FINITE AND INFINITE WAYS: The Ostberger Story by Bob Beaumont

FINITE AND INFINITE WAYS

The Ostberger Story 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 process 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. 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 in any way any part of this work; nor 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.

Table of ContentsSynopsis....................................................................................................................................................................i

The Ways and the Truths.....................................................................................................................................iii

I. The Chapters.......................................................................................................................................................v

1. Einstein and all that....................................................................................................................................11.1. Introducing the reasons for this book............................................................................................11.2. Descartes........................................................................................................................................21.3. Imaginary Time..............................................................................................................................4

2. The elements of new geometry..................................................................................................................62.1. Rotations........................................................................................................................................62.2. Three Dimensions..........................................................................................................................62.3. Standard Forms............................................................................................................................122.4. More Than Two Dimensions.......................................................................................................14

3. A story and a theorem..............................................................................................................................193.1. A Story.........................................................................................................................................193.2. And now for a theorem................................................................................................................203.3. The first rope trick.......................................................................................................................21

4. Orthogonality...........................................................................................................................................234.1. The rules of Fleming....................................................................................................................234.2. Deflection of an electron beam....................................................................................................254.3. Vectors of light.............................................................................................................................254.4. The vectors of the gyro................................................................................................................264.5. Other examples............................................................................................................................30

5. More than geometry.................................................................................................................................315.1. Studying direction........................................................................................................................315.2. Growing geometry.......................................................................................................................34

6. The development of standard forms (rank one vectors)...........................................................................386.1. Standard forms.............................................................................................................................386.2. The trigonometric forms..............................................................................................................386.3. The trigonometric standard forms...............................................................................................396.4. Applying Euclid over Ostberger..................................................................................................426.5. Amplitudes...................................................................................................................................446.6. Powers of trigonometric functions...............................................................................................446.7. Half angle tangent relations.........................................................................................................456.8. Algebraic equations.....................................................................................................................476.9. Parametric equations....................................................................................................................476.10. Imaginary geometry...................................................................................................................47

7. Four dimensions are here.........................................................................................................................497.1. Four dimensions...........................................................................................................................497.2. Observational platform................................................................................................................527.3. Quarter points..............................................................................................................................527.4. Standing in another place.............................................................................................................55

8. The development of Law Fields (rank two vectors).................................................................................598.1. Law Fields...................................................................................................................................598.2. The Newton Law Field................................................................................................................598.3. The origin.....................................................................................................................................618.4. The relations................................................................................................................................618.5. The magnetic Law Field..............................................................................................................618.6. The electric Law Field.................................................................................................................628.7. The general Law Field.................................................................................................................63

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8.8. Continuity and discontinuity........................................................................................................658.9. Matrices.......................................................................................................................................658.10. The Law Fields of number.........................................................................................................658.11. The Law Fields of thermodynamics..........................................................................................68

9. The omnipotent laws................................................................................................................................749.1. Opposites.....................................................................................................................................749.2. Absorption...................................................................................................................................749.3. Reciprocity...................................................................................................................................789.4. Conversion...................................................................................................................................80

10. Law Worlds of the first kind...................................................................................................................8210.1. Vector as pictures.......................................................................................................................8210.2. Separating direction from magnitude........................................................................................8310.3. Ordinary space...........................................................................................................................8310.4. The parallel principle.................................................................................................................8310.5. Direction only............................................................................................................................8410.6. Curvatures..................................................................................................................................8410.7. Assembling law fields................................................................................................................8510.8. Minkowski’s worlds...................................................................................................................8610.9. The grav-electromagnetic world................................................................................................8710.10. The thermodynamic world.......................................................................................................9010.11. The World of number...............................................................................................................9210.12. Other Worlds............................................................................................................................93

11. Worlds of the second kind......................................................................................................................9811.1. Worlds of the second kind.........................................................................................................9811.2. Bosons and Fermions.................................................................................................................9811.3. GEM World of the second kind...............................................................................................10211.4. The atomic elements................................................................................................................10211.5. Computer Modelling................................................................................................................107

12. Using the process for Sommerfeld’s fine-structure constant................................................................10812.1. The first application.................................................................................................................10812.2. Sommerfeld’s fine-structure constant......................................................................................114

13. New wine into new skins......................................................................................................................118

II. The Appendices.............................................................................................................................................121

A. The first rope trick.................................................................................................................................122A.1. The note 104b...........................................................................................................................122A.2. Concluding remarks..................................................................................................................125A.3. The1/4π connection................................................................................................................126A.4. Another theorem.......................................................................................................................127

B. Geometric representations.....................................................................................................................129B.1. Presentation stages....................................................................................................................129B.2. Generation.................................................................................................................................129B.3. Orthogonality............................................................................................................................129B.4. Density......................................................................................................................................129

C. The case of two zeros............................................................................................................................131C.1. Something that actually happened in a company......................................................................131C.2. Accounting errors......................................................................................................................131

D. Uses of the word“Dimension” .............................................................................................................132D.1. Note 137 Summary page...........................................................................................................132

E. Connectedness and disconnectedness....................................................................................................134F. Newtonian cases of Magnitudes with Directions...................................................................................137

F.1. DirectionZg: Directional case, Potentials orthogonal to Force................................................137

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F.2. DirectionZg: Magnitudinal case, Directions parallel to Velocity.............................................137F.3. Momentum: Directional case - the satellite, Momentum orthogonal to direction.....................137F.4. Momentum: Magnitudinal case - the rocket, Momentum parallel to direction.........................138F.5. Force: Directional case - the gyro, Force orthogonal to precession/direction...........................139F.6. Force: Magnitudinal case, Force parallel to direction...............................................................140

G. Grav-electromagnetic relations.............................................................................................................141H. Spherical surface or sphere....................................................................................................................143I. Linear algebra relations...........................................................................................................................144J. Table of delta values...............................................................................................................................145K. The magnitudinal conics.......................................................................................................................147Bibliography...............................................................................................................................................149Ostberger notes...........................................................................................................................................149Glossary......................................................................................................................................................151

Colophon.............................................................................................................................................................170

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List of Tables8-1. The Work Law Field (firstness)......................................................................................................................698-2. The Organisation Field (secondness).............................................................................................................708-3. The Reaction Field (thirdness).......................................................................................................................719-1. From the notes on social activity....................................................................................................................79G-1. The effects of the rotations (Curls) that take place in the fields..................................................................141G-2. The magnetic, electric and gravitic inverse square laws..............................................................................141G-3. Potential gradents of the three fields............................................................................................................142I-1. Algebraic Structure.......................................................................................................................................144I-2. Comparing Law Fields with Linear Algebra.................................................................................................144J-1. Definition of the delta functions....................................................................................................................145J-2. The delta values forq = 1.............................................................................................................................145J-3. The delta values forq = 2.............................................................................................................................145J-4. The delta values forq = 3.............................................................................................................................145J-5. The delta values forq = 4.............................................................................................................................146

List of Figures2-1. Cartesian frame of reference.............................................................................................................................62-2. The origin is numerically zero in every direction.............................................................................................72-3. Polar Cartesian..................................................................................................................................................72-4. The pen stand1...................................................................................................................................................82-5. An example curved geometry...........................................................................................................................92-6. An important basic geometry...........................................................................................................................92-7. Multiples of trigonometric functions..............................................................................................................102-8. The first sets of integer surds6.........................................................................................................................102-9. Elements of the form6

√a(a+ 1) ..................................................................................................................11

2-10. Two straight lines meet at right angles.........................................................................................................122-11. Two curves meet orthogonally......................................................................................................................122-12. We can apply directions along line elements................................................................................................132-13. The shortest path between two orthogonal points is a quarter rotation........................................................132-14. Different types of orthogonal elements........................................................................................................142-15. There are eight possible orthogonal curvatures of the single line element; four cylindrical and four annular

142-18. The front page of a note on sine and cosine addition and subtraction8 ........................................................162-19. A larger curved geometry.............................................................................................................................173-1. The Wun-man’s universe................................................................................................................................193-2. A rope around the Earth..................................................................................................................................213-3. A rope around Jupiter.....................................................................................................................................213-4. one unit of overlap..........................................................................................................................................224-1. Left hand rule for the motor...........................................................................................................................234-2. Right hand rule for the generator....................................................................................................................234-3. The unidirectional representation...................................................................................................................244-4. The bidirectional representation.....................................................................................................................244-5. The orthogonal vectors of light......................................................................................................................254-6. Using a bicycle wheel as a gyro.....................................................................................................................264-7. For the left hand holding the spindle; the magnitudinal part of the spin........................................................274-8. Gravity pulling down creates a torque............................................................................................................274-9. For the right hand holding the spindle; the directional part of the vectors.....................................................28

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4-10. The orthogonal surface vectors of the gyro..................................................................................................284-11. The gyro surface produces a Law Field........................................................................................................295-1. A line minimum element................................................................................................................................315-2. A page from Ostberger’s notes on tensor calculus.........................................................................................325-4. Orthogonal curvatures in a World geometry..................................................................................................335-5. The interior Euclidean chord theorem............................................................................................................345-6. The magnitude powers ofp ............................................................................................................................355-7. The field of an electric dipole.........................................................................................................................355-8. The field of a pair of parallel wires.................................................................................................................366-1. The directional form; the functions arise as directional ratios.......................................................................396-2. The magnitudinal form; the functions arise as line length.............................................................................396-3. The magnitude form of tangent......................................................................................................................396-4. The magnitude form of secant........................................................................................................................406-5. Magnitudes of circular functions of the outer angle.......................................................................................406-6. Magnitudes of circular functions of the the inner angle.................................................................................416-7. Directions of the magnitudes for the outer angle in the first quadrant...........................................................416-8. Applying Euclidean theorems to the trigonometric standard forms...............................................................426-11. The inside powers of circular functions belonging to the outer angle..........................................................456-12. The outside powers of circular functions belonging to the inner angle........................................................456-13. A copy of the half tangent study from the Ostberger notebook...................................................................466-14. The normal form...........................................................................................................................................477-1. A World point.................................................................................................................................................497-2. Cylindrical elements.......................................................................................................................................497-3. Annular elements............................................................................................................................................507-4. Measuring the directions of the surface..........................................................................................................507-5. Numbered octants...........................................................................................................................................517-6. Rotations associated with the Lorentz tranformation.....................................................................................537-7. Rotating the World..........................................................................................................................................557-8. Curves in the planes of“str aight” nulls.........................................................................................................567-9. Curves not in orthogonal planes.....................................................................................................................567-10. Curves in the moving and rotating orthogonal planes..................................................................................578-1. The generalised Law Field.............................................................................................................................598-2. The Newton Law Field...................................................................................................................................608-3. The magnetic Law Field.................................................................................................................................618-4. The electric Law Field....................................................................................................................................628-5. The general properties of a Law Field............................................................................................................638-7. The number operations Law Field (secondness)............................................................................................678-8. The interpretation of number Law Field (thirdness)......................................................................................678-10. The organisation law field (thermodynamic secondness).............................................................................698-11. The reation Law Field (thermodynamic thirdness)......................................................................................709-1. An absorption matrix......................................................................................................................................749-2. An associative matrix with missing elements is still completable.................................................................759-3. An associative matrix with a quadrant magnitude (determinant) of 10..........................................................759-4. The absorption bottle5.....................................................................................................................................769-5. The absorption identity matrix.......................................................................................................................779-6. The Law Fields of Matrices............................................................................................................................779-7. Cardinal and Ordinal numbers........................................................................................................................7810-1. Equal vectors are parallel, and of the same magnitude................................................................................8410-2. Curving an element.......................................................................................................................................8510-7. The extravariant Thermodynamic World......................................................................................................9110-8. The Extravariant Number World..................................................................................................................9210-9. 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........................................................................................9410-11. The first issue of the incomplete Fluidic Law World.................................................................................9411-1. A minimum line element..............................................................................................................................9811-2. Two kinds of particle statistics.....................................................................................................................9811-3. The four fold infinite transformation which births the extravariant World from the intravariant................9911-4. Assembly of the representations of Fermions and Bosons.........................................................................10011-5. The anti-commuting assembly of the intra and extra Grav-electromagnetic Worlds that creates the

representationof atomic elements4.............................................................................................................10211-6. The first set of orthogonal anticommuting elements on the intravariant surface........................................10311-7. Counting electrons, the triplet states...........................................................................................................10511-8. Counting electrons, the first multiplet state................................................................................................10511-9. A sketch of a three electron state................................................................................................................10612-1. Finding the new universe............................................................................................................................10812-2. The one unit rope trick................................................................................................................................10912-3. The Planck Trick.........................................................................................................................................10912-4. The complete set of Red geometries of the inner product..........................................................................11012-5. The complete set of Blue geometries of the outer product.........................................................................11112-7. A magnitude essential to Quantum Mechanics..........................................................................................11212-8. The delta geometry sets are part of the Hydrogen atom solutions (q= 1)14..............................................113A-1. A rope around the world..............................................................................................................................122A-2. For each meter of circumference ................................................................................................................122A-3. A rope around Jupiter..................................................................................................................................123A-4. The largest possible curvature of the rope...................................................................................................123A-5. Curvatures at infinity...................................................................................................................................123A-6. One unit.......................................................................................................................................................123A-7. Half a step to infinity...................................................................................................................................124A-8. One step to infinity.......................................................................................................................................124A-9. One extra unit of circumference internally..................................................................................................127A-10. One extra unit of circumference externally...............................................................................................127E-1. Finding the new universe.............................................................................................................................135F-1. A Schwarzchild gravitational field...............................................................................................................137F-2. The satellite operates in the momentum-static plane...................................................................................138F-3. The rocket is an exchange of momentum device.........................................................................................139H-1. Elements of a sphere’s surface.....................................................................................................................143H-2. Elements of a spherical surface...................................................................................................................143K-1. Sketches of the conic sections expressed in Magnitudinal and Directional form.......................................147

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SynopsisThereis 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 andenlivens the life that has been created over the millennia. Life which is agenerate, whose entropy is negative andwith which, our tiny minds venture to see out into the universe, trying, desperately, to understand the order bothbelow us and above us.

The Sun is remarkable because it too is an agenerate part of nature. From Hydrogen it generates Helium in afrenzy of fusion. It is as if the hydrogen were coming in through some four dimensional hole in the universe to befused into the beginnings of agenerate life in the Milky Way. Forming the lesser Hydrogen into the greater Heliumand expending vast amounts of radiative energy in doing so. This Hydrogen is the beginning of this process andyet Hydrogen is the least of all the elements we know in the universe. It is an atom with a hole in it where theneutron is supposed to be. It consists of just one electron and one proton and because of its asymmetric form itlikes to live with a partner which is similar to itself as the diatomic moleculeH2.

But hydrogen also lives within us. During the process of digestion the Krebb cycle extracts energy from conversionof Adeninine-diphosphate into Adeninine-triphosphate and in doing so leaves a Hydrogen atom to be split into itscomponent 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 theUniverse around us. Riemann, Planck, Einstein, Minkowski, Hilbert, Sommerfeld, Dirac, Heisenberg, Bohr,Schroedinger, Feynman, Hawkin, Penrose all contributed to the mental pictures of the four dimensional worldwe live in. The Hydrogen atom was described and its properties exposed with ever increasing correlation to ex-perimental evidence. Sommerfeld discovered a constant which was as universal to Hydrogen as the speed of lightis to the universe. A constant that is the basic measure of the scale belonging to the Hydrogen atom. A constantwhich 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 Hydrogenatom. 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-structureconstant which has hitherto been the subject of theoretical calculation and laboratory experiment throughout thelast century.

He has used the processes of geometry to reason. He calls them simply,“Directions” and says that they can bestudied in just the same way that we do numbers. That an algebra of Directions can be formulated which leads toeven 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. Hediscovers a four dimension geometry and even n-dimensional ones too. He shows that there are at least a pair offour 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 of Euclid, Ostberger creates a library of geometric solutions which eventually build into acomplete picture of the subjects that we study. He culminates with a model for the chemical elements in a deeplycomplex 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 aquestion, when is your yin equal to your yang?”The result is the selection of one number from a table of newconstants he calls the“Delta values”. This selection yields this Universal Constant to an asymptotically endlessaccuracy of decimal places.

This book is one man’s attempt to tell the story of the creation of this remarkable theory which took Ostbergerthirty years. It is sometimes difficult to comprehend but the geometries are clearly set down for us all to see. The

i

Synopsis

whole book is backed by the notebook of Telle Ostberger which it is said gives mathematical detail of all that isincludedin 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 allkinds. We simply have not recognised them as such. But now, the numerical size of an object will take its placeamongst 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 thatincludes their respective directions. So when we look at a star we are actually looking around the corner into auniverse 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 tellsthem where their next energy level is? Where are the Directions of money? Can we transport ourselves into anorthogonal universe?

It may not be an international best seller but then that is not the object of this book. The object is to expose thisman’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 tofollow 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 enoughto 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 TruthsI am mad. I am completely insane, round the bend, over the hill, half a coin and waiting for the yellow coat toeven think about writing a book like this and expect anybody to read it. In fact, any publisher would also be madto imagine he could find an audience for such a subject as this. I would even venture to suggest that I might dobetter to wait for the next extra terrestial arrivals and hand it to them. For the subject matter is about as cranky assome 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 thatthey have been dropped off on this planet by accident. That their intergalactic bus is about to come back and pickthem 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 realisedthat they are the would-be saviours of mankind. In Ostberger’s case, he had a new language that could be easilylearned and applied to everyday activities in a way that would enabled every person on the planet to see the wholepicture 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 tomankind. The Truths may be important but they vie with the Ways. Ostberger has found a few of the Ways.

The Rope Trick ofAppendix Ais just one example of manydirectionaltheorems. The only number involved isthe number 1. Yet we can deduce from this theorem a great deal about our universe and thewaywe must observeit. There are no straight lines in our observations of the universe which leads to the idea that we should studyonly 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 fromthe “Gottingen club” and all the mathematicians from virtually every University, who in this millennium hasunderstood the message of these old sages? Have they said that we must learn about curvature? Are we guilty ofbetraying the next generation?

Then there are the absorption matrices which give a complete picture of our accounting processes. They are sosimple. How could we have missed them? They are fun for children like chequers or chess, yet they are aboutlearning 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 noliterature that I could find except one,The School Mathematics Project book Zof the Nuffield Foundation(http://www.nuffieldfoundation.org) forO levelStudents. There it is passed over as an example of a surface thatcannot 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 is saying that our lives are pervaded with magnitudes and their directions, the Truths and the Ways. Theyare implicit in all studies not just mathematics and physics. This is theyangand theyin of the Eastern wisdom.His discovery is that we can as easily study the yin or Directional process as we can the yang or Magnitudinalprocess. He has begun the study of the yin process and begun to show its relationship to the yang process. The Di-rections are studied as geometric forms. In particular orthogonal geometric forms. The Magnitudes are studied ashieroglyphics, particularly mathematical ones in the form of equations. For every directional form there is a cor-responding magnitudinal form, if we can find it! They are the left and right hand of the same body of informationthat 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 theenormousrepresentations at the end. It is with great difficulty that I have found a connective path at all amongstthe 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 yearsbefore, mixes with people half his age, teaches, talks and dances his way through life with peace of mind andlightness of heart.

He has opened a Pandora’s box of work to be done that will take more years to complete than universities havestaff to cope. The other books, which relate to the Ostberger notes, are not about mathematics or physics. Theyare about human activity, a subject which intensely interested this man to the exclusion of his further researchesinto the mathematics and other technical studies. We have lost a valuable contributor to this new science simplybecause we could not hear.

