. I refer to those which have their magnitude and directions separated by 0, π/  π/22, π  and  3π/  3π/22. 5. The b basis asis of Inte Integratio gration n by parts.

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Appendix Appen dix B. Geometric Geometric represent representation ations s TAO The following partition the processes of creatin creating g geometric represent representations ations in my experience.

B.1. Presentation stages 1. A yin representa representation tion of direction only. only. A gener generalise alised d geometry which is much like a guess. An undirected geometry. 2. A repres represent entati ation on of stage stage 1 with with the yang magn magnitu itudes des on it. The shape or form whic which h is expec expected ted.. An undirected geometry. geometry. 3. A represent representation ation which extends extends stage 2 to include the yin of the magnitude. magnitude. That is the directio directions ns that are attached to the line lengths which, at this stage are not defined in magnitude. A directed geometry. 4. A complete representation which includes all the above sstages tages and the yang magnitudes which finalise it. A set of scale factors relate the whole to our measurement of reality. A fully directed geometry with sufficient magnitudes to permit a unique representation.

B.2. Generation 1. Absorption. Absorption. 2. Reciprocity or inversion. 3. Real/Imaginary conversion. conversion. 4. Intravariant to extravariant extravariant condensation.

B.3. Orthogonality Representations having their Magnitudes and associated Directions: 1.   0  radians apart. 2.   π/2 π/ 2 radians apart. 3.   π  radians apart. 4.   3π/2 π/ 2 radians apart

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 Appendix B. Geometric repres representations entations

B.4. Density 1. No directions in the line. (e.g. plain geometry geometry,, position vectors.) 2. One direction in the line. (e.g. ordinary vectors.) vectors.) 3. two directions directions in the line. (e.g. the tensors tensors of rank 2 in the Law Fields.) Fields.) 4. Multiple directions of greater density. (e.g. the Ricci and We Weyl yl Tensors Tensors and the geometric Worlds of the first and second Kind.)

130

 

Append App endix ix C. The case case of two two zeros zeros C.1. Something that actually happened in a company

Both accountants and engineers who work with computer systems are well versed in handling two different zeros in the system. One is zero minus and the other zero plus. It would not be possible to work without them. In accounting there are things called  “transactions”. They are the operators of the accounting world. They make the systems work by saying what must happen to the money and the goods. When goods enter a company a “Good   Inward Note”  transaction is prepared The accounting transaction double enters thus, Debit Stock at goods inward inspection account Credit Supplier suspense account. (Which says that we owe money to the supplier so we will hold an account for him ready to pay. Its part of the “payables” on the balance sheet) On the other hand if goods leave the company a  “Goods Despatched”  transaction is raised which reverses these entries. Now a tool which was critical to production was sent to a subcontractor for heat treatment. The tool was mislaid and a search for it began. The marketing department pointed out that advertising worth hundreds of thousands of  pounds had been bought to a strict time schedule pressure built up and panic set in. A new tool was commissioned through an outside agent who could produce it in a very short time. But at a very high cost. Some of the sales were met but many were missed. Date Stock at Goods Inward Supp. Suspense. Legal action ensued which culminated in the finding of the tool. But not until the accountants had produced the following records to the solicitors, Date Stock at Goods

Inward Supp.

Suspense.

4th Aug

0 Cr

0 Dr

and the reverse entry in the accounts of the heat treatment contractor showing quite clearly that the tools had been sent to, and had arrived at, the company. Compensation was settled out of court. The total value attached to these zeros was very large indeed.

C.2. Accounting errors When an accountant reconciles his balances and finds an difference of say, 4, he knows that this could represent an error of millions because the double entry will allow any constant to be added to the difference. So he needs

numerical  “tricks”  to help him with his accuracy. The absorption matrix (Note 100) is a great help. It explains how the numbers can be found even when there are errors.

131

 

Append App endix ix D. D. Uses Uses of the wor word d  “Dimension”  D.1. Note 137 Summary page 1. Dimensions Dimensions in units of measure. measure. It is common practice to call the units of Mass, Length and Time associated with a quantity, the  “dimensions” of the quantity. For example we say that the dimensions of force are  M LT −2 .

2. Dimension Dimension of a drawing. drawing. The ord ordina inary ry length length measur measureme ement nt which which scales scales the va vario rious us attrib attribute utess of a dra drawin wing. g. Partic Particula ularly rly an eng engine ineeri ering ng drawing but also in geometry.

3. Dimensions Dimensions of a Riemannian Riemannian kind. kind. The number of orthogonalities required to analytically define a set of elements which associate at any number of points in space. In an n dimensional space there will be n + 1 points. I refer to these n dimensions as being “magnitudinally separated”  because there must be some size, distance or numerical value separating each pair of orthogonal points. One might refer to this kind of dimension as  “magnitudinal in character”  or of a “magnitudinal kind”. Each of the 12 elements of a world geometry of the first kind constitute a dimension in this sense because they are all mutually orthogonal despite being spatially separated. Likewise each of the 40 elements of a geometry of the second kind also constitute a dimension. In three dimensional curved space there are three orthogonalities. The size of the space is determined by the number of orthogonal correspondences there are between the points 1 on the three orthogonal lines. If  we move to a space which is orthogonal to this then we may find another set of orthogonal line elements. Moving through another orthogonal rotation we can come to another set. We now have nine Riemannian dimensions. And so on. A World of the first kind has ten. But these are generic dimensions when the Law Fields are applied. Each law field restricts the correspondences of the  π  element to a law such as reciprocity. Note that in a Cartesian space the lines are at right angles and so each point on one line is orthogonal to all other points on its neighbouring line thus making it an highly overdetermined. Only in a curved space

can we sort out this overdetermination of the spaces. Furthermore we may rank the spaces by the Law Field technique. These principles are no more than that of the Orthogonality theorem (P.A.M.Dirac page 32 XXX) of Quantum Mechanics. It is this theorem that is required to bring String Theory in to line with the Ostberger work.