There are many who can reason the future Ways of physics and chemistry and accounting and engineering, for that iswhere this road is leading. They will be able to predict the properties of composite materials, make models of atoms andwalk inside. They will be able to travel the universe for years just as Columbus travelled the ocean. They will be ableto make products the size of the smallest dot and the largest city. They will be able to treat illness and learn longevitywith the skills of the great medical masters who have been buried. But they will not even enter the gate of this beginningunless 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

Chapter 1. Einstein and all that

1.1. Introducing the reasons for this bookBooks that relate mathematics to physics and physics to our universe usually centre their attention on that part ofphysics which deals with Einstein’s Relativity. Yet the reality is that the world of Relativity is not more than onethird 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 volumesand pressures of substances raised through temperatures by stuff called heat to produce a thing called work. Thestate of organisation of the working device or substance is negative entropy and a thing called the absolute orKelvin 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 hascreated such as a jet engine, a heat exchanger or a household boiler1. The second comes from the internal affairsof substances such as gases, liquids and solids2. In the former we find that the higher the temperature the greatercan be the organisational efficiency. In the latter the reverse is the case, the lower the temperature the greater isthe efficiency of the substance. In reality the two processes actually work in opposition to each other. Thus in acar engine the gases are made to disorganise themselves by exploding which in turn conveys a certain amountof organisation to the engine in producing work. Such devices are degenerate because they disorganise nature inexchange for something which a human being eventually discards. Thus a car which is manufactured with a greatdeal of human ingenuity, materials and fossil energy disorganises not only man himself but also nature. Natureinevitably retaliates by reorganising the waste into some new form which is foreign to the environment in whichman first created the car. Thus we may consume all the oxygen by burning it in engines which drive machinery tocut 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 havea remarkable ability to sacrifice their life for the ecology. We cannot be degenerate for ever. Yet like all animalsbefore us we will, no doubt, wait for the storm to be upon us before we consider that we ought to have made ashelter.

There is another great World of physics which is not pop chart topping. It is the world of Fluidics. In this world wehave the great laws of Stress and Strain and their ratios, Young’s Modulus of Elasticity, The Modulus of Rigidityand Poison’s Ratio for materials and the laws of structural mechanics described by Mohr’s circle and all theresulting equations. There is the compressibility of fluids, their motion, their viscosity and all the stream lines andpotentials that gather in the equations of Bernoulli. There are the great tensors of stress and strain and vectors ofpotential. This is a world describing the ways in which our universe flows. It is the creations brought about in theuniverse through the combinative building up of the two worlds of Thermodynamics and Grav-electromagnetics3

which we see in the great study of physics.

1


A sketch of a very dense geometry which Ostberger says encapsulates the laws of Physics. It’s presented in moredetail in Chapter 10.

This book is about the work of Telle Ostberger who sees the world, not in terms of the magnitudes of things, butinsteadin terms of their directions. He has discovered that the directions of space are just as much a reasoningtool as are the magnitudes. He has cemented his ideas into the language of mathematics and begun to show thatperhaps all mathematics can be approached in this way as well. He argues that it is possible to see the universein two ways, not one. We are able to examine the Ways of the universe as well as the Truths4. The ways arethe Directional5 studies that Ostberger has begun and the Truths are the Magnitudinal5 studies that have been thesubject of mathematics since the passing of Euclid.

Ostberger takes the principles of Euclid expands them into a whole new paradigm of mathematical thinking, aboutthe 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 Cartesiancoordinates. We may use them to construct our local material environment with vehicles, buildings, possessions andthe like but we may not apply them to nature. She has her own language of curvatures. There are no straight lines andshe does not recognise the concept of distance measured in metres. The curvatures fold over and over transformingthemselves into ever more dense forms of our universe. Only the Laws of nature are straight, as we perceive them, and

2


they do not exist. The Laws are our way of seeing very special exceptions in the world around us. They are an invariantpart of nature 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 thesubject one or the other never quite makes it. The synaptic gap is either joined by a yang magnitude or left open by a yindirection.

In a 3-dimensional world Rene Descartes decided that we should adopt a system of coordinate measurementconsisting of three axesx, 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 entrenchedin 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 Directionswhich belong to them. They are a part of their make-up and they have relationships as Ostberger shows. Whenwe observe light we also observe with the Directional aspects which belong to light. It is a curvature as Einsteinshowed. Light travels along geodesic curves to our eye. The line of sight from the intergalactic body to Earth isa curve belonging to the electromagnetic phenomena that we know as light. Its rate of curvature is given by theLorentz Transformation.

But change did take place through the work of Bernard Riemann in the middle of the 19th century. He constructeda 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 thencame to his peers and said look I can calculate the motions of the universe from Riemann’s work. He producedfrom the curvatures of the calculus three effects:

• The advance of the perihelion of Mercury

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

• The red shift of light 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 butthat 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 proximityto the Sun. When the Sun was eclipsed by the moon the image light could be observed. He predicted that the starwould be seen just a few minutes earlier than the straight line of sight would predict. His calculation fitted theobservation.

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In the third he calculated that light from bodies far away in the heavens would be shifted towards the red end oftheelectromagnetic spectrum as it travelled towards our planet.

What Ostberger observed was that all these phenomena are Directional effects. His calculations from the curvaturetensors of Riemann’s geometry produced directional results. The Mercury orbit changes its direction in a givendirection according to some rate. The light changes its direction around an object with a dense gravitationalfield. Radiation changes its directional frequency through space. However Ostberger saw the third one slightlydifferently.

The current wisdom says that the red shift is like a Doppler effect. When a train travels towards us standing at thelevel 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 toan interpretation of space looking like the train. The farther away an object is perceived to be the greater the redshift and so the greater the velocity of the object away from our witness. This in turn leads to the Hubble constantwhich 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 whichgoes round the Earth. It rotates once6 for each complete sidereal cycle around the Earth. What he says is that thegeometries in his notebook, which are introduced in this book, show that the red shift is due to the curvature ofthe 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 describewhat 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 inreal time or an infinite player7 living in imaginary time is a question which nature will not answer for us. We mustfind the answer for ourselves. Or perhaps it needs no answer and we will learn to live as both by alternating fromone 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 timewhich represented four dimensions.

The time component (ict) was made imaginary by attaching the imaginary numberi as a coefficient. The timeitself was scaled by the magnitude of the velocity of light8. The concept of time was now connected to that ofmotion. (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 realand go positive9. Steven Hawkins10 says,

When I tried to unify gravity with Quantum Mechanics, one had to introduce the idea of ’imaginary’ time. Imaginarytime 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 ourimagination 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 onethat we can measure the easiest. What we have done is to take one rotation of the Earth, divide it into 24 bits, thendivide it into 60 smaller bits, then 60 smaller bits for a second time to obtain a unit of measure we call the second.

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We have measured a Direction in space. The accuracy of this always needs correcting because the rotations of theEarthare difficult to measure exactly. The Earth not only rotates on its axis but also rotates around the Sun. Itselliptical orbit moves to and fro and the Sun is moving and so on. An exact2π revolution is practically impossibleto measure but fortunately the remarkably exact laws of nature permit us to measure the rotations mathematicallyusing the heavenly bodies. It is the fact that the heavenly bodies, and the rest of the universe, follows these lawsso precisely that titillates our imagination and drives us to scientific exploration. What drove Ostberger to furtherexplore the Directions of nature is the realisation that they too are precise in their relative juxtaposition as we shallsee in this book.

These Directions are the curvature of space which we can model in the laboratory. Even more than this we canmodel 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 aninstant 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 yetcontrolled 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 continuumto dis-continuum and back to continuum again alternating the processes of continuousness with those of dis-continuousness.

The key to all this is a rotation of one quarter of a circle. This is what Ostberger searched for in the study ofDirection, and his notebook is filled with examples of this11. Orthogonality is the quantum of Direction.

Notes1. Systems made by man; the machine, as Carse ([Carse87]) calls it. Historically, so far, these are degenerate.

2. Systems made by nature; the garden, as Carse calls it. These areAgenerate.

3. Ostberger used three new words and they are all in this first chapter.Intravariant, ExtravariantandGrav-electromagnetic. See the glossary of terms for full definitions.

4. New testament John Ch. 14 v 6.

5. Ostberger uses the wordsDirectionandMagnitudewith a capital letter to talk specifically about magnitudinalanddirectional phenomena. Both terms are defined in the glossary.

6. The reader is left to think about this.

7. [Carse87]

8. Our connection to nature is through our senses. One of these is sight which uses the electromagnetic phe-nomenain a very narrow band of frequencies. This mathematical phenomena scales everything we observethrough the eye, even if it is transformed by other devices, with the large number3 x 1010, the velocity oflight in meters per second.

9. In the equation−ict becomes−ic(it) which is +ict.

10. [Hawkin88], chapter 9.

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

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Chapter 2. The elements of new geometry

2.1. RotationsOne example of rotations in space is the measurement of time. In the last chapter I mentioned the two methods ofvisualising time, one real and one imaginary. The former describes time as a magnitude, a number that somehowexists in the real world. The latter describes time as a Direction in space and this accords with Steven Hawkin’sdescription 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 whichoperate at the speed of light. The Caesium clock will produce299, 979, 200 vibrational rotations whilst the ra-diation travels one metre. That is because the speed of lightc is 299, 979, 200 metres per second. One metre istherefore0.000000003335640952 seconds. This accords with Einstein who used the scale factorc in his fourdimensional equation. The directional rotation of the Caesium atom and the direction rotation of the Earth areconnected by the laws of physics through the measurement we call the second. The atomic clock is ticking awayat 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 thefield of Newtonian mechanics. From it he deduces that“every momentum has a contra momentum”and otherNewtonian laws (Chapter 8).

2.2. Three DimensionsAt school we learned about three dimensions through the process originally devised by Descartes in the 17thcentury. Three straight line axes were set at right angles and labelledx, y, z. We used them for all our mathematics.They are extensively used in structural and mechanical engineering, physics, chemistry, surveying, navigation, andfor all our local creative activity on Earth.

Figure 2-1. Cartesian frame of reference

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The essence of the Cartesian frame of reference is that it is taken to be numerically zero at the origin no matterwhich direction we approach it from. The negative direction is a translated axis ofx, y, z. There is no reason toassume that this is right or even useful. We have simply not been given reason to use anything else. But Ostbergergives us a reason to approach a point in space with values other than zero. After all we are well aware that somesets of ideas and objects do not contain the number zero. Velocity is one such example. We cannot find a placein the universe and say this is absolutely stationary. Velocity and reciprocal velocity have the property that theymeet at the number one. A place at which we must identify a normalising number, usually1. Another example istemperature and yet another is the set of reciprocal real numbers.

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

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

Figure 2-3. Polar Cartesian

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The boxes that describe the Cartesian space are not always convenient. In navigation, for example, we need aradarto see objects in a polar frame of reference. But this too is only for our local creative activities. It still hasall 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. Thetheorems of Euclid form a basis on which to create geometries which describe the mathematics which is hiddenin its hieroglyphic language.

First, a simple demonstration. The page opposite shows a pen which is held in a stand by a magnetic ball. The penis used as the radius vector representing the polar description of a 3-dimensional Euclidean space1. At (a) the penacts as the position vectorr1 of any chosen length in the z-x plane and together with the angleθ1 describes a setof points within the first right angle. At (b) the pen rises in the x-y plane to produce a set of points(r2, θ2) withinthe next right angle. At (c) the pen falls forward a right angle to produce a set(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. Inmathematics we would refer to this as a new position vector because it is in a space which is at right angles tothe original vector (pen). We can now begin the whole process all over again repeating the four motions of thepen from (d) but with the clip rotated ninety degrees. We can do this four times, each time rotating the pen ninetydegrees. In the quadrant of the space2 that is before us in the picture there is a four fold set of points.

Figure 2-4. The pen stand1

What is happening here? The angleθ4 is creating a new vector which allows us to rotate the pen with its new

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position another four times in the space. Butθ4 is an internal rotation whereas the other three are external rotations.Yet they are all rotations and Ostberger’s work is about rotations in space, particularly ones that seem to go toinfinity, so we cannot ignore it.

But there is another question. Where is ther4 that belongs toθ4?

The pen stand is not the easiest method of displaying the four components of the space. There are clearly someproblems in measurement. Not the least of which is the fact that we cannot seer4. We can seeθ4 and we can askthe question“Is θ4 orthogonal to the other threeθ’s?” We need also to ask what we mean by“orthogonal” inrespect 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 orthogonalrotations using the cube as a measuring instrument. It was clear at the outset that two intersecting lines could beorthogonal at one point in space but perhaps no other. If a curve had a series of points which were orthogonal toa series of points on another curve then there was something special about the two curves. But how would theybe measured? One method was to associate the known mathematics with the curves and prove orthogonality bytrigonometric functions. Another method was to use Euclid’s geometric theorems as proof.

Figure 2-5. An example curved geometry

One thing was clear Cartesian-type straight lines had infinities of points in a one to many correspondence whenthey were at right angles. There were so many points a quarter of a rotation apart that they represented somethingvery special indeed. Ostberger eventually concluded that straight lines which were at right angles were so specialthat they did not exist! We will look at this inChapter 4. What did exist were the curvatures which had severalpointsin one to one correspondence. The circular geometries shown here are particularly tractable and have manyqualities that make them a sensible starting point for the study. These are both geometries in the plane and someasurement of orthogonality does not present a problem. TheFigure 2-5is very close to the shapes producedby Radical Circles except that the line~AA is not a straight Cartesian axis and that makes the shape algebraicallycomplicated. However an algebraic description of a shape is very different from using the geometry to representa vector. The same shape of a geometry may have many uses in representations (see example inFigure 5-6).

The lower geometry (which Ostberger calls“of the first kind”) is perhaps the most simple form to use but firstwe must rid ourselves of our Cartesian frame of thinking. If there are axes then they will form themselves in thecourse of constructing the representation. Let’s look at a couple of examples from the Ostberger notebook.

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Figure 2-6. An important basic geometry

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

Figure 2-7. Multiples of trigonometric functions

Some of the trigonometric functions are shown in the geometry ofFigure 2-7. It uses the geometry of the first kindin Figure 2-6. We see here that the intervals between successive circles is constant with respect to the origin. Thismeansthat any scale between1 and nearly zero is possible using the sine function here. This can form a basis forworking with vectors. The amplitude of any sine or cosine function is clearly identified as associated with one ofthe circles and can be related to a curvature. All amplitudes are possible4. This becomes a library standard formtogether with all the other trigonometric functions5. It can be used to construct further geometries because thespace becomes unique if the nomenclature is followed.

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Figure 2-8. The first sets of integer surds6

Figure 2-8is an example of the mixed use of ordinal and cardinal number. Here the line lengths in the geometrymeasurethe roots of all integers. If the circles are of a continuous nature then it is apparent6 that all real numberroots lie on the horizontal line at unity. By picking out groups of lines from point to point one may find severalkinds of root functions expressed as the length of a line element. In this case a straight line element as this is inEuclidean 2-space.

Figure 2-9. Elements of the form6√a(a+ 1)

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Figure 2-9is 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 ofthe studies do not pay off. At least for the present. But we never know when a piece of mathematics, in this casegeometric mathematics, will be useful. So we need to build up a library ofStandard forms.

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

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

The earliest standard form is an examination of two straight lines meeting at right angles. We can imagine thateachpoint amongst the infinity of points on the blue line can be measured as being orthogonal to every point onthe red line at right angles to it.With an infinity of points on the blue line there is a square of infinite orthogonalconnections in each quadrant. This representation leaves us with a many to one and a one to many relationship ofpoints. 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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On the other hand two curves can meet orthogonally at a point (Figure 2-11). But we still need to discuss inwhatsense they meet orthogonally. Two curves of equal curvature meeting orthogonally will have a whole set oforthogonal points in one to one correspondence. On the other hand if they are unequal curvature such asFigure2-5 then the meeting point is special. It may not be unique for there may be a pair of points as inFigure 2-11.

Figure 2-12. We can apply directions 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 thespacethe orthogonality manifests itself (Figure 2-12). We can apply directions along the line elements so thatthey become unique in the sense of having an inward or outward relationship.

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

We can establish rules about the juxtaposition of the lines in space and say, for example, that the shortest pathbetweentwo orthogonal points is a quarter of a rotation (Figure 2-13) or that the shortest path between two pointswhichare 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 thenone of the ends must be at infinity. This is analogous to saying that we cannot form a number system with anyless than two characters. The binary system is the minimum that will allow a complete set of magnitudes to be

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represented. But what do we mean by“complete”7? The two magnitude characters0 and1 of the binary systemare to be replaced by two Directions in a line element in the Ostberger geometry.

Figure 2-14. Different types of orthogonal elements

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

2.4. More Than Two Dimensions

Figure 2-15. There are eight possible orthogonal curvatures of the single line element; four cylindrical andfour annular

Just as we may embed a vector in mathematics into a one, two, three or n-dimensional space so we may embed aline 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) in thex

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plane and four in they plane.

In each of the curvatures we have numerical scales in four senses. Two scales along the line element (forwardand backward) and two scales orthogonal to it (in the radial and the normal). This amounts to saying that theMagnitudes can be associated with the Directions in four ways. In the first way the magnitude and direction are0π-apart, in the second way they areπ-apart; the orthogonal way isπ/2-apart and in the case of the normal theyare2π 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 inthe next chapter.

Figure 2-16. The old vector rules

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

Figure 2-17. The new vector rules

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

But what this means for mathematics is that Ostberger is saying that vectors may have reverse magnitudes. Forexample a positive one in the forward direction and a negative one in the reverse direction. Both the Direction andthe Magnitude can be reversed.

What Ostberger has discovered is really quite simple. Directions are just as much a tool of reasoning as are themagnitudes. Mother nature knows this and makes thorough use of the fact. She does not easily yield up her secretsbut Ostberger has discovered that Magnitudes and Directions are separable entities. He has left us with a legacywhich will serve us well in the next millennium. Our modelling techniques can now include pictures. But theycan 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 vectorspaces. But the Tao is not there if the yin or yang is missing. Neither is it there if they are found because whenthey are found and placed together there comes another yin or yang to be found which is inside out to the previousand 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 stillmissing the Directions that belong to nature.

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Mathematics is primarily a yang study. A study about the magnitudes of things. About trying to establish a numberthatwe can attach to something. But Ostberger invites us to commence the yin study of Directions so that we mayestablish the directions in life’s rich tapestry.

To quote from his notebook,“for what we have ignored is that the Laws of Nature follow very strict rules ofDirection, her yin, as well as of magnitudes, her yang. The idea that everything in nature is going to be describedby a number that can be extracted from a mathematical equation is as strange to me now as it was when I was astudent. Although mathematical equations contain components of both magnitude and direction the later is easilylost in the craving for a number; for the Magnitude has components of both direction and number and so does theDirection.”

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

Figure 2-18defines the attributes of a vector of position like the pen. The length of the vector represents themagnitude.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 radiansbut can be other measure such as the delta values in the atomic structure9. The other is the direction of the directionwhich we measure with geometry such as Euclid and Ostberger. There are four aspects to the vector ofFigure2-18, not two as our presence mathematics currently presumes. This results in the possibilities shown inFigure2-17.

TheOstberger work sets out to represent all four aspects. By selecting one of the aspects first we may bring ourreasoning to bear on the problem that we wish to represent. We may select the Magnitudinal aspect first in whichcase we delve into our existing hieroglyphic mathematics to seek a magnitude solution. It is likely that we willtravel via the eigenvector theory towards our solution. On the other hand we can select the Ostberger process bylooking 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 followthe processes which nature herself seems to use. She oscillates between a magnitudinal (yang) process and adirectional (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. Theprocess is endless. It seems that our right and left cerebral hemispheres are a gift for this process.”

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However, it would be impossible to foresee all aspects of a representation before we start and so we have to creepupon it just as we do in mathematics by using the lesser calculus such as simple differential equations to build upinto a denser one with operator nomenclature.

The Ostberger process is broken into four main stages.

1. A yin representation of direction only. A generalised geometry which is much like a guess.

2. A representation of the stage 1 above, with the yang 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 areattachedto the line lengths which, at this stage are not defined in magnitude.

4. A complete representation which includes all the above stages and the yang magnitudes which finalise it. Asetof scale factors relate the whole to our measurement of reality.

Figure 2-19. A larger curved geometry

Each stage is a slow process and none have been formalised in Ostberger’s work. He simply recognises the needfor 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 ismirrored in the other symmetrically. However, I suspect that the mirror is asymmetric just as we find in physics.”

Notes1. The simple mathematics is given in[note103].

2. By using the full 360 degree notation ofθ andthe positive and negative values ofr we can describe the wholespherical space in terms ofr andθ.

3. See[note140].

4. Ostberger made notes on imaginary geometries.

5. SeeChapter 6and the[note5xx]series of notes.

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6. For the mathematics of this please read[note586].

7. The extremely powerful Godel’s Theorem (the incompleteness theorem) will lead us, in later chapters, to asystemof geometric representation which has no singularities, but“time must be a Direction in space”insuch a system.

8. See[note117].

9. Delta values are explained inChapter 12.

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Chapter 3. A story and a theorem

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 otherwun-men. They were all confined to one dimension because they had two heads and because the top half of thembreathed 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 madetheir nests. Whenever they went walkabout they always travelled in straight lines on account of them being onedimensional and having two heads. They would go outward with one head thinking and come backward withthe 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 hishome and every body accepted that that was the way it was.

Figure 3-1. The Wun-man’s universe

Also, as you can imagine, the habits of their predators was made very easy. All they had to do was to home in onthestraight lines of the wun-men and consume them to extinction. And we know that that is true because thereare 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 asentry to measure the directions of the fly fish as they went walkabout. If the wun-men did not return then it wastheorised that there was a hole in that direction or that the poor little chap had dropped off the edge. No wunmanwould go that way in the future. The sentry was know as the fly fish angler. But alas the angler was to be theirdemise 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 theother wun-men.“How can he survive with only one head?”, they would ask,“he will never be able to get backhome”, they said.“he will always need some-wun with him to show him the way home”, they said.

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So, as he grew up he was never jolly because no wun would join with him and he was jibed and jeered as juvenilesdo.But he was a determined little fighter and whatever the other wuns would do he would do too. He soon got thenick-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 alittle so that he could see where he had been. He would then do a thing he called“backwards”. It was so excitingthat 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 togo “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 tofind 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 backthat 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 goout 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’simpossible; and with only one head its doubly impossible. If you go out that way you must come back that wayand if you go out this way you must come in this way. Otherwise you drop off the edge. That is an establishedscientific 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 heknew. He knew it was true for this was his second discovery. Wun could go out in one direction and arrive back inanother direction and wun could go out in two directions at the same time. There really was a second dimensionin 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 thewun-men was the two-man; and the only survivor of the two-men was the three-man; and the only survivor of thethree-man was the four-man and so on until we human are the last survivors of the hu-minus one-men. And theone-man was where we began.

It was the wun-men who discovered, through a genetic mutation, that in order to survive ann-dimensional worldwe need to know about ann+ 1-dimensional world.

The moral of this story is that our survival in this 3-dimensional world depends upon our ability to understand a4-dimensional world. That is what Ostberger is giving us.

3.2. And now for a theoremTo lighten up the studying of these new ideas Ostberger invented a rope-maker who was all too often involvinghimself in problems with ropes which he didn’t understand and so tried to find the answer in mathematics. Heusually 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 outsetit is difficult to imagine how we could construct a process of reasoning with geometrical forms which has a sound

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

3.3. The first rope trickA rope-maker is contracted to make a rope to circumvent the Earth. The mean radius of the Earth is 6.37 millionmetres making the circumference some 40 million metres.

Figure 3-2. A rope around the Earth

The rope-maker was overjoyed with his contract. He contrived a machine to make the rope at great speed andcalledupon 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 endstogether and celebrated his success. But no sooner had he finished celebrating than he received irate telephonecalls from around the world saying that the rope was far too long.

The rope-maker had a brain wave. He drew the ends of the rope taught and measured the spare rope. It measuredjust half a metre. He could not understand how this was possible, so he went to his local university for help incalculating the problem.“It’s far too trivial for us”, they said, “we don’t get involved in trivia like that”. So therope-maker did his own calculations.

What he discovered was quite remarkable. The half metre overlap (Figure 3-4) that laid on his workshop floortranslatedinto 15 cm of slack in the rope all the way round the Earth! The phone calls were right. There was a lotof slack in the rope. He could not believe that the half metre of rope that lay on his workshop floor could produce15 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 hiscalculations 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 wonderif it will have to be as accurate?”

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Figure 3-3. A rope around Jupiter

He did some more calculations. To his surprise he came up with the same answer as for the Earth rope. Thedistancethat the rope would stand off from the surface of Jupiter would be 15 cm for each 1 metre of error in thelength of the rope.“How can that be?”he exclaimed.“Does this mean that the stand-off height is a constant nomatter how big the planet! Is it always 15 cm per metre?”

Figure 3-4. one unit of overlap

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

With nobody to tell about his discovery the rope-maker slipped his notes into the workbench drawer and proceededhome to bed.

The rope-maker had indeed discovered something quite useful and this is given inAppendix A. You do not haveto be a mathematician to understand the arguments, although you will need some skill to extend the arguments tocases 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 thearguments and produce the answers. There is also another rope trick in the appendix and more in the notes.

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Chapter 4. Orthogonality

In this chapter I want to look at the way in which Ostberger slowly arrives at the idea of orthogonality in physicsand then transcribes that idea to the mathematics.

4.1. The rules of FlemingClench the fist of your right hand and point your thumb upwards. Now point your first finger away from you andbring your index finger out so that all three fingers are at right angles. This is your very own personal orthogonaldirection 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 meticulousexperimentation and is essential to the calculation of such fields at a point in space. It puts into a formula thedirectional relationship between the current in a wire and the field outside it. However, it is not easy to see in theformula 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 electriccurrent as an input to extract motion as an output. If you take the motor apart (one with coils on the rotor) andplace your index finger (representing the current) in line with one of the copper wires and then rotate your wristso that your thumb (representing the motion) into the line of motion of the motor then your forefinger will tellyou the direction of the magnetic flux at that point in the motor. Of course, its a silly thing to do, so we do it withour heads instead.

Figure 4-1. Left hand rule 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 everandever, and ever. It will simply never go away. It is as fixed as the stars or the laws of the heavens that positionthe planets to the last decimal part of a millimetre. So Ostberger thought that he had better see if there were anymore 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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Figure 4-2. Right hand rule for the generator

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

Figure 4-3. The unidirectional representation

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

Inward A process of putting motion in to get current out using a magnetic field as a converter is agenerator.

Outward A process of putting current in to get motion out using a magnetic field as a converter is amotor.

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Figure 4-4. The bidirectional representation

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 noteswhich suggest that he despaired that he might not be able to see the agenerate processes of nature in the newprocess. But we will see later that he succeeded in showing the remarkable processes which are agenerate innature and ourselves.

4.2. Deflection of an electron beamIn a cathode ray or television tube electrons are produced in the gun at high voltage and they motion towards thephosphorescent 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 reliablyfollow these rules. But what are the rules and how do they manifest themselves in geometry?

4.3. Vectors of lightThe 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 beidentified 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 con-cludes that the directions belonging to the vectors of light are the surface vectors which are wrapped around themagnitude vectors show inFigure 4-5.

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Figure 4-5. The orthogonal vectors of light

We can identify three vectorsE theElectric intensity (force),B the Magnetic Flux density andv the velocity ofpropagation. These are vectors of a magnitude character and we can ascribe a number to them according to theparticular manifestation of the light. They are related precisely byFigure 4-5. The three vectors are orthogonalin space and we can identify them with a right hand rule just as we do Flemings rules. Again the precisenessof this relationship is never in question. Our experimental measurements indicate that they remain consistentlyorthogonal, in the manner ofFigure 4-5throughout the universe.

Ostberger asked himself,“Is this velocity the same kind of velocity that we have seen in the electric motor andgenerator?”He also asked himself how these vectors could remain orthogonal in a space of curvatures. It took ajourney of many years to find the answers.

It was on this journey that he identified the surface vector ofFigure 4-5as belonging to a group of tensors whicharepurely directional in character. He later named themElectric DirectionandMagnetic Direction. They were theDirectional component parts of the vectorsE andB. These Directional vector components manifest themselvesasPotentialsin physics. The connection between the two is by far the most remarkable discovery that Ostbergermade. Two great four dimensional geometries containing the same complete set of vector elements appear tobe inside out to each other. In fact they are orthogonal as we will see in later chapters. Ostberger called themintravariant andextravariant.

4.4. The vectors of the gyroThis example of orthogonality is very much down to earth. The Earth is a gyro body and it obeys the mathematicsof the universe. Or is it that the mathematics obey the rules of the universe3? 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 spokesto spin the wheel and there you can feel the forces of nature in the gyro.

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Figure 4-6. Using a 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 lefthand.The wheel does not fall to the floor. What does it do? It rotates.

Looking down in the Direction of the gravitational acceleration it rotates clockwise. It will rotate clockwise foreverybody 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 thepictures and pass on to the next chapter.

Figure 4-7. For the left hand holding the spindle; the magnitudinal part of the spin.

It is easiest to look at the spin of the wheel first. Hold your right hand out with your orthogonal indicating digitsin position. Point your forefinger along the spindle of the wheel and follow the rotation of the spin with a righthanded screw motion. Construct an axial vector with a ring around it in the direction of the screw motion (Figure4-7). This is the spin component with its velocity around the ring.

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Figure 4-8. Gravity pulling down creates a torque

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

Figure 4-9. For the right hand holding the spindle; the directional 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 willhave to point your forefinger up in the air. The right handed screw turns the way of the procession as we haveobserved (Figure 4-9).

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Figure 4-10. The orthogonal surface vectors of the gyro

We can now assemble these findings on to a single surface, theFigure 4-10. Here we can identify three axialvectors in the representation each with its orthogonal phenomena connected thus,

1. The axis of the Precession with its associated surface Direction.

2. The Spin with its associated surface Velocity.

3. The Torque with its associated surface Force.

This is a second stage diagram of the type mentioned at the end ofChapter 2. There are no magnitudes and sofurther work must take place to make a stage 3 representation. But the directions will remain in their relativejuxtaposition. In fact, we will see later that the three surface vectors form what Ostberger called aLaw field(Figure 4-10). The reason why this is special will be discussed inChapter 8.

All these vectors are well known. But, perhaps it is not so well understood that the Precession produces a Directionsurface vector. What kind of phenomena is this and to what group of phenomena does it belong?

Figure 4-11. The gyro surface produces a Law Field

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The Law Field helps us to group the phenomena as we will see inChapter 8. Thus all forces of a Newtonian kindcanbe aggregated on to the Law Field by mappings. So they are a group of phenomena. So also are the groupof Newtonian velocities4. The third group is a group of Potentials which is here associated with the directioncomponent of the whole field. It is called Direction as a general term because, as we shall see, there are many ofthese. It is a class with the termflux perhaps.

The precession is associated with this Gravitic Potential. What Ostberger showed was that this Potential belongsto 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 accustomedto the idea as we work out our misconceptions in terms of these directional phenomena.

4.5. Other examplesThere are other examples both in thermodynamics and fluid studies where orthogonality is the key to separatingthe 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 potentialsaround an electric charge, the vectors in a transformer, the force and potential in a Schwartzchild solution, thevelocity, 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 groupsto our social activity too, such as accounting.

Notes1. In Chapter 10these three fluxes will appear on the surface of a World geometry.

2. Now we see the relationship to the clip on the pen inFigure 2-4.

3. This is an example of intra- and extravariance.

4. SeeFigure 8-2.

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Chapter 5. More than geometry

5.1. Studying directionIn studying directions I have come to realise that they are very real and tangible entities that enter into the structureof 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. Thisis not so. Numerical answers can be found amongst the geometry of curvatures that are just as precise as the oneswe find by formula, but these two concepts are not in competition. They need to be in cooperation, for sometimesit is better to use the Direction process and other times it is easier to use the hieroglyphic process. They are botha part of Mathematics.

In the final result we are looking for the truth and a way of expressing the truth. Both processes speak in theirspecial 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 andoften try a directional sketch to achieve the result. The mathematics often hunts the directional eigen-vector as apath 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 cannotwhittle it down to a specific entity if we have already used the generalisation successfully. If one generalisationworks then we may test it again to see how many times it does work. After a while we gain sufficient confidencein the process to be able to accept the generalisation.The trick is to get the generalisation right first time to avoidextensive researching of mistaken direction. The examples of orthogonal phenomena in the last chapter wereintended to show this process at work. We can go further. If we can relate the generalisation to a known piece ofmathematics such as tensor calculus and see that it fits then we can gain confidence at the start.

Figure 5-1. A line minimum element

How many dimensions must a line have for us to be able to see it? Let’s look at a one dimensional lineA in Figure5-1. It is not there because we need to have another dimension to see it. The line must have a small part of aseconddimension if we are to see it at all. SoFigure 5-1B is a 2-dimensional line displayed for us to discussone of the dimensions. If we now curve the line we are producing effects in 2-dimensions. This is the minimumdisplay 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 commentsthus,“If string theory were to take account of both of the two most important theorems in Quantum Mechanics itwould accord with my own discoveries precisely. String theory has ignored the orthogonality theorem.”

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So Ostberger built up geometries on the basis of curvatures starting with examples like those inChapter 2andmoving on to more condensed forms of geometry which we will see inChapter 10. It took thirty years in all.

Themost 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 suffixeswhich are conventionally attached to the character of tensor calculus were, in fact, the Directionsi, j andk, of thisgeometry. In a note on the subject1 he identified all the permutations of tensor calculus as products of this space.The picture at the top inFigure 5-2is one of the many diagrams he drew for that note. Interestingly he identifiedtheEinstein 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 andDirections respectively and hangs his potentials on the Weyl tensors.

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

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

32


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

Figure 5-3. Permutation of a world geometry in tensor 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 theinterface3. 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 contravariantspace whilst keeping theij junctions and all the others orthogonal. That is one possibility. Now suppose we havea whole set of thej circles spreading over the covariant surface. (See Ostberger’s sketch in the bottom left handcorner ofFigure 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 wereverse 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 theij 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 enormoussince 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 thei andjdirections in turn (the small pictures inFigure 5-3).

Thepicture gets bigger because we can hang all three permutations in theFigure 5-3on to a singleWorld geom-etry. A sketched example is shown inFigure 5-4.

And the picture gets bigger still because we can use each of the orthogonal states of this geometry to representa group of phenomena. This is what Ostberger did with his process of Law Fields which we will look at insubsequent chapters. The result was geometric representations so large that it becomes difficult to handle thesolutions they contain, in much the same way as in the Schrodinger equation.

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Figure 5-4. Orthogonal curvatures in a World geometry

But the picture did not stop there. An even bigger picture was obtained by Ostberger. He looked into the geometryof Figure 5-4and imagined that he could stand inside the covariant components which encircled him, the onesthatare outside the World. He imagined that all the surface components of the World (and there are two furthersets that I have not mentioned here) were expanded to infinity and collapsed around him. The result was a newWorld 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 inChapter 10.The new World is clearly different. The first he calledintravariant and the second he calledextravariant. Theformer addressed the problems of classical physics and the latter those of Quantum Mechanics. The extravariantWorld was of discontinuous character and the intravariant World of continuous character. They required differentmathematical treatment. We will look at this later.

5.2. Growing geometryEvery geometry can be used many times for different kinds of representation. Take as an example the simplestkind of geometry inFigure 5-5. It is Euclidean and undirected. It is the simplest form of one of the theorems welearnat school.

Figure 5-5. The interior Euclidean chord theorem

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Now apply this to some geometries which are constructed inFigure 5-6. These geometries are also Euclidian. Theyarenot even bidirectional, yet to describe them with algebra would require a considerable degree of mathematicalskill for there are many circles in them arranged in a special displaced order. So, algebra is not the first choice forthis process. Let’s look at the geometry.

Figure 5-6. The magnitude powers ofp

The two sets of circles are to show how different magnitudes are obtained from exactly the same geometric formsimply by changing the choice of scalar in each set.

How did he arrive at these magnitudes? Consider the Euclid’s theorem ofFigure 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 areequal 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 theeffect obtained by exchanging the values ofp and unity. Each value ofpn in the geometry ofFigure 5-6’s righthandside is measured from the centre line and clearlyp0 is unity. Comparing the two geometries, one is for thevalues ofp less than one and the other side for values ofp 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 ofChapter 2.

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Figure 5-7. The field of an electric dipole

There is another kind of geometry which is well known, the field an electric dipole (Figure 5-7). This time thegeometryis of a vector character. The dotted lines are equipotential lines and the others are the strength of theelectric 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 ofthis kind representing the field of a pair of parallel wires with currents flowing (Figure 5-8). This is the kind ofgeometrythat 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 shapeswith the pole at the limit point).

Figure 5-8. The field of a pair of parallel wires

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

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to say that the study of the hydrogen atom displays such a geometry. InChapter 12the geometry which producesOstberger’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 ofOstberger’s work as is any other subject. It is self analytic. InChapter 10the structure of subjects is based on theprincipleof the Law Field which brings together the biggest possible concepts into a geometric form. The straightline 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 arrowswhich point a way and there is no difference between forward and backward. The theorems of our school daysdescribe these undirected line elements. From here we move on into the study of position vectors. These aredirected line elements. These position vectors have transformations which operate on the vectors, making themenlarge, 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 nowadd another set to this, the Interpretive set4. 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 ofvectors would appear to be complete. The "scalars" are vectors of a zeroth kind. The vectors of force, motion andmomentum are those of the second kind5.”

Curved vectors of the second kind are not commonly used in mathematics. Indeed their existence is scarcelyadmitted yet they are quite valuable tools. Vectors having multipleheadsand tails are certainly new and notyet related to tensors as Ostberger has done. These are the vectors of the third kind. These four kinds form amathematical 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:

• Extended Klein’s concept of geometry as consisting of the group of all transformations over the group of all geometriesto include a further group, the group of all Interpretations. The Interpretive group is over the Transformation groupand the Transformation group is over the Definition or Formative group

• Exchanged planes and straight lines for curves and curved surfaces making the former a very special case of thelatter.

• Extended 3-dimensional geometry to 4-dimensional.

• Incorporated the principle of orthogonality to all directional study.

• Added the principle of conversion from intravariance to extravariance.