4. Dimension Dimension of the Einsteinian Einsteinian kind. The number of orthogonalities required to define a set of elements at one point in space. This seems to be limited to four. Ostberger says four is suffice. One might refer to this kind of dimension as being of a “directional kind”  since there is no magnitude needed to identify the set. The elements are  “directionally separated”. However any number of groups of four can be generated into higher spaces. In the World geometry of the first kind there is one group of four. In the World geometry of the second kind there are four groups of four. One may imagine that a geometry of the nth kind has 4n  4n groups of four. Such geometries would be extremely dense representations since the geometry of the second kind is sufficient not only to describe the whole of 

132

 

 Appendix D. Uses of the word “ “Dimension” Dimension”

the chemical table in respect of the electron’s activity as we perceive it but also in respect of the proton and neutron activity as well.

Notes 1. Points do not exis existt in the curved spaces spaces of the Worl Worlds, ds, only new worlds. worlds. But what othe otherr language sha shall ll I use? The points of two orthogonal lines from a new world or quasi-world which we recognise as a new particle in physics. Hence the Eight Fold Way.

133

 

Appen Appendix dix E. Connected Connectedness ness and disconnectedness Science has a great fondness for the speed of light. Much more so than the other constants of physics that are equally as interesting but less often brandished in the public eye. The Sommerfeld fine-structure constant 1 for example is fascinating because it is derived from a geometry, a pure shape in our universe. This shape is a disconnected one. It is a quantised geometry and yet there is a connection to the set of Delta values 2 that ensue from it. That connection is the particular rule used to draw the geometry in the first place. The rule is the connection and not the geometry. The idea of connectedness is very important because it underlies the whole structure of our universe. But it is not just connectedness, it is disconnectedness too that drives the structure of form. The two vie with one and the other. Chaos bursts into the world and becomes a part of some new order. Wars bring about the human need for peace. The genes swap about to produce rational and irrational humans or males and females. Asymmetry vies with Symmetry to produce the rules of quantum mechanics. Even the interpretational arguments in science vie extra variant views with intravariant ones. Finite people vie with the infinite in our society. Lorentz Connectedness is all around us and we may be on the side of believing it or on the side of disbelieving it. It does not  matter, for the disbelievers are the disconnected and so are still a part of the connected picture.

In the Grav-electromagnetic World the Lorenzian effects are a part of both the intra and extra Worlds. The Lorentz transformations apply equally to the  “Quantum Mechanical view”  as they do to the  “classical”. In notes 1110 and 1111 the geometry of the Lorentz transformation shows the connection between the relative velocity of two

observers. The geometry shows the well known changes in the measurements of mass, length and time. As the speed of light is approached the mass is observed to increase to infinity, the length to contract and the time dilate. But do they really? The Ostberger work shows two aspects to the Lorentz transformation. When the World geometry expands there is a  directional  effect as well as a  magnitudinal  one. The Magnitudinal one is the formulae for calculating the observational effects and the directional is ignored. It is the directional effect that will come into its own when we travel into deep space. Then we will need to know which curvature we are on in order to return home. Just as the intrepid explorers of the mid-millennium needed to know that the world was spherical in order to sail their way around the latitudes. The idea that directions must be taken into account is not new. Nevertheless many minds will have difficulty embracing such a concept. Some will not. That is the bigger picture to which human beings must succumb. Let’s take a journey into space. Our craft is large, very large, We have a family of people who are well versed in the ways of the extravariant and others who are intravariant by nature. Our energy source is from matter and we have a constant force motor some 20 kilometres behind us. It produces an acceleration of 10 metres per second every second. Just in excess of 1g. We have a department for steering and a mass repulsing shield forward of the control deck. We seek gravitational fields that will directionally accelerate us without our motors. We are observing the Lorentz effects of the matter around us and plotting its curvatures. We are calculating the geodesics along which we need to travel. There are departments for all other aspects of sustaining life. We are all walking about in as if we were in a gravitational field our heads pointing in the direction of the craft’s motion. The question is, What happens when we reach the speed of light? The speed of light is about 300,000 kilometres per second. Let us stand on the control deck and calculate how long it will take us to reach the speed of light.

134

 

 Appendix E. Connectedness and disconnectedness

speed = acceleration x time Our acceleration is 10 meters per second every second which is a very comfortable one g acceleration. We all live with our heads pointing forward. After one year we will be travelling at 315,619,200 3 metres per second, well in excess of the speed of light. On the 348th day (earth rotations) we have a party because we are at the speed of  light. Its a “disconnection”  party.