Notes1. [Note1254]

2. [Note1250]

3. For the mathematical these aregij , gij andgij andtheir inverses.

4. SeeChapter 9and[note220].

5. SeeAppendix B.

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Chapter 6. The development of standardforms (rank one vectors)

6.1. Standard formsOne of the tools that is particularly useful is the library of Standards Forms1. Many equations can be representedin geometric form, many solutions can be geometricized, so it is helpful to create a library into which all the usefuland 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 aretools at every level of the company. The international corporation regards countries and governments as tools, theconglomerate regard sites and buildings as tools, the company regards computers and milling machines as toolsand the operator regards keyboards and setting clamps as tools. Tools operate at every level.

TAO

In the process of building up these geometries I have established a small library of geometric standard forms. I wouldlike 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 monthson a mathematical subject far removed from that which was relevant to the standard form. I have done no more thanscratch 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 formsThere 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 ofequations. 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 these2 occupies a space about the size of this book. Yet Iam 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 learningmethods into simple forms. Yet the time spent discovering these simple forms is immense and will only beunderstood 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 processof 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 theirapplications.

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

Figure 6-1. The directional form; the functions arise as directional ratios

We may also establish these same ideas in a Magnitudinal form. ConsiderFigure 6-2. Here a triangle is boundby a circle of unit diameter. The angle at the circumference is always a right angle and the sine and cosine arealways as shown. All the principle values of these two elementary functions are contained in the motion of thefirst half circumference. (But note that the angleθ has only rotated a quarter of a cycle.) Here I will only show thefirst quadrant of the functions but in the notes are the complete set together with a computer animation of all thefunctions3.

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

These two representations are yin and yang. Neither can claim exclusivity. They are compliments of each other andserve useful purpose for different applications. The fact seems to be that this property of pairing Directional formswith Magnitudinal ones continues indefinitely. It has to be researched on every occasion. When a representationis discovered there will be the yin or yang counterpart somewhere. It is not always obvious or easy to find.

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Figure 6-3. The magnitude form of tangent

In Figure 6-3the tangent functions are added. InFigure 6-4the secant is added. The line lengths shown are alwaysthemagnitude of these two function when measuring the angleθ as shown.

Figure 6-4. The magnitude form of secant

We may continue in this way until we have produced the combined geometry shown inFigure 6-5. Here a rotatinghalf vector is to be seen carrying with it all the elements in the picture as the pointP traverses through the firsthalf circumference. The angleθ traverses 90 degrees.

Figure 6-5. Magnitudes of circular functions of the outer angle

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What is to be understood here is that each line element representing a function is accurately the correct length.Thelimitation is the accuracy of the process used to construct the representation. The Direction of the function isalso correct with respect to the angle atO1 in Figure 6-5orO0 in Figure 6-6. The nomenclature must be followedwith precision.

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

What is the difference between theFigure 6-5andFigure 6-6? At first sight there seems little difference exceptthat the angle of measurement has been changed fromθ0 atO0 to θ1 atO1 (seeFigure 6-7). What the studentwill find, upon careful study, is that theFigure 6-5is a discontinuous geometry whilstFigure 6-6is a continuousgeometry. They can be assembled together in many different ways.

Figure 6-7. Directions of the magnitudes for the outer angle in the first quadrant

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

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simply because we have drawn it. It may not be consistent or necessary or sufficient. Only when the geometry hasall its stages (Appendix B) defined can we accept a given geometric configuration. In this case we need to definethefinal stage, the stage of applying Directions to the magnitudes i.e. the line lengths.Figure 6-7is the final stageof production for the Trigonometric geometry. It makes it mathematically unique because all four aspects shownin Figure 2-18have been defined. The convention here is that one unit of amplitude is defined upward whichmeansthat in the bidirectional space of the representation minus one unit is downward in the same space.

6.4. Applying Euclid over OstbergerWe may increase the depth of our understanding of the trignometric standard form (TSF) by applying Euclideantheorems to it. This is an example of using a lower density4 geometry and applying it to an higher one. The lowerdensity, the theorems, are undirected and unspecified in magnitude. The higher density, our TSF, is directed andspecified. We can apply one over the other. Look at the figures inFigure 6-8. Here are just five examples of theuseof Euclidean theorems to extract more information from the TSF.

Figure 6-8. Applying Euclidean theorems to the trigonometric standard forms

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We have:

Figure 6-8a. An amplitudeA appliedto the TSF of the outer angle. The student would see that this has theeffect of scaling all functions relating to the initial scalar. This represents a big mathematicalmove. Were the amplitude applied to the TSF of the inner angle then this is simply a variableamplitude in a geometry of continuity. It is the fact the the outer angle TSF is a discontinuousgeometry that makesA have such a big effect. InFigure 2-7of Chapter 2, for example,Awould have the effect of changing every amplitude in the set. It is like the difference betweena multiplier and an operator in mathematics.

Figure 6-8b. Pythagoras’ theorem applied gives a well known relation. Compare withFigure 6-8e wheretheexternal chord theorem is applied to give another relation.

Figure 6-8c. Applying Pythagoras yields this common relationship which can be seen to be satisfied forall angles including those greater thanπ giving the negative magnitudes inFigure 6-8d.Simply reversing the triangle inFigure 6-8c does not satisfy the conditions set. It would havetheeffect of changing the convention used. That is unacceptable. The formula atFigure 6-8dis currently considered to be trivial. Here it is shown to impart different directions to therepresentation. It belongs to a different region of the circle.

Figure 6-9. A proof of the reciprocal relationship between tangent and cotangent

The interior Euclidean chord theorem shows that,cot t. tan t = 12, which proves the reciprocal relationshipbetween them.

Two more examples before leaving this chapter. InFigure 6-9I have assembled two copies of the unit TSF, oneabove 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 bythe large circle. We may apply the internal chord theorem to yield the fact that there is a reciprocity between tanand cot. An obvious fact in a new context.

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

The Euclidean chord theorem shows that,a cos t.a sec t = a2, which says that there is a reciprocal relationshipbetween cosine and secant,cos t = 12

sec t , which provides a simple proof of a well known relationship from earlierstudy of Euclid.

In Figure 6-10the external chord theorem is used to find a similar relationship between cos and sec. Here we canapplythe 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 andbackward directions are the reciprocal of each other. Here are two reciprocal representations which may be usefulin the future.

It is currently common practice in mathematics to assume that the forward direction is positive and the backwardis negative and that they do not overlap. This was Descartes creation. I now see that this has, unwittingly, been agreat hindrance to mathematics. In this chapter I have seeded the idea that other possibilities exist, not the least isthe idea that positive and negative may be represented as opposing and overlapping.

6.5. AmplitudesIt should be quite evident that combiningFigure 2-7with the TSF of the outer angle will provide representationsof all discrete increases in amplitude. The same with the inner angle will provide all the continuous increases inamplitude5.

The geometric standard forms for frequency6 and multiple angles7 are particularly interesting because the geom-etry creates a unit circle rolling through the space of the multiple angle. Its meaning is elusive.

6.6. Powers of trigonometric functionsIt 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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the discovery process. I reproduce one of the resultant library forms for the powers of sine and cosine inFigure6-11. This is reproduced from the Ostberger notebook.

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

By extending the process of discovery here to the outside region of the bounding circle we find the secant andcosecantpowers inFigure 6-128.

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

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6.7. Half angle tangent relationsTheserelations are well known in A-level mathematics but Ostberger extends them. The page inFigure 6-13isa copy from Ostberger’s notes9 which geometrise the half angle tangent relations and extends them to interestingand potentially useful geometries. The geometry shows that there are directional and magnitudinal cases of theserelations and connects a unique and infinite series of them. The mathematics and geometries are some 40 pagesin all.

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

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6.8. Algebraic equationsFromthe earliest days of algebra processes had to be developed for finding the roots of equations. The methodsof factoring, finding solution formulae and trial and error are all a part of the history. That it is possible to use ageometric method may not be new but such a process has not been accepted as enhancing the understanding ofthe algebra process. What does seem to be useful is that the geometric process using lines in space can lead moreeasily to bigger concepts. Thus, if we solve quadratics10 and cubics11 with a Directional process then it might bepossible to solve quadrics and other higher powered polynomials12.

6.9. Parametric equationsThe representation of parametric equations lends itself easily to the direction process.Figure 6-14is the normalform x cosα + y sinα = p for the Cartesian equation3x + 4y = 7. It is a specific case of the more generalform described in[note90] (14 pages). It is particularly useful in the learning process because it relates, in aproperpictorial way, the relationship of a Cartesian inner space to its outer space. It is, however linear andtherefore of little value in handling the curvatures of later chapters here. The circle is the same circle of theunit TSF. To relate this mathematically the student will see that the parametric equation must be the negativecase−x cosα − y sinα = −p. The reason for this is the labelling of the pointO0, which is essential for themeasurement of the angleα as shown, makes the sine and cosine negative and in the second quadrant.

Figure 6-14. The normal form

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6.10. Imaginary geometryBecausethe square of the imaginary quantity

√−1 producesa negative magnitude we need to examine just

how geometries with imaginary values actually work. The Pythagorean hypotenuse is not necessarily the longestlength. Ostberger has made a study of these13. 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 constructingpictures which are mathematical representations. Eventually we will arrive at two demonstrations of the processand 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.

Notes1. We need to bear in mind that this book has been created because there has been no way of expressing Os-

tberger’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 mathe-matics. The resistance to new work is prevalent in all professions during the twentieth century and no mech-anism has been set up to allow new work to entry the wisdom of mankind. Such wisdom is still the domainand privilege of the high priests.

2. The trigonometric notes are found in notes 110-120, 501-540 and 1200-1221.

3. This animation is interesting because the vectors cover the region enclosed by the circle twice, yet the anglethetais covered only once through 360 degrees. It is a Riemann two-sheet.

4. SeeAppendix B.

5. [Note504]

6. [Note112].

7. See[note512],[note513]and[note514].

8. These two geometries are the culmination of five months work.

9. See[note1206],[note1207]and[note1210].

10. See[note98].

11. See[note96].

12. Ostberger predicted that all polynomial solutions lie in a single bidirectional plane although there is no evi-dencethat he proved this.

13. [Note1230]

14. The Hydrogen atom is presented in a separate book. It examines the process of arriving at the model throughthe Schroedinger equation and compares an almost identical model arrived at through Dirac’s equations ofvectors.

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Chapter 7. Four dimensions are here

If it were not for the fact that a four dimensional geometric representation was possible I do not believe thatOstberger would have continued his studies. This is now introduced so that the student may also be motivated.

7.1. Four dimensionsThere are several uses of the word“dimension” in mathematics and so it is essential to iron out their meaning.Appendix Ddoes 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 universewe will always live in disharmony. It seems that we must understand a fourth dimension to be able to live stablyin a third dimension. It is fortunate that we have a geometric picture that allows us to do this and that Ostbergerdescribes 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 aspectwith just sufficient mathematics so as not to discourage a wider audience1.

Let’s look at the surface only of a spherical geometry like the one shown inFigure 5-4. We may colour theelementsred, green and blue.

Figure 7-1. A World point

The line elements in this sketch have no thickness and so cannot easily be said to be orthogonal. To describe thissurface 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.

TheFigure 7-1needs four coordinate numbers to describe, uniquely all the possible points. The question is“ar ethey orthogonal coordinates?”.

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Figure 7-2. Cylindrical elements

Look more closely at the junction of the line elements,Figure 7-2. The three coloured elements meet at a point2

on the surface. If they are cylindrical elements they look likeFigure 7-2. If they are annular elements they looklikeFigure 7-3. In each case there are two small sets of orthogonal coordinates formed by the meeting. One set isformedby thenormalsto the elements and the other by thebi-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 betweentheannular and cylindrical sets. But with the coloured elements present we can see that inFigure 7-2the thicknessof the elements are in the direction of the bi-normal whereas inFigure 7-3it is in the direction of the normal.

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Figure 7-4. Measuring the directions of the surface

Let us take a trip over the surface with an orthogonal indicator,Figure 7-4. A little practice and the student soondiscovers that the surface is rather strange. for one might expect that the indicator would rotate about its own axisas it is cast around the surface. In fact, what happens is that the red, green and blue directions of the indicatorremain in the same direction all the time. Wherever we go over the surface the indicator faces the same way. Tryit 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 ofChapter 2. There we saw that there were four possible spaces in the onequadrant.Each plane was defined by a 90 degree orientation of the pen until the pen was back home after twelvemoves 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 fouroutward. These spaces are the quadrants of the world. There are eight quadrants on the world which correspond tothe eight spaces of the pen stand. The permutation of the directions is4 x 3 x 2 = 24 which is 3 ways on each of8 quadrants. The three ways are toward each pole of the world from theW0 point. TheW0 point is that one whichsubtends 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 atall. It still lies in a fixed position as if constrained in a field like a compass3. The answer lies in supplying thesurface elements of the world with some directional information. This will then produce a result that resemblesthe parametric representation ofFigure 6-14but in three lots of two dimensions known as the Hessian form4. Theway of identifying each of the eight quadrants is by rotating the world surface and observing the indicator changeits spacial orientation. The juxtaposition of the three colours red, blue and green are different in each quadrantwhen the rotation is made (for exampleFigure 7-5). These eight quadrants correspond to the eight quadrants ofthepen stand. Four with the outward pointing pen and four with the inward pointing pen.

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Figure 7-5. Numbered octants

There are several ways of defining the magnitudes and directions belonging to the world surface. Each has its usesbut some are special5.

The curvatures here are described by the Gaussian curvatures6. The three coloured elements have curvature whichare 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 becurved. These will turn out to represent the geodesic nulls of tensor calculus and the null identities associatedwith 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, therotation of the pen. In the world surface we can also see a magnitudinal component of the fourth element. Look attheFigure 7-4. Consider the motion of the world as it enlarges; and there is only one, non trivial, transformationof this geometry, enlargement. As the enlargement takes place the orthogonal indicator at theW0 point wouldremain in the same juxtaposition if the pole elements were straight (inset inFigure 7-4). However they are not.They are curved. The orthogonal indicator rotates with the curvature of these polar elements. It is aninternalrotation of the orthogonal indicator at theW0 point.

There is another rotation taking place as the enlargement proceeds. This is to be seen in the Ostberger sketch inthe lower left ofFigure 5-2. As the world expands the tangent and the normal to the surface rotates albeit thattheplane of rotation is a geodesic plane8 and not one of the planes of the coloured elements. This is an externalrotation.

So what is the total rotation of the world geometry in this fourth dimension? Ostberger says that in proceedingfrom the first (unit) circular element to the last curved element before infinity the rotation is almostπ/2. It consistsof almostπ/4 for the internal rotation and almostπ/4 for the external rotation. Is it orthogonal?

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

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7.3. Quarter pointsThereare certain places in the geometries that seem to give better answers than others. The quarter points arethose that are 45 degrees from the measuring elements. The most important quarter point is theW0 point as thisunderlies the basis of the measurement of the whole space9. ConsiderFigure 7-6. If this is representing a vectorspacethen 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 observerview-point B”, and the other at the infinite element on the same axis, generally called thez axis.

If this is representing a vector space then the upper geometry relates to a stationary observer watching the quarterpoints moving out into space as we move at constant speed. They track a straight line as inFigure 7-6. If, however,theobserver-point is moving (which is an accelerating observer) the quarter points rotate and the track is a curvedline. Asymptotically they rotate almostπ/4 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, whichincludes a human being, travels into space at ever greater speeds his physical character changes. The old interpretationsays that he contracts in the direction of his travel.

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Figure 7-6. Rotations associated with the Lorentz tranformation

The new interpretation says that he does not; his physicality does not change. What does change is the measurementof our observation. We record correctly that we see a contraction of the parallel length because we see through oureyeball into our head that he is moving out of our local observational region where everything is measured equally in alldirections.

What the journeying human being is doing is to join in with one of nature’s laws, the law of velocities11, for velocity is apart of nature. We have no copyright on it. It belongs to nature. If we join in with her we must accept her rules and travelon a curve into space. Her rules say that the traveller appears to contract. That is only an observation not a reality. It is

54


an observation and therefore imaginary, not real.

Thejourneying human being observes us the same. He sees a contraction because velocities are relative and he does notknow which of us is moving or stationary. Neither do we.

Another effect of the Lorentz transformations is the slowing of clocks. The new interpretation says that two clocks sepa-rated by a velocity do not change their time, but we must take account of the fact that we will observe and measure theother clock as being slower by the amount of the Lorentz ratio.

Another effect is the change of wavelength of light that approaches us from the universe. The lengthening of the wave-length means that we observe light to be shifted towards the red end of the spectrum. The old interpretation of this is thata Doppler effect is taking place and that the further away we observe the more red shift there is and therefore the fasterthe objects are travelling. This leads to the Hubble constant which says that our universe is expanding at a phenomenalrate. That we are on the surface of a spotted balloon and the farther out we see the bigger the balloon. The bigger theballoon 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 identifies the Lorentz Transformation with a rotation in Minkowski space i.e. the space of my worldgeometry. The form of representation is independent of the particular directions in which the chosen set of orthogonalaxes happens to point, so the Minkowski representation ensures that the equations of relativistic mechanics is independentof those in which the Lorentz observer is involved. All equations moving with constant relative velocity thus use equationsof 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 dimensionalMinkowski world rotation that I have described. This rotation is real.12 It is a part of nature and we cannot observe ouruniverse as if we belonged to a world outside of it. We belong to this universe and so the rotation must be taken intoaccount. The red shift is a measure of this rotation. As we view our universe we are looking round a corner. The greaterthe red shift the more we see round the corner. At the speed of light we see almost 90 degrees round13.

Looking out on our universe in radially straight lines is analogous to observing the Earth as if it were flat. Those dayssoon 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 weobserve are related to the different curvatures of our observations?14

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

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Figure 7-7. Rotating the World

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

Figure 7-8. Curves in the planes of“straight” nulls

To understand the transformation that takes us from intravariant to extravariant (which we will see again later intheLaw Worlds) we need to stand at a position somewhere outside the world surface. What we see is the externalelements which are orthogonal to the surface extending out into the space around the World (Figure 7-8). Butdo these curves belong to a common surface? If they are the curves belonging toFigure 7-9then they are notin common because they are parallel to the sames planes from which they eminate. However, if they are curvesbelonging toFigure 7-10they are part of a common surface because they areorthogonalto the planes from whichthey eminate. The circles in the illustrations attempt to show this. These curves come from a geometry in whichthe pole elements are curved. The three external curved elements now rotate as the World enlarges (seeFigure 5-2showing a section through the enlarging World.)

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Figure 7-9. Curves not in orthogonal planes

Figure 7-10. Curves in the moving 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 whichall of the imaginary region that envelopes the outside of the one I am looking at15 will pass over my head as theworld enlarges. If I can find a way to stand on the outside of this new world I will see a world with an imaginaryregion inside which is enveloped by a real region outside.16”

He did not work out that these two world were orthogonal for many years. He simply thought that they were insideout to each other.

It is difficult to show this transformation on a single piece of paper. There are six pages in the notes.

Notes1. [Note1120],[note1121]and[note1122], amongst others, contain the mathematics of the geometry.

2. With the discovery of the extravariant geometry that assembles back on to its counterpart the intravariantgeometry(in Chapter 10) the need forpointsin a curvature space completely disappears.

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3. I have written it this way to show this as an example of extravariance. We do not see the world existing becauseour indicator shows no change at all when passed over the surface, yet we may sense its presence. But whenshown in this form it is clear that the world exists and possesses the 3 dimensional properties mentioned.

4. The Hessian form is[note409].

5. [Note250]

6. [Note1021]

7. For exampleDiv Div u is identically zero.

8. The geodesic plane is the one that would pass through the centre of the world if it had one. Itdoesnot havea centre in the conventional sense because the pole elements are curves. In fact it will pass through a limitpoint. 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 accords with mathematics.

10. [Note1110]and[note1111].

11. Velocity is one of the elements of a Law Field. SeeChapter 10.

12. It is real in an intravariant space and imaginary in an extravariant space.

13. SeeAppendix E.

14. Ostberger makes notes of this. He reverses the roles of the intra and extra worlds in a geometry of the secondkind 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 development of Law Fields(rank two vectors)

8.1. Law FieldsTAO

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 LawFields just as Ostberger constructed them and then show how they work.

There are two ways of seeing the process. One is the up-view, which is to show by mathematical reasoning thatthe Law Field diagram works and the other is the down-view, which is to show that the results produced aresuccessful. 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) inFigure 8-2. The second the general case inFigure 8-1which will serve as the pattern for further examples.

Figure 8-1. The generalised Law Field

You need to recall that straight lines were extremely special. They are so special that they do not exist in therepresentationof magnitudes. Only the directional component of a vector can be represented by the straight line1.Even then, the straight line is so special that it represents the exception to the rule. It turns out that the exceptionis the rule!

8.2. The Newton Law FieldConsider Newton’s first Law,“Every body continues in its state of rest or motion unless a force is applied tochange 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 line2. And thatis how Ostberger represented it. It later turned out that this same straight line was in fact the gravitational potential3

of physics. He labelled it DirectionZg 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 theradius or the curvature4 (which is the reciprocal of the radius). But measuring along the line of the potential yieldsnothing. It is this nothingness that makes the phenomena a candidate for being described by the idea ofabsorption(Figure 8-2). Absorption takes the total representation of both the positive and negative directions and says thatthey must always be zero. This is described later inChapter 9in several different ways including a matrix method.

Figure 8-2. The Newton 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 ofreciprocityor Inversion. It took Ostberger many years to realise that recip-rocal velocity was an admissible idea. He admits that he included it in the first instance because all the other LawFields contained the same pattern and not because he deduced it. But later he was able to show that it is the correctdeduction.

We must bear in mind that there seems to be two kinds of velocity. One belonging to the World of Fluidics andthis one. Here the velocity creates a thing we call momentum and that gives us the clue to the reciprocal nature ofvelocity. In the World of Fluidics the velocity is internal and may have a different property5 to this one. This oneimplies that,“to every momentum there is a contra-momentum6”.

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 acontra for the energy in the Neutron. If engineers and designers really understood this law we would not have thewasteful engine designs of today7.

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The third element of the Field is evident from the Newton’s third law,“to every force there is an equal andopposite reaction”.

Newton forces are quite real. They can be measured and quantified by the real effects upon our bodies and ourinstruments. 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 hardupon the soles of his feet. We cannot get at the reaction and say“here it is”. Its measurement is an assumptionfrom 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 originUnlike 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 arrives at the numbers one and its inverse.

3. The Force arrives at the origin in a singularity. A real singularity and an imaginary one each the reflection oftheother.

The Law Field has the special characteristic that the numbers0, 1 and a singularity (near infinity) arrive orthogo-nally at the origin. And that is very special indeed.

8.4. The relationsAt 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 leads to a velocity.

2. A rate of change of a velocity with 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 seemto 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(seeAppendix F).

I need to remind the reader that there are no Magnitudes in the Law Field. It has no size. Neither will there bemagnitudes in he Law Worlds later. It is not until we begin to use the process that we need to apply the magnitudesin 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 FieldThere is a general pattern to these Law Fields which will become self evident as we progress. In the magneticLF8 there is an intensity of magnetic origin which we designateH which exhibits a forceFm (Figure 8-3). Thereis also a magnetic potential which has the same absorption properties as the potential in the gravitic LF. There isalso a flux, we designateB, which has properties which are isomorphic tovelocityin the gravitic LF. There is alsoa potential, one of a magnetic kind.

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Figure 8-3. The magnetic Law Field

This Field is on a different level to the gravitic one. It is a field offirstness9. The Newton Field is one ofthirdness.

8.6. The electric Law FieldThe field of secondness is the electric Law Field. It is shown inFigure 8-4. All its properties are well establishedin the literature.

Figure 8-4. The electric Law Field

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This is a field of action. We cannot grasp the secondness fields. They are the operators; the players in the game.Thetransformers, 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 performancewill rest.

Like all the other fields they have rates of change and enlargements which are not shown in the LF diagram butcan be worked in the form of the Law World where the curvatures become evident.

We will assemble these Law Fields into Law Worlds inChapter 10. In the meantime let’s look at the general LawFieldof Figure 8-5.

8.7. The general Law FieldRemember that in these studies the Directions of the things that we study are just as important as the size orMagnitude. This is the case in reality. For example, suppose my friend and I were going to London and we wantto 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 unlikelyto do though, in our present way of living, is to consider the map reading skills of each other. We would assumethat we both knew the directions. But what if we did not know the direction or we did not have a map or a mapdid not exist? Would our assessments of each other’s transport be of any value? If one of us did not know thedirection to London then the bet has a high probability of being won by the other person. The direction becomesthe 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 wherethey all are. Indeed it is doubtful whether any of them are recognised as directional phenomena at all, and yet it isclear from our betting game that the Directions of nature may be more important than her magnitudes. So too forthe directions of our social activity, perhaps.

The Law Field acts as a guiding tool. Ostberger demonstrates its use in many examples, the most significant ofwhich are the representations of the Hydrogen10 and Helium atoms. It brings together in a single diagram threeessential principles. Each principle is represented by a line element orthogonally. Each has a contra-principlerepresented bi-orthogonally (opposite).

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

The origin of the LF has six magnitudes depending upon the direction of approach. Along the Absorption11 axistheorigin is approached in 0+ or 0- (Figure 8-5c). Along the Reciprocity or Inversion axis the origin is approachedin 1 or 1/1 (=1) and along the Conversion or Real/Imaginary axis the origin is approached in a real or imaginarysingular point12.

I will explain these inChapter 9. It is to be understood that theseaxisprinciplesare very much the starters, theprologue to the drama. We must find a more exact basis on which to tease out a solution to a problem or to makea representation. Thus in Quantum Mechanics we find the Unitary transformations13 which fit the LF and whichcan 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 inFigure 8-5a is not to be confused with the Cartesian representation which does not

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appear in the geometries at all except to make the connection between the current mathematics and the Ostbergerprocess.When mathematics, in the sense of present-day equational mathematics, is used with the geometries thenthe Law Field diagram has to be transformed (rotated for example) in the space of the room to permit the ordinaryEuclidian space to be related to the Law Space. In doing so, it becomes apparent that some of the Law spacecannot be measured (represented) because we have run out of dimension (Einsteinian). Another way to see this isto understand that the Law World fully occupies the space of 4 dimensions. There is no room for any more lineelements.

Any attempt to occupy the space with any other dimension such as a Euclidean line element extinguishes theavailability of some other element in the geometry. This makes it unavailable to sight or size. Measurementcannot be made. This is Heisenberg’s principle.

8.8. Continuity and discontinuityMany of the phenomena we see around us are divided into Continuity and Discontinuity. We humans are examplesof 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 areseparate bodies, we live separate lives with laws of discreteness14.

Continuity and Discontinuity vie with one and the other in life15. So also in the geometry as we shall see inthe later chapters of this book where extravariant Worlds anti-commute with intravariant ones. Here in the LF(Figure 8-5b) one axis is always continuous and another discontinuous. There is a third which is the real and theimaginaryof which we are all consciously aware. There is another which is seen in the 4th dimension: the processof Condensation which converts the imaginary from outside to inside and begins a new process of discovery.

8.9. MatricesWith the discovery and use of the Absorbing matrices Ostberger puts the remaining kinds of matrix into the LFform.

In mathematics the most commonly used matrix is the reciprocal or inversion kind. These are used for solvingmultiples of linear equations, for example. The absorbing matrices are new and are explained inChapter 9and[note100]. The Conversion matrices are few; they contain both real and imaginary elements in the same matrix.OnCondensation Matrices Ostberger writes,“I cannot see how these will work particularly as we are not at easewith 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 numberThese are the Numbers that make the things we call Magnitudes, the sizes of things, the bigness or smallness, theamplitude or the length. These are the concepts with which we measure our universe, that lend uniqueness to thecurvatures of space whether that be a Gibb’s ensemble in thermodynamics or a Loan space in Finance.

This kind of Number is calledOrdinal. They describe the magnitudes16. There is another kind of number calledCardinal. They tell us, not about the bigness of things, but about their state of order. They are labels.

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

1. Integer number are wholesome. They cannot be broken down into parts. In a binary system there are onlytwo. 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 otherto the negative rationals.

3. The point at the origin is no point at all. It turns out to be a world of number.

4. The set of irrational numbers is special because there is a contra set which are reciprocal. They are contin-uousin their nature. Our decimal point expresses the nature of this reciprocity.

5. Imaginary numbers are all prefixed with the square root of minus unity. Although they areimaginary theyare 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 spadefrom a pack of cards (13/52nds). They label the cars on a race track and specify their winning order but they saynothing about their size. These cardinal numbers belong to the extravariant number World17.

I am not going to describe these here. The notes give a description and definition of the fields. No doubt they willcause discussion and maybe dissension but that is what Ostberger is about; causing debate and discourse. Whatwe have here are the sets of number that are well recorded in the literature.

• N, the set positive integers

• Z, the set of integers

• Q-, the set of negative rational numbers

• Q+, the set of positive rational numbers

• R , the set of real numbers

• C, the set of complex numbers

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• 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 isomorphicpatterns 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 operations Law Field (secondness)

In one of his early notes Ostberger exposes his concern at educating children into believing that all operationscommute,thatAxB = BxA except in higher mathematics; when in fact the case is completely the reverse. Onlywhen we are using pure numbers does the commutative law work. In all other casesAxB is not the same asBxA. Thus 3 bags of four apples is not 4 bags of three apples. The bags may be as important as the apples. Yetwe are teaching children to think solely of the magnitudes and launching them into a yang world without a yinperspective. 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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Figure 8-8. The interpretation of number Law Field (thirdness)

The third LF isFigure 8-8. This is the field with which we interpret number. Thus we lay them out in indicialorderusing the addition operator in a base of 10, 2,345 is2x103 + 3x102 + 4x101 + 5x100.

We could use any base. For computing we us a binary system with base 2. At school we learn about bases, numberrings and clock numbers. These appear on the Law World which we will see inChapter 10.

The Imaginary halves of these three Law Fields are worth mentioning. In the first field there is a whole regionof the complex numbers. In the second field the imaginary region belongs to the operators which are themselvesimaginary such as is used in the study of electric currents wherej is the usual symbol. In the third Field there is aquestion mark. What are the imaginary bases of number?

8.11. The Law Fields of thermodynamicsI can remember spending many hours as a student plotting engine diagrams. I later saw these in practice in theaircraft industry where the performance of turbines and jets were displayed on three dimensional diagrams. Iwondered 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 levels18 and had curvatureshelped me to understand why so many thermodynamic devices deviate from the theory as the parameters ofvolume, pressure, frequency and temperature become extreme. The distortions are an inevitable part of the naturalcurvatures belonging to this World.

I am now going to show in a couple of pages the essence of Ostberger’s work in Thermodynamics which tookhim 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 something very final, something that would remain at the foundations of future educa-tion 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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I despaired that I would ever get it down on paper. Only with the advent of the computers did my happiest realisationscometo 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 earlier in this book. Volumes do not move discretely; theymove 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 World as fluxes belonging to threedifferent generations.

• Dalton talked of pressure probabilities. The react to the real pressures in a statistical way.

• The fourth field her is Enthalpy.

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

Volumes remaining and consumed Absorption

Pressures real and probable Conversion

Temperature 1 of the first kind and its reciprocal Inversion

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

Table 8-2. The Organisation Field (secondness)

Entropy internal and external Absorption

Internal energy U and its statistical counterpart? Conversion

Absolute (Kelvin) Temperatureof the second kind and its reciprocal Inversion

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Figure 8-11. The reation Law Field (thermodynamic thirdness)

Table 8-3. The Reaction Field (thirdness)

Mass fraction participating and remaining Absorption

Chemical Potential and its likely reaction Conversion

Temperature of the third kind and its reciprocal Inversion

TAO

There were several aspects of the World that surprised me and it therefore took several years for me to settle on the factsas I had found them. Not the least was the fact that there were three different temperatures on three generic levels. Theyare each orthogonal and have a reciprocal which accords with Onsager’s “Reciprocity Theorem”. The pressure elementaccords with Dalton’s principles of probability pressures in so far as there is seemingly always a real pressure whichreacts 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 thepotential to work.

There are many questions to ask about these Law Fields and that is Ostberger’s intention; that question shouldbe asked and the subjects discussed. He saw his work as a foundation for discussion in the search for order, notas a fete a compli. The ultimate test is the application of the process. For that reason he chose to represent theHydrogen atom. But even in that he says,“... this is not proof that the Law Worlds are all in order. Only time andmore 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 areperhaps all of the character of potentials. That is if we regard Pressure and Internal Energy as a potential. Thissuggests that the World of Fluidics may have elements of a potential character as possessing the property ofInversion; 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 suggeststhat 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 shouldtry to use them as a basis for our mental state of order regardless of whether they are perfect or not. Indeed weneed to ask even more serious questions about our approach to the universe. The end of the twentieth centuryhas seen the meeting of ideas about life from the East and the West. This seems to be characterised by either anintravariant or extravariant view of life.

In the intravariant view we see our ourselves as the masters and owners of nature, controlling new developmentsand 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, for without it control of an opponent is allthe more difficult. It follows that our domination of nature is meant to achieve not certain natural outcomes, butcertain societal outcomes.”

In the extravariant view we see ourselves as a part of the universe and belonging to it; as servants of the universehelping 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 differenceswithin itself. The unique and surprising are not suppressed in some persons for the strength of others. The geniusin you stimulates the genius in me20.

Notes1. SeeAppendix A.

2. There are no straight lines in space because there are always forces.

3. We understandPotential in terms of the effect it has on the Newton Force. The gradient of the gravitationalpotential is the gravitation force (seeAppendix F).

4. The measurement of this element is not simply through its single curvature as a law line. The law line appearsto us in all three dimensions. The proper measure is vector function and Gaussian curvature by the vectorgradient (seeAppendix F).

5. The notes relating to the World of Fluidics are not well developed, but it is clear that a velocity associated withfluids exists and that it has an independent nature to the gravitational kind. The former is internal whilst thelatter is external. That is to say that the velocity that we associate with gravity makes the whole mass moveand 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 areevident. (Could it be that we can account for the slowing of e-m radiation by the existence of these two kindsof velocity?)

6. There are copious notes about the application of this Law. The engineering designer will think very differ-ently when this Law is borne in mind. It makes the current reciprocating engine into an antique device. Italso makes the current rail transportation seem primitive. In the[note9xx]series notes several Carnot cycleenginesand Sterling engines are suggested which use this principle. The Sterling engines also make use theThermodynamic ’lasing’ principle which appear in the theory.

7. The[note9xx]series notes describe designs which make use of the contra-momentum principle.

8. [Note1702]

9. Other than the Taoists I found only one philosopher whose work accords with these concepts. Charles Saun-dersPierce (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 1891. Pierce introduced the ideas of firstness, secondness andthirdness 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 theconception of being or existing independent of anything else. Second is the conception of being relative to, theconception of reaction with, something else. Third is the conception of mediation whereby a first and secondare brought into mediation.”

10. The representation of the hydrogen atom is explained in a separate document of some 60 pages. The Som-merfeldfine-structure constant is derived from a special geometry in[note665]and [note666]and shown,summarised,in Chapter 12. The Delta values which seem to be useful in scaling the atomic elements aregiven inAppendix J.

11. These principles are discussed and quantified inChapter 9.

12. Shown in theFigure 8-5as + or - infinity, the different types of singularity are the subject of a note XXXXwhichone? XXX.

13. For example, A = A-1, where A is a square matrix.

14. Accounting is a process of discreteness. The law endeavours to resolve civil dispute with discrete Acts andRulesof procedure. The[note20xx]series of notes are about the use of the Law Fields in social activities. Ithasbecome evident to many people that we are not an animal that lives easily alone. Like necrosis in the bodycells something of us dies when we live alone. When we live together we feel a greater warmth and securitywhich 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. SeeChapter 10, also[note257]and[note258].

18. Generic levels, meaning firstness, secondness and thirdness.

19. The notes on Fluidics are[note1750], etc. The World of Fluidics is given inChapter 10.

20. [Carse87]

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Chapter 9. The omnipotent laws

In this chapter I am going to briefly examine the magnitudinal aspects of the Law Fields. We will look at theirrelationship to well established mathematical procedures. The first example is not well established. It is new.

TAO

It comes from my personal experience in accounting. I realised that I was able to express the whole edifice of finance interms of a new and simple kind of matrix which I called “Absorption matrices”. They are easily missed by academicswho have not spent years applying their subject in a real commercial environment.

9.1. OppositesThere were certain geometries which Ostberger studied that disappeared when assembled. They could disappearby virtue of having opposing directions or by there opposite magnitudes, but the fact remained that they mathe-matically disappeared. This was primarily due to the fact that the representations had both positive and negativeattributes in the same space. They overlapped in opposite directions. A conventional Cartesian graphical repre-sentation 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 geometrically1 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 outof space and leave even more behind. Well, that is exactly what the geometry does. When a part is taken away thepart that was previouslyabsorbedre-appears.

I doubt that there is a philosopher who has not made good use of the concept of opposites; sages and prophets too. Thereare so many examples and so many philosophers, sages and prophets that I would scarcely have the pages to write them.

In Taoism opposites merge to form the Tao. In Buddhism too the Koans are given to the students by the masters tohelp them understand the two opposing concepts thoroughly. The more the student uses his experience to merge the twoconcepts the more he understands a world in which opposites play a common part. Opposite ideas can be merged intonothing forming the structure of nothingness.

This is my experience too. Not only does the physics of nothingness (the void or vacuum) have a structurewhich is not yet revealed but the reverse process applies to our social activities and we can form structure outof nothingness. A simple example is the rise of Local Exchange Trading Systems2 in which money is createdfrom nothing. Before the system begins there are no Brights in Brighton or Trugs in Lewes. But a year later thereare thousands of monetary units being used by the people.

9.2. AbsorptionWhat Ostberger did was to incorporate the principle of opposites into the geometry. But first he expressed it in theform of new matrices3. He called theseAbsorptionmatrices and the spaces absorption spaces. It was these spacesthathe found were disappearing.

Let us look briefly at these matrices.

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Figure 9-1. An absorption matrix

Figure 9-1is a simple absorption matrix. It has the property that every line horizontally and vertically aggregatesto zero. It is a zero matrix. Its Identity is zero. We do not have to add or subtract the numbers, we merely have totake 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 handleones with hundreds or thousands of rows and columns. But that does not make the process any the less under-standable.

Figure 9-2. An associative matrix with missing 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 withoutdestroying its structure.Figure 9-2is the matrix ofFigure 9-1yet it can be determined completely. Notice alsothat the magnitude in each quadrant is thedeterminantin 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 fourquadrant magnitudes are equal.Figure 9-3is 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 elsewhere4.