Figure E-1. Finding the n new ew universe

We are now disconnected from the folks back on Earth just as was Columbus. There is no grav-electromagnetic energy connection and we really are worlds apart. The Lorentz connection no longer applies because we cannot observe each other by electromagnetic means. We are  macroscopically disconnected . We have become part of the fabric of the universe. We are like being in a fluid 4. The new explorers will not be back for many years and may even colonise new territory. They must be self  sustaining or perish. Like Columbus they have to be a self sustaining system or find landings which are hospitable. They also need to know how to get back. For that they need to know how to plot their position in the universe, not only with four coordinates but also with double entry. The absorption matrices will be required to position the craft in both the forward and the backward directions of all its curvatures of travel. The accountant and the physicist must talk at the same table. Back here on Earth we have now realised that the stars are not where they seem to be. The light from the stars arrives on curves. The curves come from different Directions. Each curve is part of a 4-dimensional set belonging to the Laws of the universe. We have to sort them out in just the same way that we sort out the directions of the atomic structure. TAO To construct a map of the universe as if every line of sight of every telescope around the Earth is straight out into space  forever  fore ver is as arrogant as constructing a map of the Earth as if it were flat. It is a part of our present-day present-day arroga arrogance nce that  believes that we have some kind of power over nature.

135

 

 Appendix E. Connectedness and disconnectedness

Notes 1. See  Chapter 12 2. See  Appendix JJ.. 3. One day is 60 x 60 x 24 x 365.3 seconds seconds = 31,561,920 31,561,920 se seconds conds.. The speed is 31,5 31,561,92 61,920 0 x 10  ms−2 4. Ostbe Ostberger rger make makess notes on the possible effects effects of being in a fluid of the univ universe. erse. The Laws Laws of fluidics apply and so communication may be possible through those laws. Because we have traversed a geodesic bend and are now orthogonal to our kith and kin there seems to be some peculiar possibilities that we may have to face. For the present however they are best left in the closet.

136 

 

Appendix F. Appendix F. Newtoni Newtonian an cases of Magnitude Magnitudes s with Directions A brief look at the six cases mentioned in the text. Each of these six cases have components in Euclidean 3space. There are two case each of the  Direction, the Momentum and the Force . The first case is having the directions parallel and the second is having them orthogonal. A further two cases of each of these phenomena in

the extravariant World will also exist. These are in the microscopic World of Quantum Mechanics.

F.1. Direction   Z g : Directional case, Potentials orthogonal to Force The gravitational Force is equal to the Gradient of the gravitational potential.

F g  = gradV  gradV g   =  iδV/δx + jδV  jδ V /δy /δy +  + kδV/δz The potentials  V    are the orthogonal case of the use of   Z g  the Direction belonging to the Newton Field. Here we have the potentials   V  orthogonal to the gravitational force  F . The gradient is the measure of the slope of  the potential in the region considered. The greater the slope the greater the gravitational pull. It is the same on a contour map in two dimensions; the closer the lines appear on the map the greater is the slope of the hillside.

Figure F-1. A Schwarzchild gravitational gravitational field

F.2. Direction   Z g : Magnitudinal case, Directions parallel to Velocity The common vector notation for the position of masses in space

137 

 

 Appendix F. Newtonian cases of Magnitudes with Directions

F.3. Momentum: Directional case - the satellite, Momentum orthogonal to direction

Figure F-2. The satellite operates in the momentum-s momentum-static tatic plane

The mass is constrained to move in a curved path. It is the rate of change of the Direction that creates the force. In this case a centrifugal force. In the case of a bob on a string it is a centripetal acceleration down the string which constrains the motion.

F.4. Momentum: Magnitudinal case - the rocket,

138 

 

 Appendix F. Newtonian cases of Magnitudes with Directions

Momentum parallel to direction Figure F-3. The rocket is an exchange exchange of momentum device

The motion of the craft is exchanged for the energy of the particles. To lift the craft vertically the rate of change of momentum must be greater than the acceleration due to gravity. We may express the laws of 3 and 4 here by an alteration of the common expression for the second of Newton’s laws. We may say that, “The rate of change of momentum of a body is proportional to the force. When the change in the motion is directional the force acts at right angles and when the change in the motion is magnitudinal the  force acts parallel.”

This leads to the invention of a   best trajectory  take off for momentum exchange devices to clear the Earth’s escape velocity. By combining both of these laws together a launching device can be proposed which maximises the benefits of the momentum laws (note No 930).

F.5. Force: Directional case - the gyro, Force orthogonal to precession/direction The attributes of the gyro1 were explained briefly in  Section 4.4 (see 4.4  (see notes 1010-4). It epitomises all rotating devices that are in the field of an external torque such as a wheel fixed on one side only, a bicycle wheel that can turn in a headstock, a train turning a corner 2, a car accelerating round a bend and so on. The internal torque is the one that drives the flywheel. A torque is a  curving force  and in the case of the gyro sets up a miniature law field set on a spacial surface (Section (Section 4.4). 4.4).

139

 

 Appendix F. Newtonian cases of Magnitudes with Directions

Several interesting devices can be seen to be possible from this view of the gyro. In particular Note 966 combines the internal and external torques.

F.6. Force: Magnitudinal case, Force parallel to direction The internal torque drives the gyro’s flywheel. Most of the twentieth century transport is driven this way with some extremely inefficient designs of engines. We ar aree le lead ad to the the co conc nclu lusi sion on th that at,, “ to every momentum there is an equal contra momentum” and and we ca can n se seee this this as the counterpart to the law which says that,  “ to every force there is an equal and opposite reaction ”  and there will be a corresponding statement for the Directions that says  “ to every Direction there is a contra-Direction.” Several significant improvements in prime mover design are contained in the Notes series 900 simply from understanding these laws. These laws come from the intravariant World representing the macroscopic laws. The corresponding set of six in the extravariant World case represent the microscopic laws. These are the cases in which the Magnitudes are antiparallel and anti-orthogonal (270 degrees) to the Directions.