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Figure 9-3. An associative matrix with a quadrant magnitude (determinant) of 10

These seemingly trivial matrices are extremely useful. An academic person who has not lived in the world ofcommercewould have missed the point of them entirely. Indeed because they are empty sets they do not appearon the mathematical scene at all. Ostberger called these“Matrices of the first kind”. The matrices of the secondkind 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 ofFigure 9-1means could express the bathing of babies,

4 volumes is the first baby

6 volumes is the second baby

2 volumes 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 absorption bottle5

The Archimedian principle is quite fundamental to our learning process yet we are stumbling over this truth andpassingon. The displacement of water in a jar (Figure 9-4) is the exact counterpart of double entry accounting. Iftheair is treated as credits and the water as debits then the movement of the surface gives the balance. For eachvolume of air that is changed so the same volume of water is changed in reverse. Except for compression (whichis 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 wefind 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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In accounting the basic tenet is that,“No credit exists without its corresponding debit”, and this is somethingthat the physicist must learn for it applies equally to the directions of space and all of those phenomena that areDirectional 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 therope-maker found.

When the student taoist was ask to consider the sound of one hand clapping he was asked to discover these simplefacts so that they may become the foundation of his understanding. We do well to heed his experience.

Figure 9-5. The absorption identity matrix

The identity for the absorption matrix is given inFigure 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 minusare very real entities. They appear in directional studies quite frequently. They also appear in computer systemsand accounting. What happens is this. As a magnitude dies to0+ its Direction changes from plus to minus andthe magnitude is born again at0−. The pole or zero point of a magnitude is an indication that the direction ischanging 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. En-tropy appears in the World of thermodynamics inChapter 10. It is the state if disorganisation of the thermodynamicsystem.Its counterpart is negative Entropy,S−i , which is the state of organisation of the system. It is clear thatin any organisable system the boundary between organisation and disorganisation moves in an absorptive way. Achange towards organisation is the same change away from disorganisation, itdouble enters. But it does so in adifferential manner and so such a matrix needs to be expressed asdS+

e with dS−i .

Another guise is the volumeV , with which we measure space in thermodynamics. We may divide any thermo-dynamic system into two parts. The Volume Occupied and the Volume Remaining. These are absorptive and areexpressed in terms ofdV + with dV −.

In physics the Ostberger Worlds show three kinds of Direction; Magnetic Direction, Electric Direction and Gravi-tational Direction. All possess the property of absorption, yet they are all of a different character because they areon different levels. It turns out that they are thepotentialsof our mathematics.

In accounting the two zeros that occur in the system are of paramount importance. InAppendix Cis an actualexperience in which the outcome of a discordant exchange between two companies over a tool that was missingwas 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 themasters to help the student understand the two concepts that are born out of nothing.“Listen to the sound of onehand clapping”, says the Master,“Feel the space in the pool when the foot is withdrawn”. These are the teachingsof wisdom. Teaching the student to see two sides to every event, just as the accountant must and, in the future, thephysicist must too.

Absorption does not just apply to the nuts and bolts of a system, it also applies to the operators as well. Itapplies 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 9-6. The Law Fields of Matrices

There may be four Kinds of matrices.

1. Absorption Matrices. New but simple and useful matrices with meaning to their rows and columns as de-scribedhere and further in the notes3.

2. Reciprocal matrices. The ones we use to solve linear simultaneous equations and which form groups theoryatuniversity. They are well established in the literature.

3. Conversion matrices. The upper triangle is imaginary whilst the lower is real. The Pauli matrix is one exam-ple.

4. Condensation matrices. A new kind of matrix which encompasses normalisations. There are no notes ofthese.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. Thenthese are the groups of Ordered matrices.

9.3. ReciprocityThe second Direction of the Law Field is divided into two parts by the number one. In fields of greater complexitythis 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 philo-sophically deep idea as I had no experience on which to hang my understanding. There are cardinal numbers andordinal numbers. Cardinal numbers label things. They provide a means of putting things into a state of order. Thisleads to the ideas of probability, chance and statistics.

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Figure 9-7. Cardinal and Ordinal numbers

Ordinal numbers, on the other hand are those that give magnitude or size to things. These two aspects of numberarenot to be found in a single World of number. The cardinal numbers belong to an extravariant World whereasthe ordinal numbers belong to an intravariant World. Such a big idea could not possibly be conceived by myinexperienced mind at primary school. Yet it is not long before we can appreciate the difference of these ideas anduse 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 identitywe say that it must have a reciprocal. Even more than this we say that every element of the group must have thesame identity. Thus the rational numbersab suchas 2

3 , 34 , 4

5 and 67 have reciprocals such as32 , 4

3 , 54 and 7

6 andtheidentity to the group is the number1.

What we probably fail to observe is that the number1, in this instance, is not an ordinal number but a cardinalnumber. The cardinal number1 identifies the group and allows it to be almost infinite in both the number ofmembers and their magnitude, both of which are ordinal numbers. This is explained by the application of a Worldgeometry 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 ofthe aimless. We are all different and we all seek different identities. We are members of the golf club, the physicsInstitute, the liberal party or Greenpeace. And each of these has aims and objectives and vector themselves towardthem. It is these aims, which are incorporated into the constitution of the group, that determines the eigenvectorof 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.

Oneof the fields of human activity that uses the reciprocal relation is that of Agreements. Every day of our life wemake these things we call Agreements, but do we stop to think objectively about them? What are they? How dothey occur? Do the different types of agreements fall into any kind of pattern? The more we ask, the more there isto know.

The law deals with agreements and breaks them down into their natural groups. It deals with the making andbreaking of agreements. It deals with how they should operate and how they should be interpreted. What we allowand what we forbid is a part of the eigen-vectored journey of the human race8.

The principle of Reciprocity litters the technical literature. There is“Onsager’s reciprocity” which describes thetemperature process in Thermodynamics. In the Law Fields this is interpreted as the special property that we canattach to Thermodynamic Direction which we call Temperature. There are three different types of temperature atthree generic levels9.

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There is also: Electrical network reciprocity, various mathematical reciprocity Theorems, the reciprocity region forneutronenergy, the law of reciprocity in photography, reciprocity in acoustics, the Maxwell and Betti reciprocitytheorem, 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 aroundus. It must also be in our heads.

9.4. ConversionWithout imagination neither we nor any other part of nature would exist. Even the smallest part of nature thatwe understand, the hydrogen atom, has its imaginary parts and the mathematics of Quantum Mechanics whichdescribes it is itself imaginary. Indeed, in mathematics there is probably as many theorems about the imaginaryas there are about the real.

There are many amongst we humans for whom the idea of something imaginary has no meaning and does notform a part of life. Yet again there are more for whom the imaginary is a God outside of themselves and cannotbe reached except in death. But there are a rising few who have a god inside of their being which forms an equalpart 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 anactivity that has gone on for thousands of years and looks set to continue for another thousand. Such people arethe salt of the earth. Yet these people must live a part of their lives in the imaginary world of planning. They mustplan what they will take to market. The greengrocer must assess what he thinks he can sell according to what theweather might be or what the people might want or what the telly might advertise. What is going on in his mindis 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. Thewhole company is involved in planning the future. Everybody is guessing what might happen next year. It is allpie in the sky.

Every business spends a great deal of money preparing for the future and when it is all done the accountantwill translate it into a Sales and Profit Forecast10. The accountant will not call the forecast moneyimaginary;instead he will call itnotional money and as the year passes he will offer a tracking system that compares thenotional forecast with the actual performance in hisvariance accounting. But it is all imaginary and the processis conversion.

We have seen that one of the best known examples of conversion is in the third law of Newton,“to every forcethere 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 isaccompanied byrisk. Risk is purely notional yet it is considered essential to spend considerable amounts of realmoney assessing the risk of a project. Risk analysis is a very serious study and demands a person of considerableexperience, 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 thatthe geometry is complex. The element of Risk is theeigen-linewhich draws together all the resultant risk vectorsthat could be assembled into the space which is the whole of the imaginary half of the diagram.

Notes1. Ostberger considered that nature contracted herself by this process. In the World of the second kind this is

exactly what happens.

2. LET-Systems, new money. Michael Linton and Angus Soutar realised that money could exchange hands with-out the use of banks. Individuals trade their skills by writing their own personal cheques in a local currency

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(Truggs in Lewes, England, and Brights in Brighton, England). An account is kept of the income and expen-ditureof individuals. The system has reached most parts of the planet.

3. [Note100]describes these matrices and their properties. They are shown to be the basis of accounting andusedfor a complete accounting system in the book entitled XXX“The Information System belonging to theManufacturing and Distribution Sector of an Economy”.

4. See the book entitled“The Information System belonging to the Manufacturing and Distribution Sector of anEconomy”.

5. See[note124]

6. An example of a World geometry of the second kind is given inChapter 11. The intravariant ordinal numbersarethe structural framework for three orientations of extravariant cardinal number. This means that the groupcan be identified by any cardinal number, its statistical operator or interpretation.

7. The[note20xx]series notes are about Social order. The[note23xx]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 Temperature Law World with the Agreement Law World gives an excellent insight intotheunderstanding of these three generic levels.

10. The[note23xx]series notes are about finance. The Law Fields of finance are isomorphic to those shown in thisbook.The book XXX “Systems in the Manufacturing and Distribution Sector”also shows the isomorphismbetween finance, internal data systems and mathematics.

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Chapter 10. Law Worlds of the first kind

10.1. Vector as picturesIt is now time to look at the constructions that Ostberger called World geometries after Minkowski1. But first aword 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 WestSussex and began studying geometry from a caravan. I simply felt the need to spill out all the pictures that were inside ofme. And I did. There seemed to be endless amounts of geometric study to do and never enough time to do them. The ballreally began to move fast when the ideas of curvatures took over my studies. I had suddenly abandoned the stricturesof my imprisoned education and launched myself into a new world. From then on I could see that I would never haveenough time or money to complete the gigantic task of studying all the Directional properties of all the mathematics thathad ever been done in history. Many tasks would have to be left undone, many gaps would have to be filled in by otherworkers in the field. I began to aim for the mathematics of the Hydrogen atom because that was the best known and mostexact application of mathematics. Riemann and Einstein would be companions on the way and Dirac would be the manI had to emulate directionally.

There is a poignant statement by P.A.M. Dirac2 which captured my imagination: “In the case of atomic phenomena nopicture can be expected to exist in the usual sense of the word picture, by which is meant a model functioning essentiallyalong classical lines. One may, however, extend the word picture to include any way of looking at the fundamental lawswhich makes their self-consistency obvious.”

A picture is exactly what I was aiming for. It would not be an ordinary picture, it would be a vector picture. I was goingto 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 whichnobody else had either. For I had already learned that vectors can produce some funny looking effects on paper that donot always follow our instincts. For example it was not my instinct to believe that a single point on space could havethree separate numbers attached to it or that the point would turn out to be no point at all but a kind of inside out Worldthat I was creating at the time. My Cartesian training had firmly implanted the idea that one point in space had onevalue.

Yet there was something left from my A-level days that haunted me. One question that had never been answered. Whenwe represent a vector in mathematics we draw a line, put a point for the beginning and an arrow for the ending and labelthe length as a representation of its magnitude. Then, when we want a negative vector we simply turn the vector round toface the other way. My question was, “why was a negative vector always facing the other way?” Why could I not havea vector whose magnitude changed from positive to negative. Why was it always the direction of the vector that had tochange?

I resolved to accept that I was right and that vectors could have negative magnitudes as well as positive ones. Thispresented me with a headache that I would rather had left behind. The solutions of mathematics were doubled. I wasfurther resolved to continue.

My headache would be compounded by the fact that I had accepted curvatures as the basis of my geometry giving aneven greater scope for sailing into uncharted waters. The pain was even worse with the realisation that the pen standsaid that there were four times as many points in a Euclidean space as I had been led to believe. But my resolve wasunattenuated.

Now in Quantum Mechanics of the type that Dirac constructed the Vectors are not just the stage hands they are the wholecast. 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 isthat vectors have both magnitude and directions. There are no vectors without both. Dirac calculated the probabilities,produced some excellent numerical results and even showed that a negative electron must exist. But, where, I ask, arethe 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 thestage? Or is it that the magnitudes come out to see us, leaving the directions in some kind of hiding place?

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If Quantum Mechanics was a vector study then both the numbers which issued from the probability equations and theirrespective Directions were available for me to find. So I set out to find them.

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

The basis of the Directional process is that we seek orthogonality3 in space. In particular we seek orthogonality ofpoints in space in the dimensions defined by the pen stand and later the World geometry. They are not always easyto find when a geometry is compounded into a new space. We must get used to orthogonal conditions appearingunexpectedly. It becomes very easy to make assumptions about the amount of a rotation in space; as historyhas proven. Historically we have ignored the internal rotational of a line element and thereby passed over theopportunity that Ostberger has revealed.

10.3. Ordinary spaceThe space of a Law World is not that of ordinary space. The Law World is embedded in ordinary space. So, thebest way to visualise it, in the beginning, is to imagine that the World can be rotated in the room in which it sitsas a model. The rotation in the room is then the ordinary Euclidian space. But, we need to remember that theroom-space is also 4 dimensional in the same sense as the World model. That is to say that the pen stand appliesto 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 andthe 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 thevelocity of a car as, say, 17 Kph. We then represent this on paper to a scale of 0.5 cm per Kph. There are twomagnitudes and two directions here:

• The velocity of 17 is a Law World Magnitude.

• The velocity is undefined in Direction. It simply is not present.

• The direction of the scale is relative and on the 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 otherDirections of the Universe which are part of the natural Law. It becomes a part of the physics around us. Its effectson other phenomena, such as the generation of a De Brogli wavelength, becomes evident.

10.4. The parallel principleWhen we represent a vector in a vector space any parallel vector having exactly the same magnitude is a repre-sentation 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 themagnitudesfrom the greatest definable curvature at the pole to a fixed value at the great circle of the same colourin the same plane. In the enlargements there are an infinity (less the last) of possible fixed values belonging tothe great circles. It is therefore possible to fit the curvature of this space with all the values of numbers from0through∞ to∞− 1, but they need to be of the same class of continuous number. The other coloured surfacesare then available to define the other classes of number; the discontinuous and the complex. There is only oneWorld geometry in a purely directional space and all possible curvatures are present somewhere. This is theproperty of the Riemann unit Spherical surface. Enlargement is the only operation available to us. Rotation isnot available because it must be along one of the curves already defined and therefore simply produces a parallelvector. Translation also produces a parallel vector because it too must be along one of the geodesic pole elementswhich is already defined. Thus it follows that if we can describe an atom using a geometric representation thenany identical atom is described by the same representation.

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

10.5. Direction onlyIn a social environment which teaches a predominantly yang view of life it will be difficult for most people tospring their magnitudinal jails and see the World geometry in terms of its Direction alone. Particularly in theWestern hemisphere where we are taught to believe that size is what matters. It is not so. In what follows thereare no magnitudes unless they specify themselves or we apply them in searching for solutions. They specifythemselves by producing natural numbers. The relation of2π in a circle is one example. It is a purely geometricconstant and entirely universal. So ise, the natural constant of growth. So too is the rope-maker’s constant4, thereciprocal of2π and the Sommerfeld fine-structure constant for Hydrogenα, as we will see inChapter 12.

The world geometries are constructions of Direction only. There are no magnitudes. Only when we begin toformulate a solution do we begin to see the numbers that specify the magnitudes appear in the shapes. Someelementary examples were given in earlier chapters, e.g.Figure 5-6.

TheLaw World geometries are no more an answer than the Schroedinger or Bernoulli equation. They provide thetool with which to approach an answer. A tool which lends guidance to an otherwise desperate situation. The trueunderstanding comes from their application. It is hard work despite its visual presence.

10.6. CurvaturesNature is as much about shape and form as it is about number and size. This is particularly evident in biologywhere function and form are often visually separable. She does not produce a single tree that is so big that it overshadows 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 wouldnotbe 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 curva-tures. 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 mathematicscreated by Riemann.

10.7. Assembling law fieldsThere is an excellent analogy of the way we currently use geometry in mathematics which relates it to the Ost-berger process.

Our current use of geometry is like laying out the components of a motor car in a long line. Each component issomething different and not recognisable as belonging to any particular car. There are a few, expert individualswho can relate the component to the the Volkswagen that it really is. The students and the rest of us simply learnthe 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 thatthey can be easily related to the car as a whole or in parts. Starting with the saleable product he identifies preciselyevery generation of every family of parts which go to make up the car. With the advent of computers he now alsoreverses the process and constructs the“Where used next level up”data which every buyer of spare parts for hiscar can see on the stockist’s computer. Every change of component by date and model is now tracked and madeavailable 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. Theyare just like the engineer’s“Where used next level up”generic map of the car.

Figure 10-2. Curving an element

TAO

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

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Figure 10-3. Early sketch joining 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 toB. The constantk is the productof the number of turns in the circuit and its area.

3. The static termsE.D andH.D are zero. Energy is dynamic.

In Figure 10-3the key feature was that a changing magnetic fluxB producedan electromotive forcee whichalways opposed it. So, in some way the elementE of electric force would oppose the fluxB. But E had to bean operator onB because E is in the secondness field. Which it is. This seemed to work and gave a space forthe Poynting vectorHxD and a volume for the energydω. But the block-like construction did not measure everypoint in space of the pen stand.

10.8. Minkowski’s worldsThe 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). These are the worlds of which Minkowski1, the greatteacher at the turn of the century, was talking about in his lectures.

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

Figure 10-4. The intravariant Gravelectromagnetic World

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

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

• This diagram is illustrative only. The external elements are 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 asonecould 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 sizeof 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 aEuclidean 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 threeForces. 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

Firstness a Direction possessing the property of Absorption

Secondness a Flux possessing the property of Reciprocity (Inversion)

Thirdness a Force possessing the property of Conversion

and that is the general pattern for every Law Field and Law World. There is a fourth field in each of the spaceswhich, 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 themanner 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; ayin process. In the other type the scales produce rings in the surface which belie the scale that is applied, the yangprocess. 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 propertiesof the extravariant World.

If all the elements of the intravariant world are transformed through an infinity (a singularity) the result is theextravariant World. The two worlds contain the same elements precisely. CompareFigure 10-4andFigure 10-5.

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Figure 10-5. The extravariant Gravelectromagnetic World

• The exterior region is inrealspace.

• The interior region is inimaginaryspace.

• This diagram is illustrative only because the external elements are not shown in their correct orthogonal ori-entation.

How much of a rotation in space is required to make this transformation? The answer is almostπ/2 of all theelements5, which is rather remarkable since the imaginary region has swapped from being external to being inter-nal which we would normally consider to be aπ transformation. Yet, looking at the two worlds we see that thesurface elements of the intravariant World (Figure 10-4) have become the external elements of the extravariantone(Figure 10-5).

Whatis 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 is2πapartfrom the spin down electron. Yet its spin vector direction suggests that it is onlyπ apart. The exact geometricmeasurement of this is given in the notes6.

In the World of the second kind the extravariant World is assembled back on to the intravariant World. The vectorsof the surface of the intravariant World are then anti-commuting with those of the extravariant World. The resultsof this are explained inChapter 12. There are six ways of making this assembly.