Notes 1. The gyro repr represent esentss a quite general case. case. A torque applied applied to rotate a body with a second second moment of inert inertia ia possesses a rotational inertia. Associated with this are all the attributes of the gyro even if the resulting effects are small or resisted naturally. 2. The or original iginal British British Rail high high speed speed train, train, the HS 150, was intended to give the passengers a comfortable ride by defying the laws of the gyro on the bends. It failed! The braking system, on the same train, consisted of  external shoes mounted around a rotating drum. It also failed because it ignored the laws of frictional forces. On the outside of a drum the forces are exponential juke like the tope around a capstan. Only at very slow speeds can such forces be arrested. At higher speeds the mathematics tells us that any approaching object having a frictional effect will be thrown off abruptly. The brake shoes would judder uncontrollably.

140

 

Appendix Appen dix G. Grav-el Grav-electr ectroma omagneti gnetic c relations relations It is not my intention that this should become a mathematics book but I feel that it is necessary to introduce some of the connections that exist in the spaces. Here are some of the relationships that exist in the three fields of the Grav-electromagnetic World. Below in Table in  Table G-1 are G-1  are the effects of the rotations (Curls) that take place in the fields. The additional magnetic term marked δD/  δD/d dt comes from the rate of change in the field. It was how Maxwell explained the radio reception in an Edison earpiece. The top four are Maxwell’s equations in the electric and magnetic field. The lower one is effectively the equation of a syphon in the gravitational field. The flow in the syphon is zero if the two ends have the same potential. The path between them has no bearing on the force that creates the flow except that friction plays a part where a pipe is involved. However where motion in space is concerned a body accelerates from a place of higher potential to one of lower potential. Friction is almost absent and plays a much lesser role. Whilst the body remains in a field of constant potential (a satellite for example) there is no force to change its velocity. Table G-1. The effects of the rotations (Curls) that take place in the fields

Integral notation

 

Magnetic

H s  ds  d s  = I   =  I  +  +

C 

 

Electric

C 

 

Gravitic a

Notes: a.   0 in a Schwarzchild field,

C 

E s  ds  d s  =

  dϕ dt

− ddφt

F s  ds  d s  = 0 or   − k

Vector notation

 

 

Curl H    =  J  + Curl H   J  +

Curl E  = Curl E   =

 δ D dt

− δB dt

Curl F  Curl  F  =  = 0 or k or k

−k  in a non-uniform field. Note no 1782.

There is a discussion that centres around what kind of rotation these represent. Thus we need to ask what happens in a curved gravitational field, that is, one in which the force bends round an arc. These say that we cannot have such a force as it would imply that an acceleration could take place in the path of the arc which would lead to a singularity of infinite velocity. What thes What thesee form formul ulaa ar aree abou aboutt is th thee ro rota tati tion on of th thee fo forc rces es as the the World orld en enla larg rges es.. The The Cu Curl rlss are are the the inte intern rnal al rota rotati tion onss which produce an effect in the field. There are three  “fluxes”  involved V  , B  and  D , the velocity (density)1, the Electric flux density and the Magnetic flux density. The  “click” caused δD  δD//dt by is what would come from a circuit when it is suddenly switched off. It is the radio signal part of the equation. Table G-2. The magnetic, electric and gravitic inverse square laws

Magnetic

Electric

Gravitic

F m  =

  1  p1 p2 4πη r2

 

F e  =

  1 q 1 q 2 4πε 0 r2

 

F g   =

  1 m1 m2 4πζ  r2

The inverse square laws are shown in Table in  Table G-2 G-2.. They too tell us that there is a relationship between these fields. In this case they relate to those vectors which have their directions orthogonal to their magnitudes. (See Appendix (See Appendix F and notes).

141

 

 Appendix G. Grav-electr Grav-electromagnetic omagnetic relations

Another set of equations that relate these three fields is shown in  Table G-3. G-3. The gradient is like going up a hillside. The steepest part of the hillside regardless of which direction it happens to be is the gradient. Table G-3. Potential gradents of the three fields

a

H   = −grad gradV  V m

E  =  =

gradV  V e −grad

 

F g  = gradV  gradV g

Notes: a. The ste steepest epest sslope lope on a hill-side hill-side shown shown as contou contours rs on a map (   V   V   =  iδV/δx + jδV  jδ V /δy + /δy  + kδV/δz ). 







 

We see here that the potentials, which are a part of the Direction belonging to the Newton Law field are related to the forces that exist with with them in a simple way. The steeper the gradient of the potential (Direction) the greater the force. These three fields are hill sides in the space, albeit a three dimensional hill. What the Law field expresses is that the surface of the hill can be represented by the the isoclinic contour lines and their orthogonal risers which describe the rate of incline. What we have done is to describe the shape of the hill. We might easily be lead to believe that we have more. The negative signs may be taken to be valleys. Ostberger sees this as a difference in the standpoint of our measurement. The Magnetic and Electric fields are viewed from the outside looking in whereas the gravitational field is viewed from the inside looking out. To relate all three, two must be of opposite sign. Another way of expressing this is to realise that the negative sign means that the first two are asymmetric and will combine with separate fields in a symmetric way whereas the gravitic field is symmetric and combines asymmetrical.