Theextravariant World exhibits the properties of stability7. 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 anexpression of the omnipotent ending of our understanding. This is not so. Another World emanates from the firstto 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 overcome thatsurprise and start to work in earnest as if it were so. But even more of a surprise came the fact that there was anotherfour dimensional geometry which could be constructed from exactly the same line elements as the first and yet be verydifferent in character. But the really remarkable thing was that the latter was derived from the first because that meantthat such a process could be continued indefinitely.

Four is suffice but it births itself twice.

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

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Figure 10-6. The intravariant Thermodynamic World

1. In the intravariant World the whole of the exterior region is imaginary which is indicated by the use ofbroken 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 adifferent octant produces a different directional relationship.

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

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

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Figure 10-7. The extravariant Thermodynamic World

In nature, however there is an extravariant process taking place. This is the World ofFigure 10-7. Like the otherextra-worlds it is a recipe for stability and when combined into a geometry of the second kind8 it acts as nature’sstabiliser. 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 fromthe 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 thefacts as I had found them. Not the least was the idea that there were three different generic levels of temperature; ortemperature-like phenomena. They are each orthogonal to each other and have a reciprocity in accord with Onsager’sReciprocity Theorem. I cannot predict what this infers in reality but that is what the theory says! The pressure elementaccords with Dalton’s principle of probability pressure in so far as there is seemingly always a real pressure which reactswith an imaginary one in any contained volume.

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

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Figure 10-8. The Extravariant Number World

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 World of intravariant Number

10.12. Other WorldsThereare other Worlds which are shown in the notebook.

• [Note1750]: The Fluid World

• [Note1001]: The Fields of Engineering System Dynamics

• [Note23xx]series: The World of Finance.

• [Note2333]: The World of Commercial Operations.

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

TheFluidic World is incomplete, but is important because it completes the set of three Worlds of physics whichare depicted as a sketch inFigure 10-12.

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

Although much of the fictional discussion about our physical universe is centred around the Grav-electromagneticWorlds we see here that the other two Worlds are of equal stature in the Universe. The Fluidic World of the secondkind, which stands in the place of thirdness in the physics Universe, which is shown at the top of this sketch givesrise 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 forthe bending moments and shear stresses of physically loaded structures. The Mohr circle occupies just one of theeighteen planes in the Fluidic sketch.

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

Yes, there is a great deal more work to do in the detail structure of these geometries. But just how much work canoneman do in one lifetime and still survive on the planet? I believe Ostberger has given us a very big leap forwardand it is something to grasp and apply to our life on this planet. The fact that he has been able to show that ourfinancial 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 discover the laws of physics. They arrived at my door like with the messenger. I had no reason not toreceive them and display the process as I discovered it. The messenger was not announced, he did not say “I have comefrom God” or “I have brought you the keys to the kingdom”. Nor did he say that he knew the contents of what he wascarrying. He didn’t know and neither did I. There was, however something interior to my being that gave me impetusand 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 noidea that it would take so long and cost so much. Yet I felt quite comfortable doing work which, maybe, no one else wasdoing or even would understand.

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Figure 10-12. Sketch of the Universe of Physics

Each of the three worlds in the sketch are what Ostberger called“of the second kind”. Each of these contains sixworlds of the first kind. Each world of the first kind contains six law fields. Each Law Field has seven degrees offreedom; the last of which is half in the next. Additional degrees of freedom appear as the geometries grow. AWorld of the first kind has twenty two degrees of freedom, the last two of which have one half of their freedomin the next world. In all, one World of the second kind has more than sixty degrees of freedom. This sketchwould have more than one hundred and eighty degrees of freedom. Every degree of freedom is orthogonal toevery other. A degree of freedom is one dimension of the Magnitudinal kind. Every group of line elements, LawFields, 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 extravari-antworlds mounted on to intravariant backgrounds. They can equally be assembled in reverse. One of the notessuggests that the intravariant world when mounted on the extravariant world is representative of the macroscopicUniverse. The one that we find invitingly curious.

Notes1. Hermann Minkowski (1864-1909), the Russian born Swiss-German number theorist, algebraist, analyst and

geometerwho developed the thory of four-dimensional space-time that laid the mathematical foundation forrelativity theory.

2. See[Dirac70].

3. The half quantum of Direction isπ/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 constant2/π is a magnitudinal constant. It is the chance of a XXX? certain stick landing between twoparallel lines. Alsoπ/2 is a directional constant. It is the quantum of direction.

5. See[note1170],[note1171],[note1172]and[note1173].

6. This is explained in[note1823]in terms of measuring the rotations that are needed across a 3-dimensionalsurface to get from an up-state to a down-state of the world. The up state electron lives on one side of a Worldgeometry of the second kind and the down state electron lives on the other (Figure 11-5). They are differentelectrons.

7. Stability is where Ostberger began. He examined the Nyquist Diagrams of engineering stability and wasamazedthat stability depended purely on the direction of rotation of a locus on a diagram. The reader shouldbe aware that when the World pictures are applied to subjects which are not“below” us but“above” us suchas the Galactic Universe then the role of stability seems to reverse between intra and extra Worlds. The sameseems to apply between subjects which are“within us” such as emotions and psychology and those outsideof us such as Social Justice and Finance. This is a particularly big concept.

8. See[note1713],[note1714],[note1715]and[note1716].

9. See the[note9xx]series notes.

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Chapter 11. Worlds of the second kind

11.1. Worlds of the second kindThe Ostberger story is now becoming more difficult to write into a book that sufficient people will want to readthat 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 theWorld surface as curves but rather as straight lines with near zero curvature. This would be satisfactory for theapplications to the first atomic elements where the curvatures are thus. But even Helium will require a smallamount 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 thegeometricforms that this process demands. Mathematical journals simply don’t do it. They certainly could notconceive of depicting the geometry to be the leading part of an article that contained a mathematical text. The textwould be regarded as the leading part. Also, the line elements should show their thicknesses in each of two planesso that their orthogonality can be ascertained but it is unrealistic to believe that such a picture could be read fromthe page. Such detail must be left in the text of the notes. But a computer can display such a picture. I hope thatthere is sufficient here to convince the reader of the usefulness of the process and that one, at least, will want toexamine the Hydrogen atom mathematics and its geometry.

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

Figure 11-2. Two kinds of particle statistics

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We may think that such a small consideration does not bear relevance to the kind of systems that the statisticsdescribe,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 societyare excluded from the consequences of their actions by virtue of their wealth or their legal standing or their clubmembership then the whole society becomes unstable. Just as it does in the physics systems of Bose Einstein. TheBosonic 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 intravari-ant

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

The reality of life is that both Boson people and Fermi people1 do exist. However if one understands the way inwhich Fermi particles coalesce in physics we may be able to stabilise society in the same way that nature does inthe atomic structure. But that is another story. For now it is sufficient to understand that there are Boson particleswhich have whole number spins such as1, 2, 3 and there are Fermi particles which have half number spins such as12 , 3

2 and52 . These correlate precisely with the two Worlds geometries (Figure 11-3). The intravariant is boson-like

andthe extravariant is Fermi-like.

The transformation that takes the Boson World to the Fermi World is shown inFigure 11-4for the case of Grav-electromagneticWorld. 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 extravariantgeometry. It is suffice to say here that all the line elements in the intravariant World are the same as those in theextravariant 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 fromthe outside to the inside.

It was a great surprise to me that a 4-dimensional geometry could be rotated at all. It can be seen in[note1127]thatin the passage of enlargement to infinity a rotation takes place during the first half of the transformation from0π to theπ/4. The continued passage from infinity to the smallest case of the rope trick is a further quarter rotation fromπ/4 toπ/2. I call these two Worlds intravariant and extravariant because these would be the correct mathematical terms. Bothgeometries contain identically the same elements. They are orthogonal to each other.

It turns out, from working with the two geometries, that the intravariant one is continuous and unstable whilst theextravariant 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 orthogonalityprinciple further into the Worlds of the second kind. Indeed, it suggests that the intra to extra transformation enables anyrepresentation to be continued indefinitely. And that is rather remarkable because it implies that there is no centre to thegeometries either at the beginning or at the end. The Alpha and Omega of the universe is imbued in ourselves and thenature 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 representations of Fermions and Bosons.

Ostberger found this transformation in the seventies. It was many years before he realised that the two Worldscould assemble together under certain conditions. He had already worked on some geometry which, becauseof its absorptive properties, disappeared when assembled. He attempted several combinations of assembly thatculminated in the“World geometry of the second kind”.Figure 11-4shows how he conceived the assembly. Theextra-World was reduced to its minimum state and then assembled on to the intra-World so that the correspondingelements (of the same colour) were in opposition to each other. The external elements of the extra-World metthe surface elements of the intra-World in opposing directions. This opposition or anti-commutation led to thereciprocal elements producing a unity at the surface2. Unlike the absorptive opposition of elements that producedannihilation. The result wasFigure 11-5, the Grav-electromagnetic World of the second kind. There are six waysto assemble the two Worlds. These are labelledE, P andN with their contras back-E, back-Pand back-N3.

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 forefrontof the research.

11.3. GEM World of the second kind

Figure 11-5. The anti-commuting assembly of the intra and extra Grav-electromagnetic Worlds that cre-ates the representation of atomic elements4

A description of the geometry inFigure 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 elementwhich generally seems to be unmeasurable. The plus and minus signs are here the forward or backward directionsand not the sign of the magnitude.

The suffixm is magnetic,e is electric andg gravitic. TheZ are the potentials. TheB, D andV are the socalledfluxesof the three Fields. The number affixes count the elements of the system. ThusB+1 is the forwardelement of the extravariant magnetic field.B+0 is the reference element of the intravariant surface. To generalisethe surface elements the Greekξ, ψ andζ are used for the magnetic, electric and gravitic fields respectively inupper and lower case.

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11.4. The atomic elementsThisbook is not the place to describe the attributes of this geometry that makes it wholly suitable for the descrip-tion 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 theElectron, theProton and theNeutron. Remember this is a vector geometry and it has no magnitudes only orthog-onalities.

1. EachE, P andN is part of a pair which also includes an opposite spin partner.

2. The spin of the electron is the half-vector that sits diametrically across the gravitic extra-element (green)labelledZg.

3. The charge on the electron is carried by the half-vector which sits diametrically across the element labelledZe. It has a spin direction opposite to the Proton but in the same plane.

4. TheZm in the electron is the element that creates the Zeeman effect.

5. TheZg is the element that counts the electrons in terms of its gravitational effects. Its length, the circum-ferential length, determines the number of the electrons present. This is unlike our conventional view of thecounting process. We are accustomed to counting things as separated parts but here we count both in seriesand parallel the rings that are on the surface. Adding the rings in series makes a larger surface. Adding themin parallel creates rings providing that the conditions allow them to be accommodated on the extravariant 4-Dsurface. The pairing-particle must also be counted contemporaneously. In the case of electrons this meansthat the down spin electron which is at the“back” location must be counted too. This process is shown in thenotes to account precisely for the singlet and triplet cases in the H atom. It also has the correct attributes toaccount for the multiplets which arise in heavier atoms.

6. AtP theZe countsthe protons in terms of its electrical (charge) effects. It is accompanied by an axial vectorwhich measures the charge.

7. Any two coincident line elements creates an observable“particle”.

8. The plane of the first element, e.g.D+1, lies out of the plane of the zeroth element. This is because theorthogonality conditions demand it. We may speak of“the first element plane”. This first element is alsoskew to the second plane. It passes overE, inside back-P, under back-Eand outsideP. The other two firstplaneelements take a similar path. In this way all three maintain their orthogonality to each other and to therest of the elements. This is made more clear in theFigure 11-6where the three planes of the first tiltingelementare each clarified

9. Every junction on the geometry has three mutually orthogonal elements and is orthogonal to every otherjunctionwith the same. There are at least 36 such junctions making at least 108 orthogonal relations.

10. The intra- elements which face on to the extra-surface seem to be“silent”. That is, they lend virtually noattribute to the particle. Thus the electron hasZg silent. We may interpret this to be that the electron isvirtually unaffected by gravitational potential.

11. The neutron hasZe silent. It has no charge. It is unaffected by an electric field. In the protonZm is silentand so the proton should have no intrinsic spin magnetic moment.

12. The intravariant region is termed“orbital” in the literature and the extravaraint region is termed“spin”.

13. The Bosonic spins are of three types. They are axial vectors that measure across the intravariant interior. TheISOspin, for example, is the intravariant axial vector of theN-P plane.

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

15. TheE-state electron is a2π rotationfrom the back-E-state electron when the rotations are measured correctly

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

A vector model of the Hydrogen atom which complies with the directions of this geometry is the subject of anotherbook on the subject (available from the Ostberger website). The magnitudes which are extracted are consistentwith current theory.

It may be possible to remove certain anomalies from current theory with the realisation that the vectors in theintravariant space are infinitely separated from those in the extravariant space.

The Sommerfeld fine-structure constant has been derived and calculated from this vector model inChapter 12.Thetable of Delta values in theAppendix Jnot only holds the key to the fine-structure constant but also seems tohave 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. Afterall, it is a far cry from the Bohr model of electrons whizzing round in orbits. Yet that model worked. At leastsufficiently 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 directionalin character and they can be displayed in a geometry; the World geometry of the second kind. But this is not theend of the story. This is not the final solution. This is like a Schroedinger equation except that this equation is adirectional one. It gives the general picture and provides the essential relationships but it needs a more detailedsolution for each of the atoms.

The solutions are not easy to find. The rules of operating must be followed rigorously. Yet sometimes new rulesare found which may be sown into the fabric of the theory just as with any other theory. The essential attributesmust 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 weexplain 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 verylikely yesbecause 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 matterof relating the eigenfunctions to the arrangement of circles on a surface and getting all the quantum numbers inthe right order. In the case of Helium5 this has been done. Not all the directional permutations will be allowedin nature and so a table of the permutations showing directional reasons for eliminating some has been made byOstberger. The surprise is that there is such a great redundancy of permutation.

Figure 11-7. Counting electrons, the triplet states

To illustrate how the process works look atFigure 11-7. This is the representation of the triplet state for two upspinelectrons. The two could equally well occur at the back-Elocation and be two down spin electrons. This is atwo electron state if the circumferential measure is a single Planckh of energy. The two electrons are in paralleland form a surface. The 4-D surface area is minimised, representing the net lowest energy both magnetic andelectric in the structure. The result is that the two electrons can be assembled on the surface such that the greatestdiameter (a great circle) is not more than the next quantum size up, in this case2h. In fact, in this case (Figure11-7), the diameter of the blue element is

√2h or 1.414h. When we have a surface which measures2h across (the

great circle) then we have a potential for four electrons (two spin up and two spin down) as well as all the lesserpermutations.

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

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

Now look atFigure 11-8. This is the singlet solution. One spin up and one spin down electron. They are stillin parallel but in separated locations. Which is one spin up and which one is spin down would not easily berecognised.

What is more, since nature does not understand“metre distance”as a property, these two electrons could be anydistance apart. And that seems to lead to Bell’s theorem.

Figure 11-9. A sketch of a three electron state

Figure 11-9is a sketch of a possible surface with perhaps three electrons on it. If such is possible and complies totheinterval rules then this may be one state of Lithium.

The theory says that there must be two separate ionic bond groups. One belonging to theE location and the otherbelongingto the back-Elocation. In the formation of simple diatomic molecules such asH2 andN2 the theoryrequires that the two atoms be back-to-back with each other thus forming a kind of pseudo antiparallel electronpair. The two electrons in the diatomic pair are of the same type. EitherE electrons (spin up) or back-Eelectrons(spindown) but not mixed.

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

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

Notes1. The Fermi people are the infinite players of Prof. Carse’s“F inite and Infinite Games”([Carse87]).

2. Equivalent to saying that the metric gij

= gji = gij= 1.

3. The term used to express the reversing of the character in the pairing location of the geometry. It makes iteasierto print in the absence of the new character.

4. See[note1748].

5. [Note820](Spin Eigenfunctions for Helium).

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Chapter 12. Using the process forSommerf eld’s fine-structure constant

12.1. The first applicationWhen we are in the forefront of discovery there is no way of knowing what will come next, so it is prudentto 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 soonfind out which kind of discovery has the cul-de-sacs and intuitively stay on the main highway. Ostberger did justthis. He developed a flavour for the main highway towards his goal of modelling the Hydrogen atom. He gives hisreasons for this:

I had to find some part of nature that would lend itself to the theory in such a way as would give the opportunity ofprediction. The macroscopic universe did not do this and I could see no way of repeating the kind of predictions thatEinstein made. For one thing I had not read astronomy and for another I did not have a theory of its content. Yes, Icould see that we were in error for looking through telescopes and mapping the heavenly bodies as if they are a radialcartology; yes, this would upset the Hubblist’s and stop the universe accelerating into oblivion, but this was not goingto be a measurable observation in my lifetime. It is true that we may be able to send out high speed spacecraft and showthat new objects appear around the entrance perimeter of black holes as the speed increases, but that will be a long wayoff 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 ofpredictive consequences. This was my journey and anything that was not on the main highway would be spun aside.

I began work to find a geometry that would fit the equations of the Hydrogen atom. I had worked ordinary polynomialequations and found that the solutions lie in the coefficients; that one could construct simple shapes to solve equationswithout the use of the any numerical method. I had applied the same process to differential equations discovering thatthe coefficients created whole geometric forms. In the most difficult equations and particularly those with complex co-efficients the geometries representing the coefficients were moving in the space. They would form a step-wise series ofgeometries moving synchronously with other similar sets. By the time I had got to the kind of differential equation thatSchroedinger produced I was well versed in the technique. I was able to follow the accepted solution which divides theequation into three separable differential equations. Each pair of equations meet in a common constant. In geometricterms this meant that there was a common line of magnitudes between each pair of solutions1.

I knew that I would have to do this job myself. Nobody had been interested in the past and it would be a diversion tryingto attract anybody to be interested in the future.

I knew that the Hydrogen atoms had to have Directions. Particularly as I had established that a 4-dimensional world ge-ometry was available for the modelling process. If quantum mechanics was a vector study then there had to be directionsfor the vectors. The Hydrogen atom was described by a quantum vector picture and so the atom had to have directionsthat 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 theDifferential Equation approach and the Quantum Mechanical approach arrive at the same answers to the Hydrogenatom? I discovered that there was a very small and subtle difference between the two.

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

Figure 12-1. Finding the new universe

At the outset, it seems that Ostberger had very few clues as to the way in which geometry could serve the purposeof representing the atomic structure. One of the clues that is marked in the margin of the notes forms a relationshipbetween the Rope-maker’s first“trick” and Planck’s constant.Figure 12-2is the case of the rope trick in whichoneunit of rope is added around a body of zero size2. This is compared with Planck’s~ in Figure 12-3which isidenticalin geometric form. There is a scribble in the margin. It says,“The Planck trick... there can be only oneshape that connectsh with ~ and this is it”. The next sketch shows a shaded circle with the word“energy” insideand the words“Joulesper cycle persecond”are scribbled, with the emphasis.

Figure 12-2. The one unit rope trick

The message is that a Joule-sec is actually a Joule per cycle per second if the cycles are ignored. But in thegeometrythe cycle is a line element, and this line element is the element of Gravitational Direction, the Directionthat we call Potential in the intravariant case. In the extravariant case the same Direction is the one in which thefrequency of light rotates. The area of the circle would be a representative of the energy if the circumference werea representative of the frequency. So Ostberger, relating the trignometric frequency to circumferential measure3

had a starting point.