Notes 1. We live at a particul particular ar velocity velocity.. Our actual motion consist consist of the components components of rotation of the Earth, around around the Sun, around the galaxy, etc. So, all velocities are relative and we may see this as the relative  density  of  the velocity at which we live. As we increase in velocity this  density  changes and we become aware of new observations. Velocity is the flux of the gravitational field.

142

 

Appendix Appen dix H. Spherical Spherical surfac surface e or sphere sphere Figure H-1. Elements of a sphere’s sphere’s ssurface urface

Figure H-2. Elements of a spherical spherical surface

The difference between a sphere and a spherical surface is shown in Figure in Figure H-1 and H-1 and Figure  Figure H-2 H-2.. The line elements that can be described over the surface of a sphere make an angle at the surface which is the same as the angle expressing latitude. This is not so in the case of a spherical surface where the angle at the surface can be any which is self consistent with a representation. In the case of the Law World the angle at the surface is always orthogonal to the line element.

143

 

Appendix Appen dix I. Linear Linear algebra algebra relat relations ions The subject of Linear algebra is too deep to be discussed in any way other than a mathematical text book. However I consider it possible that we can glean the essence of the subject from a glance at some of the important theorems of the subject. Table I-1. Algebraic Structure

The  Ring

 

The Absorptive element of the Law field  forms  forms a  ring under the definition of an algebraic ring R . If a world surface is formed in which the ring is the great circle then the small circles are the  sub-rings.

The  Field 

 

A ring with a unit element is called a  field  if   if every non-zero element of  R  R  has a multiplicative inverse. The reciprocity element of the  Law field  forms  forms a field. A field is necessarily an  integral domain.

The  Group

 

The First Law Field forms a  group.

 Modulo or  Clock Numbers   Part of the thirdnes thirdnesss in the Law Law Field.

The table below shows the relationship the first Law Fields and the various classes of complex numbers and operators. Table I-2. Comparing Law Fields with Linear Algebra

Class of Behaviour Class of operators in a Behaviour Law Field Law complex under finite dimensional under the element numbers conjugation inner product space adjoint map   −1 

 

 

Reciprocal upper Unit circle iz =  = 1) (   iz element field of the: lower   (   z  = 1) field 





upper The field Real/Imaginary element lower of the: field

z  = 1/z

 

Unitary operators, complex

 

Inver sion

=

 



Orthogonal operators, real

 

 

Real axis

z  =  z

 

 

upper Imaginary axis The Absorption field element lower of the: field

Symmetric operators, complex



 

=

 

 

Conversion

Symmetric operators, real

 

z  =

−z

 

Skew symmetric operators, complex



 

=−

 

 

Absorption

Skew symmetric operators, real

144

 

Appendix Appen dix J. Table able of delta values values The table shows the first four q   values values of these constants. The particular value that belongs to the Sommerfeld finestructure constant is the value when  p =  p  = 1  and  q  =  = 1. A comprehensive list is contained in the Note 1850. There are proposals for calculating the scalars for atoms heavier than hydrogen. The value used for the fine-structure constant is that produced by the  − δ 11  term (0.085786438). Table J-1. Definition of the delta functions

+ q

q/2 + δ  p  =  p + q/2

  p( p( p + q )

− q

q/2 − δ  p   =  p + q/2

 

  p( p( p + q )

 = 1 Table J-2. The delta values for  q  =

  p( p( p + q )

+ q

− δ q

0.000000000

0.500000000

0.500000000

2.000000000

2.000000000

1

0.866025404

1.866025404

0.133974596

0.535898385

7.464101615

1

1.5

1.414213562

2.914213562

0.085786438

0.343145751

11.65685425

2

1

2.5

2.449489743

4.949489743

0.050510257

0.202041029

19.79795897

3

1

3.5

3.464101615

6.964101615

0.035898385

0.143593539

27.85640646

 p

q

p + q/2 q/2

0

1

0.5

0.5

1

1

 

δ  p

 p

 

1/+ δ  pq

 

1/− δ  pq

4

1

4.5

4.472135955

8.972135955

0.027864045

0.111456180

35.88854382

5

1

5.5

5.477225575

10.97722558

0.022774425

0.091097700

43.90890230

6

1

6.5

6.480740698

12.98074070

0.019259302

0.077037206

51.92296279

7 8

1 1

7.5 8.5

7.483314774 8.485281374

14.98331477 16.98528137

0.016685226 0.014718626

0.066740906 0.058874503

59.93325909 67.94112550

9

1

9.5

9.486832981

18.98683298

0.013167019

0.052668078

75.94733192

10

1

10.5

10.48808848

20.98808848

0.011911518

0.047646073

83.95235393

11

1

11.5

11.48912529

22.98912529

0.010874707

0.043498828

91.95650117

12

1

12.5

12.48999600

24.98999600

0.010004003

0.040016013

99.95998399

13

1

13.5

13.49073756

26.99073756

0.009262437

0.037049747

107.96295025

14

1

14.5

14.49137675

28.99137675

0.008623254

0.034493015

115.96550698

15

1

15.5

15.49193338

30.99193338

0.008066615

0.032266461

123.96773354

+ q

− δ q

1/+ δ  pq

1/− δ  pq

Table J-3. The delta values for  q  =  = 2

  p( p( p + q )

 

δ  p

 

 

 p

q

p + q/2 q/2

0

2

1

0.000000000

1.000000000

1.000000000

1.000000000

1.000000000

0.5

2

1.5

1.118033989

2.618033989

0.381966011

0.381966011

2.618033989

1

2

2

1.732050808

3.732050808

0.267949192

0.267949192

3.732050808

2

2

3

2.828427125

5.828427125

0.171572875

0.171572875

5.828427125

3

2

4

3.872983346

7.872983346

0.127016654

0.127016654

7.872983346

4

2

5

4.898979486

9.898979486

0.101020514

0.101020514

9.898979486

5

2

6

5.916079783

11.91607978

0.083920217

0.083920217

11.91607978

 p

145

 

 Appendix J. Table of delta values

 = 3 Table J-4. The delta values for  q  =

  p( p( p + q )

+ q

− δ q

0.000000000

1.500000000

1.500000000

0.666666667

0.666666667

2

1.322875656

3.322875656

0.677124344

0.300944153

1.476833625

3

2.5

2.000000000

4.500000000

0.500000000

0.222222222

2.000000000

2

3

3.5

3.162277660

6.662277660

0.337722340

0.150098818

2.961012293

3

3

4.5

4.242640687

8.742640687

0.257359313

0.114381917

3.885618083

4

3

5.5

5.291502622

10.79150262

0.208497378

0.092665501

4.796223388

5

3

6.5

6.324555320

12.82455532

0.175444680

0.077975413

5.699802365

+ q

− q

1/+ δ  pq

1/− δ  pq

 p

q

p + q/2 q/2

0

3

1.5

0.5

3

1

 

δ  p

 p

 

1/+ δ  pq

 

1/− δ  pq

Table J-5. The delta values for  q  =  = 4

  p( p( p + q )

 

δ  p

δ  p

 

 

 p

q

p + q/2 q/2

0

4

2

0.000000000

2.000000000

2.000000000

0.500000000

0.500000000

0.5

4

2.5

1.500000000

4.000000000

1.000000000

0.250000000

1.000000000

1

4

3

2.236067977

5.236067977

0.763932023

0.190983006

1.309016994

2

4

4

3.464101615

7.464101615

0.535898385

0.133974596

1.866025404

3

4

5

4.582575695

9.582575695

0.417424305

0.104356076

2.395643924

4

4

6

5.656854249

11.65685425

0.343145751

0.085786438

2.914213562

5

4

7

6.708203932

13.70820393

0.291796068

0.072949017

3.427050983

146 

 

Appendix Appen dix K. K. The magnitu magnitudinal dinal conics conics There are at least two historical references to the magnitudinal representation of conic sections. There is  Basic Physics of Atoms and Molecules  by U. Fano and L. Fano1 and D Goodstein and J Goodstein in  Feynman’s Lost   Lecture2. Both provide examples of the use of such a technique. The former use the method to derive Rutherford Scattering and the latter the orbital motion of the planets. This is the technique: We may choose any method of representing mathematical processes. We happen to follow the one chosen by Rene Descartes. I have chosen a different method because it seems to produce more useful results and is easier for the mind to assimilate.

In the Cartesian method we describe shapes by algebraic equations and draw them on pre-set axes. The equations are the magnitudes and the drawings are the directions. We have historically attached a greater significance to the magnitudes than to the directions. However, there is no foundation for believing that one should fare more significant that the other. There are many ways of representing our reasoning processes. The trick is to find the one that is most revealing and gives consistent and contiguous results. Results that can be connected indefinitely and that can be used to pass from the magnitudinal form to the directional form or vice versa, at will. In the method that I have elucidated in my Notes and has been introduced in this book very briefly I have shown how to separate the Magnitudinal form of our ideas from their Directional form. Here is an example of this process which is applicable to the conic sections. The table in Figure in  Figure K-1 shows K-1  shows the two forms. There is more than one method of passing from one to the other. It is interesting to note that the displacement of the focus in the magnitudinal form linearly from inside the circle to outside is copied in the directional form by a rotation of the plane of the shape through the cones.

147 

 

 Appendix K. The magnitudinal conics

Figure K-1. Sketches of the conic sections expressed in Magnitudinal Magnitudinal and Directional form

Notes 1.   [Fano59] 2.   [Goodstein99]

148 

 

Bibliography

[Carse87]   Finite and Infinite Games (http://www.randomhouse.com/BB/catalo (http://www.randomhouse.com/BB/catalog/display. g/display.pperl?isbn=0345341848), pperl?isbn=0345341848),  James P Carse, Ballantine Books, First edition, September 1987, ISBN 0-345-34184-8.

[Dirac70]  The principles of Quantum Mechanics , Pauli A M Dirac, Oxford Press, Fourth edition, 1970. [Fano59] Basic Physics of Atoms and Molecules, Ugo Fano and L Fano, John Wiley & Sons, New York, 1959. [Goodstein99]  Feynman’s Lost Lecture, David L Goodstein and Judith R Goodstein, W. W. Norton & Company,  Norton paperback, 1999, ISBN 0-393-31995-4.

[Hawkin88]   A brief histor historyy of time (from (from the big bang to blac blackk holes holes)) (http: (http://www //www.amaz .amazon.co on.co.uk/e .uk/exec/o xec/obidos bidos/ASIN/ /ASIN/05531 0553175211/ 75211/qid  qid  2/ref=sr_2_3_2/202-6797302-3220642) , Steven W Hawkin, Bantam Press, 1988, ISBN 0-593-01518-5.

[Penrose] The Emperor’s New Mind, Roger Penrose, Vintage Books.

Ostberger notes [Note90] “Algebraic Normal form x cos α + y sin α  = p  =  p ”, Telle A Ostberger, The Educational Trust Company. [Note96] “Algebraic quadratics”, Telle A Ostberger, The Educational Trust Company. [Note98] “Algebraic cubics”, Telle A Ostberger, The Educational Trust Company. [Note100] “Absorption mathematics”, Telle A Ostberger, The Educational Trust Company. [Note103] “The pen stand”, Telle A Ostberger, The Educational Trust Company. [Note112] “Graphing a rotating pair of sin and cosine”, Telle A Ostberger, The Educational Trust Company. [Note117] “Sine and cosine addition and subtraction”, Telle A Ostberger, The Educational Trust Company. [Note124] “Air and water bottle measuring absorption”, Telle A Ostberger, The Educational Trust Company. [Note140] “The rope maker part 1”, Telle A Ostberger, The Educational Trust Company. [Note220] “The Interpretive group and Klein’s geometry”, Telle A Ostberger, The Educational Trust Company. [Note250] “General properties of the World of the first kind”, Telle A Ostberger, The Educational Trust Company. [Note257] “The Field of label numbers (cardinal)”, Telle A Ostberger, The Educational Trust Company. [Note258] “The probability Law Field”, Telle A Ostberger, The Educational Trust Company. [Note409] “The Hesse normal form (3d)”, Telle A Ostberger, The Educational Trust Company. [Note5xx] “Series 5xx : Notes on trignometric functions”, Telle A Ostberger, The Educational Trust Company. [Note504] “Integer amplitudes of circular functions”, Telle A Ostberger, The Educational Trust Company. [Note512] “The multiple angle 2θ  2θ ”, Telle A Ostberger, The Educational Trust Company. [Note513] “The multiple angle 3θ  3θ ”, Telle A Ostberger, The Educational Trust Company. [Note514] “The multiple angle 4θ  4θ ”, Telle A Ostberger, The Educational Trust Company.

 

[Note586] “Simple surds”, Telle A Ostberger, The Educational Trust Company. [Note6xx] “Series 6xx : XXX”, Telle A Ostberger, The Educational Trust Company. [Note608] “The Red geometry”, Telle A Ostberger, The Educational Trust Company. [Note609] “The Blue geometry”, Telle A Ostberger, The Educational Trust Company. [Note610] “Squaring the circle and quantising space (blue geometry)”, Telle A Ostberger, The Educational Trust Company. [Note612] “Squaring the circle and quantising space (blue geometry)”, Telle A Ostberger, The Educational Trust Company. [Note665] “Sommerfeld’s fine-structure constant from the blue and red geometries”, Telle A Ostberger, The Educational Trust Company. [Note666] “Sommerfeld’s fine-structure constant from the first delta value”, Telle A Ostberger, The Educational Trust Company. [Note667] “XXX”, Telle A Ostberger, The Educational Trust Company. [Note820] “Helium spin eigen-functions”, Telle A Ostberger, The Educational Trust Company. [Note9xx] “Series 9xx : Inventions and engineering devices”, Telle A Ostberger, The Educational Trust Company. [Note1001] “System dynamics Law Field”, Telle A Ostberger, The Educational Trust Company. [Note1021] “Gaussian curvatures”, Telle A Ostberger, The Educational Trust Company. [Note1030] “XXXX”, Telle A Ostberger, The Educational Trust Company. [Note1110] “The Lorentz transformation geometrically”, Telle A Ostberger, The Educational Trust Company. [Note1111] “Interpretion of note 1110”, Telle A Ostberger, The Educational Trust Company. [Note1120] “4D geometry by flag method”, Telle A Ostberger, The Educational Trust Company. [Note1121] “4D geometry by colour method”, Telle A Ostberger, The Educational Trust Company. [Note1122] “4D geometry by Rubic’s cube method”, Telle A Ostberger, The Educational Trust Company. [Note1127] “Rotation in the 4th Dimension”, Telle A Ostberger, The Educational Trust Company. [Note1166] “The Kronecker delta in geometry”, Telle A Ostberger, The Educational Trust Company. [Note1170] “Intravariant to extravariant change”, Telle A Ostberger, The Educational Trust Company. [Note1171] “Worlds of the second kind”, Telle A Ostberger, The Educational Trust Company. [Note1172] “Intra on extra and extra on intra Worlds”, Telle A Ostberger, The Educational Trust Company. [Note1173] “Looking down microcosm, looking up macrocosm”, Telle A Ostberger, The Educational Trust Company. [Note1206] “Half angle tangent (geometry) relations”, Telle A Ostberger, The Educational Trust Company. [Note1207] “Half angle tangent. A further study”, Telle A Ostberger, The Educational Trust Company.

 

[Note1 [No te1210 210]] “Ex “Exten tendin ding g the hal halff angle angle geomet geometry ry to all va value lues”, s”, Telle elle A Ostber Ostberger ger,, The Edu Educat cation ional al Trust Trust Compan Company y. [Note1230] “Geometry of imaginary character”, Telle A Ostberger, The Educational Trust Company. [Note1250] “Introduction to tensor calculus”, Telle A Ostberger, The Educational Trust Company. [Note1254] “The Einstein tensor”, Telle A Ostberger, The Educational Trust Company. [Note1466] “Draft Code of conduct for Universities and places of learning which are servants of the tax-paying public”, Telle A Ostberger, The Educational Trust Company. [Note1702] “Biosavart and Lenz’s laws”, Telle A Ostberger, The Educational Trust Company. [Note1713] “Law Worlds, intravariant”, Telle A Ostberger, The Educational Trust Company. [Note1714] “Law Worlds, extravariant”, Telle A Ostberger, The Educational Trust Company. [Note1715] “Combined intra and extra Worlds”, Telle A Ostberger, The Educational Trust Company. [Note1716] “Some collected predictions”, Telle A Ostberger, The Educational Trust Company. [Note1748] “Gravelectomagetic World of the second kinds (extravariant)”, Telle A Ostberger, The Educational Trust Company. [Note1750] “Fluidics”, Telle A Ostberger, The Educational Trust Company. [Note18xx] “Series 18xx : Notes on the Hydrogen atom”, Telle A Ostberger, The Educational Trust Company. [Note1823] “Rotation from spin up to spin down =  2π  2π ”, Telle A Ostberger, The Educational Trust Company. [Note1850] “XXX”, Telle A Ostberger, The Educational Trust Company. [Note20xx] “Series 20xx : Notes on social order”, Telle A Ostberger, The Educational Trust Company. [Note23xx] “Series 23xx : Notes on finance”, Telle A Ostberger, The Educational Trust Company. [Note2333] “The stable universe of commercial operations”, Telle A Ostberger, The Educational Trust Company.

Glossary A Absorption Th Thee proc proces esss in wh whic ich h a quan quantit tity y or a vect vector or di dire rect ctio ion n di disa sapp ppea earr by virt virtue ue of have have eq equa uall an and d oppo opposi site te prop proper erti ties es of exactly the same character by definition. Positive and negative rational numbers of the same magnitude absorb. Irrational numbers, ones that have not cut-off or ending cannot be said to absorb.

151

 

Glossary

Absorption matrix A two dimensional dimensional array of numbers numbers or quantities quantities in which the row and column vectors absorb absorb.. The determinant of such matrices is always zero.

Accounting Painting a picture of trading goods, money or services with numbers.

ADP Adeninine diphosphate. The half way stage between Adeninine phosphate and Adeninine Triphosphate in the Krebb cycle of the digestive system, the process by which we extract energy from food. During the stages the Hydrogen atom is split into its proton and electron for recombination.

Agenerate The concept of building up a process or system towards its state of  Condensation  Condensation, condensing it and repeating the process again at its next level ad infinitum. This is a negative entropic process. The complementary process is  degeneracy, which is entropic in nature.

Algebraic forms The use of alpha-numeric characters to describe the shape of a mathematical concept in a Cartesian space.

Anti-parallel Being directionally π  radians apart indefinitely. This applies to lines, surfaces and volumes.

Asymmetric A form that cannot be divided by a mirror.

Atomic element One of those in the periodic table.

Atomic representation

The geometry consisting of an intravariant World superposed by an extravariant World in six positions labelled E, P, N and back-E, back-P and back-N.

152

 

Glossary

ATP Adeninine Tri-phosphate. See  ADP .

B Back-E The location of the spin down electron in the atomic representation.

Back-Location A convenient way of writing a reversed character which appears on the geometry. It avoids using a reversed character font.

Back-N The location of the spin down neutron in the atomic representation.

Back-P The location of the reverse spin proton in the atomic representation.

Blue geometry A colloquial label given to one of the 600 series geometries that is a suitable candidate for representing the directions in the hydrogen atom.

Bohr, Neils Creator of the electron orbit model of the atom.

Bosons Particles that occur in physics which have whole number spin.

C Circel A term used by Ostberger to describe a complete set of circles in a geometry. There must be a known connective linkage between the circles.

153

 

Glossary

Colour nomenclature The nomenclature adopted by Ostberger that uses blue for the vectors of formative or firstness character, red for the vectors of operator or secondness character and green for the interpretive or thirdness character.

Condensation Contracting a complete set of a system, in general in groups of four, into a single component of the Definition (or firstness field). This begins a next-level-up system.

Connectedness A mathematical term defined in the subject of  complex   complex variables. It does not imply continuity necessarily for a set of layers or Riemann surfaces can be stepwise connected.

Continuity Without break. Mathematicians have their definition.

Contra direction The direction of a line element which is in the same place as, but opposite to its co-direction.

Contra force The force, represented by a line element, which meets in opposition to its reaction to produce no effect upon its surrounding.

Contra momentum Any momentum which gyrates about some centre which its co momentum into equilibrium. Our body motions by the contra momentum action of two flexible parts.

Contravariant A complete set of tensors (vectors of ran 2 or more) which vie with their covariant neighbours.

Conversion The process of transforming from the real to the imaginary.

154

 

Glossary

Covariant A complete set of tensors (vectors of ran 2 or more) which vie with their covariant neighbours.

Credit The con contra tra posting or entry in a book book of ac acco coun unts ts.. It is nega negati tive ve by co con nvent ventio ion n an and d ofte often n sh show own n red. red. Li Liab abil ilit itie iess are primarily credit numbers as are Sales.

Curl Another term for Rotation in vector analysis. It used to be called  Rot . The term includes any kind of rotation and is mathematically well defined.

Curvature Measure by the reciprocal of the radius at the point of curvature. Ostberger regards nature as a process of  winding up the curvatures.

Curved vectors Why not? Satellites have elliptical ones.