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Figure 12-3. The Planck Trick

The first breakthrough for Ostberger seems to have come out of one of the many library Standard Forms that hehadcollected. 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 start. In essence these geometries are very simple in so far as they arewell known shapes. They appear in algebra5 as Radical circles. They appear in system dynamics as“M-circles” 6

which relate open and closed loop responses. They appear in magnetic and electric fields and many other placesin mathematics and physics, but here they appear in a new guise because thesered 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 product

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If we look at the two chords in the centre we see that an application of the inner chord theorem mentioned on page4 XXX of Chapter 3results in the square root formula7.

This is easily referred to as thered geometry. All the elements arrive at a pole situated on the infinite element andthe measurementsp are horizontal.

There is also ablue geometry. In the blue geometry ofFigure 12-5we may take the outer chord theorem andapply it to obtain the same formula! This time the measurementsp are vertical. All this data must be taking intoaccount. In mathematics we would not see these small, but important differences. The blue geometry shape is alsowell known in mathematics, but here it is quantised.

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

Now, suppose we assemble these two geometries orthogonally, as inFigure 12-6. A well known shape is produced,except that it is quantised. The whole space is littered with quantised values and quantised (orthogonal) points.

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Figure 12-6. The red and blue geometry assembled orthogonally

The red8 geometryand blue9 geometry assembled orthogonally.

The red elements can be used to represent the magnitudes of the vectors10. The blue elements can be usedto represent the Directions of the vectors11. There is a two-fold infinity of trigonometric functions which canbe arrayed geometrically in this space. Functions of all amplitudes, powers and phases can be represented12.Although these orthogonal geometries are well studied in terms of algebra, the trigonometric functions containedin 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 repre-sentative of the directions of the vectors and the red elements are representative of the magnitudes13. By using thetrigonometric geometry inChapter 6this geometry can be used to represent Fourier components.

Supposewe assemble these two geometries in parallel! Or suppose we assemble them antiparallel!Figure 12-7istheantiparallel assembly.

Figure 12-7. A magnitude essential to Quantum Mechanics

√p(p+ 1)

Ostberger studied this assembly and produced the table of constants inAppendix J. These constants come frominsertingquantum values into the formula inFigure 12-7for −δqp and+δqp. These are hisdeltavalues.

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Why are they significant?

1. These are natural constants.

2. They have arrived out of a geometry that has shape without reference to magnitudes.

3. They reflect the angle of tilt of the first element of the World of the second kind.

4. They contain the magnitudes essential to Quantum mechanics.

The geometry in theFigure 12-8has some peculiar properties. As shown it represents all the geometries withinteger values in the lower half plane.

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

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1. Despite the fact that it is circular it is none-the-less quantised. The values ofp andq areinteger and halfinteger.

2. There is a special connection between the areas of the green geometries and those of the red and the bluegeometries.The area formed by corresponding circles of equal values ofp is always equal to the area withinthe green element circle regardless of the values ofp.

3. The whole geometry is scaled by~.

4. The magnitude ofδ is less then the magnitude of~! The argument for“hidden variables” is here partiallyvindicated. I say partially because the values ofδ are linear measure whereas the values of~ are an extractionfrom the valueh which is an area representing the energy. So the argument is not so much a question ofwhether there are hidden variables but rather one of whether we can measure something less than energy.

5. There are two half states of the green circle.

6. To help our imagination we may regard the green circle as a ball which when depressed opens the gateformedbyX. TheXR is the close state and theXB is the open state.

7. Theredandbluegeometries are antiparallel. They have opposite directions.

8. This geometry was used by Ostberger as the first part of the solution to the Hydrogen atom. It fits into theWorld model on page 9 XXX ofChapter 11in theD plane.The exact method of fitting it to the orthogonalconditions is given in the notes.

9. The values ofq area part of the extravariant region which is shown green here. The values ofp are a part ofthe intravariant region in red and blue. The two parts anti-commute and cannot be joined in a classical sense.

10. In the Hydrogen atom the value ofq = 1. In this geometry when the integer values ofp have reached∞− 1then a new set is available with the value ofq = 2, and so on. We can now extend the study of atomicstructures to integer valuesq > 1; the green circles.

12.2. Sommerfeld’s fine-structure constantLook at the table of delta values inAppendix J. Look at the first value, whenp = 1 andq = 1. It is 0.08786437626.This is noticeably close to the square root of the fine-structure constant of0.085423666, but Ostberger ignored itfor 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 thescale of the model, has a tilt and he had ignored it. He had been trying to fit the delta geometry into theD0 plane.But when he applied it to theD1 plane and calculated the projection of the delta geometry on to theD0 plane hearrived 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 values15.

Thefundamental scalar of Hydrogen is derived from the fact that some natural length of magnitude exists,−δ11 , which is

spacially rotated through an angle whose magnitude may also be measured as−δ11 . The direction−δ1

1 and the magnitude−δ1

1 are orthogonal.

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

TheFigure 12-9shows the final stage of the calculation. The value of−δ11 whichis shown inFigure 12-8becomes

the sine of the tilt angle. It follows that the cosine is√

1− −δ11 . This sine-cosine product is a chord (likeC in

Figure 5-5) and the fine-structure constant is an orthogonal chord (likeA in Figure 5-5)16.

Onehas 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

115


geometric picture is particularly difficult. It eluded Ostberger for many years until he assumed that the two halfstateswere Magnitude and Direction. The Magnitudinal representative, the blue geometry is the leader of the firsthalf 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 thenYang. The one case has the potential to satisfy the differential equation (Schroedinger) approach and the other theQuantum Mechanical (Heisenberg) approach. Either case yields the fine-structure constant. So that students mayfollow him Ostberger made notes on his intuitive assumptions, particularly the ones that worked.

I find, in general, that every yang solution or representation is followed by a yin one. In the Hydrogen atom it wouldseem that every successive energy level alternates as a yin and then yang solution. In this case the yin is the directionalpart 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 applythem to known physical results that could lead us into blind alleys. Thus, making assumptions about the way inwhich these processes work is essential to its development. However it will add nothing to our understanding orthe 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 polarisation) it maybe remarked that the main objective of physical science is not the provision of pictures, but is the formulation of lawsgoverning phenomena and the application of these laws to the discovery of new phenomena. If a picture exists, so muchthe better, but whether a picture exists or not is only of secondary importance. In the case of atomic phenomena no picturecan be expected to exist in the usual sense of the word “picture”, by which is meant a model functioning essentially onclassical lines. One may, however, extend the word “picture” to include any way of looking at the fundamental lawswhich makes their self consistently obvious.

The geometries of Ostberger are certainly not along classical lines. They are an extension of the word picturewhich includes the measurement of vector curvatures. They are a consistent“way of looking at the fundamentallaws” which exposes the truth of their nature. It is the truth that we seek in order that we may understand the paththat 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 fromOstberger which may ease the burden of understanding mathematics for those who are not conversant with itsidiosyncratic language.

I have minimised the mathematics by leaving a trail of references to the notes.

Notes1. The geometries used are given in the[note18xx]note series. In particular the notes 1870-90 give the detailed

mathematicsand associated geometric constructions. This is being issued as a separate booklet of some 60XXX pages.

2. SeeAppendix A.

3. [Note112]

4. [Note6xx] series.

5. The algebraic versions are graphical shapes. It is useful and necessary to know the relationship between theOstberger geometry and the graphs because the equations are expressed in the algebra. In algebra we describethe 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-circles.

7. All Euclidean Theorems can be applied to a true vector geometry to produce a true result.

8. [note610]

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9. [note612]

10. [Note608]

11. [Note609]

12. Refer to the[note5xx]series notes.

13. This idea is too rich to express here but is clarified in the notes.

14. Notes 1850-65

15. [Note1850]

16. The detail is given in the[note666]and[note667].

17. [Dirac70] (page 10)

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Chapter 13. New wine into new skinsIf I do not stop this book here it could become as voluminous as the notes themselves. The notes are numberedfrom 0 to 3000 although there are many spaces left vacant for future additions. In all there are probably half thisnumber.

The notes begin with the words,“Some people write a diary which is a record of the past. I have written thisnotebook which is something for the future”and that is how he sees the world for he says again,“there is alwayssome part of today that has something for the future”. He sees today as a part of the future and lives to continueit. Others see yesterday as part of today and wish to consume it. This is like the thermodynamic Volume thatobeys the absorption rule (Figure 8-9). Ostberger views the“Volume Remaining”but some view the“VolumeConsumed”or as professor Carse1 says,“There are at least two kinds of games. One could be called finite, theother infinite. A finite game is played for the purpose of winning, an infinite game for the purpose of continuingthe play”.

These two kinds of games are the ones that people play; and so does nature with her Bosons and Fermions. It isnot that either is right or wrong in the universe but rather that one must have preference over the other if the flowof life is to be maintained. The intravariance and extravariance of the world determines its stability. That is theOstberger discovery.

There is a subtle difference between the two ways which cannot be observed except in action. It is the manner inwhich nature acts that tells us which is Boson or Fermion. So also for the finite or infinite player, it is the actionthat 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 book2 in which he divides up the Riemann tensor into two compo-nents,“Riemann = Weyl + Ricci”. This is a division of the magnitudinal and directional aspects of the Riemanntensor. The Weyl part, with its ten components, are directional in character whereas the Ricci part, with its tencomponents, are the magnitudinal part.

This is easily related toFigure 2-18. The Weyl describes the“dir ection of the magnitude”whereas the Riccidescribes the“magnitude of the magnitude”. The other two parts ofFigure 2-18belong to Ostberger. Forming the“dir ection of the Directions”is making the geometric shapes and their orthogonalities. Finding the“magnitudesof the Directions” is seeking the answers such as the examples given of the trigonometric representation or theDelta geometry. All four attributes are essential to our understanding of the universe around us. Now that we havebeen presented with the remaining two we have one heck of a lot of work to do! This is in complete harmony withSteven Hawkin’s remarkable view of such a discovery:

[Hawkin88]page 175

However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not justa few scientists. Then we shall all, philosophers, scientists and just ordinary people be able to take part in the questionof why it is that we and the universe exist.

Steven Hawkin’s remarkable insight continues,

[Hawkin88]page 168

Seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity. Nowadaystens of thousands of university graduates do, and many millions of people are at least familiar with the idea. If a completeunified theory was discovered, it would only be a matter of time before it was digested and simplified in the same wayand taught in schools, at least in outline.

and again his remarkable insight into the future yields,

[Hawkin88]page 169

A complete, consistent, unified theory is only the first step; our goal is a complete understanding of the events around 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 mayhave thickness in two dimensions and be of the simplest circular kind. But it differs from String Theory in twoessential attributes. The first is that it is in the language of lines in space that can be drawn on to paper andcomputer. The second is that the Orthogonality Theorem of Quantum Mechanics has been placed foremost in theprocess 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 Book3.

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 theyare orthogonal. This means that if there are two states for which a measurement of the vector is certain to givetwo 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 thatwe can attach to the eigenvalue of the state. In this theory the Orthogonality of the state is the first objective of thestudy and the numerical value the second. This is not the case in String Theory where the reverse process takesplace.

The second part is the fundamental equation and the fundamental conditions which are on page 87 of Dirac’sbook3 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 firstelement(Direction) produces the second element (velocity). Providing both Magnitudinal and Directional effectsare considered separately.

Eddington, back in 1928 foresaw that the fundamental quantum conditions were of greater significance thansimply a formula. He predicted that we would be able to base some of our understandings on something otherthan number. He virtually predicted the Ostberger discovery. I have placed Ostberger’s scribbled notes along sidethe two key equations of Quantum Mechanics on the page below. He notes also that there will be two directionalcompliments to these. One is clearly the orthogonality conditions which comes from the Orthogonality Theoremof Quantum Mechanics4. The other, I do not know what he meant.

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Chapter 13. New wine into new skins

Sir Arthur Eddington5 wrotein his “The Nature of the Physical World”the following passage in 1928, page 209.It virtually predicts the the contents of my Ostberger’s notes.

I venture to think that there is an idea implied in Dirac’s treatment of [the above equation] which may have greatphilosophical significance independently of any success of this particular application. The idea is that in digging deeperand deeper in to that which lies at the base of physical phenomena we must be prepared to come to entities which, likemany things in our conscious experience, are not measurable by numbers in any way; and further it suggests how exactscience, that is to say the science of phenomena correlated to measure numbers, can be founded on such a basis.

Notes1. [Carse87]

2. See[Penrose]page 271.

3. [Dirac70]

4. “Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal.”[Dirac70]page 32. It is difficult to believe that every eigenvalue is going to be orthogonal to the next because that wouldmeanthat there are vast numbers of orthogonalities in the space. But Ostberger shows that such large numbersof orthogonalities can described and depicted for study.

5. A Friend.

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II. The Appendices

Appendix A. The first rope trick

A.1. The note 104bIn the following I consider the possible quantised extremities of a connected curvature. That is to say a continuouscurvature that can change in integer intervals no matter how small the interval. I look at the largest possiblecurvature and then the smallest possible curvature. I conclude that a straight line cannot exist in a space whichis to be used for the measurement of a magnitude. This infers that a magnitude space has no zero. A directionalspace 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). Let us say it is 40 million metresroundthe Earth and the rope just touches the Earth around its entire circumference. Let us now add one metre ofarc into the length of the rope. How far from the Earth will this rope be if it stands away from the surface equallyall round?

Figure A-1. A rope around the world

Let the diameter of the original circle (the Earth) bed1 and the diameter of the next up integer circle bed2.Likewise the circumferences arec1 andc2. Then,

c2 = c1 + 1 (1)

The distance that thec2 circle stands away from the Earthh, is given by:

d2 − d1 = 2h (2)

Sincec = πd in general we may express (2) as,

1/π(c2− c1) = 2h (3)

But from (1) the difference of thec’s is a unit of arc so that,

1/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 therope by one metre it will stand away from the surface a distance of nearly 160mm all the way round! This is notour normal comprehension. There is something wrong with the proportions of our thinking and that is why thestudent must play with string and pencil to acquaint himself with the reality of an otherwise difficult magnitudinalconcept.

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

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Given that the rope is 40 million metres in length and that we have added only 1 metre to its length our sensibilitiestells us that the stand-off is likely to be of the order of1/40, 000, 000th It is not. It is1/2π times the distanceadded in.

Figure A-3. A rope around Jupiter

But it is important to note that the one metre added is a length of arc whereas the distance of the stand-offh is alinear measure.

Consider now a rope around Jupiter that is some 446 million metres round. Stretching the rope by one metremakes the stand-off 160mm. Again! The reason is because the same simple mathematics applies.

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

Consider now a rope around the Universe (Figure A-4). We have established that for each unit of arc added intothecircumference of a circle there is a1/2π displacement radially (and therefore orthogonally) to the circle. Sothat, if we were standing on the curvature that fits around the universe and someone adds a unit of length into thearc of that curvature at a point that we cannot easily observe then we will motion with that curvature a distanceof 1/2π orthogonally to the curve. Providing, that is, there is continuity in the curve. Without any other line ofreference we could not detect the displacement because we are moving in conjunction with an infinity.

Now consider a rope at infinity. Theinfinite elementis the straight line inFigure A-5. Whether this is by definitionor by reason does not affect the argument. When the curvature is zero the radius is infinite; both are extremities.

Figure A-5. Curvatures at infinity

Consider the curvature that is last before the infinite element; the one inFigure A-5that has its centre of curvaturesomewhere below the infinite element. A unit of arc added into this will cause a displacement1/2π upward. Nextconsider the first curvature after the infinite element. A unit of arc added in to this will cause a displacement1/2πdownward. Where in the space are these displacements? Either the two curvatures are1/2π apart or they are twice1/2π apart. Which is it?

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Figure A-6. One unit

To answer this consider the smallest possible circular path in whichd1 is zero. That is we put the rope aroundnothing (Figure A-6. We cannot have a smaller circle since there is only one unit around its circumference. Todivide this unit into smaller parts is to divide the arc unit into the same smaller parts and we are doing no morethan scaling the whole sheet of paper including the infinity. Whichever rod we use to measure thec’s we use thesame rod to measure thed’s. We cannot pretend to change the unit of measure of the smallest arc without changingthe largest arc likewise. It would be cheating on the continuity principle of the circles.

Figure A-7. Half a step to infinity

So what happens to the maximum curvatures inFigure A-5? Do the displacements cross over in the distanceh makingthe space between them bi-directional as inFigure A-7? Or is it likeFigure A-8in which the spaceremainsuni-directional but the distance between the two maximum curvatures is2h?

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Figure A-8. One step to infinity

If the Figure A-8is admissible then the infinite element exists and the two curvatures each have a step of one unitto infinity. If the Figure A-7is admissible then the two curvatures have a step of one unit of~

1 to each other andthe infinite element does not exist.

A.2. Concluding remarks1. The paths shown here are circular and simply connected. What happens if the paths are not circular? In note144 I deal with the non circular paths using line integrals and the Residue theorem. However I can leave my pathscircular 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 therebyallowing me to develop non-circular paths later as part of the solution finding process. Secondly I need to stay withthese circular paths because I am going to seek orthogonal conditions which are based on establishing rotations inspace. I must therefore find the rotations first and then apply any distorting mappings afterwards. Thirdly I needto be able apply the shortest distance axiom to the geometry and since the shortest distance between to orthogonalpoints 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 withthis 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 infinity then there is only one direction in the line. (Analogous to the character“1” innumber theory.)

....if neither end of a line is at infinity then there are two directions in the line. (Analogous to saying that we havetwo 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 analogousto 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 forwardwe can measure a counter-action backward. This also demands bi-directional line elements.

The question here is, which of the two geometriesFigure A-7or Figure A-8is the one which exists at infinity?

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I have considered the arguments, made numerous mistakes, and concluded that both are permissible. Both arenecessaryand then they are sufficient to make the representations. Indeed, it turns out that both are required. Theyare“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 acceptthis principle and make progress whilst allowing somebody to prove it otherwise3.

What has brought me to this conclusion?

My representation of the Hydrogen atom required the calculation of the Sommerfeld fine-structure constant froma geometry (Note 666). In this geometry the use of both theFigure A-7andFigure A-8are necessary. Both arerequiredin the atomic representation. TheFigure A-7is the magnitudinal vectors and theFigure A-8the last ofthedirections 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 isπ/2 at the point ofintersection and nowhere else connected.

In Figure A-8the smallest circular path and the largest circular path have no intersections that can be orthogonal.The two meeting points either meet at zero orπ-apart depending upon the directions of the elements. HoweverFigure A-7does have a possible orthogonal condition at the intersection of the last curvature before infinity andthecircular path with radius1/2π. i.e. between the last curvature and the first curvature.

But is it orthogonal?

I have used Dirac’s~ to label the distance h/2p here to emphasise the significance of dealing with concepts atinfinity. But ~ is used in Quantum Mechanics solely for the quantum of energy per cycle, 6.602 x 10-34 in theGrav-electromagnetic world. There is no reason why the same principle cannot be used in other representation inThermodynamics, for example.

The fact is,~ is not local. It belongs to measurements near infinity. Quantum Mechanics seems to be taking therelationship between h and~ as indefinitely linear. It cannot be. As the geometries grow~ is also subject to thethe 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 thisrelationship. The next circle contains a small element of curvature and so the relationship is an approximation.

A.3. The 1/4π connectionThe area ofFigure A-6is 1/4π. It is a constant and has no units of measure. It forms a constant relationship withh and with other well known constants of the grav-electromagnetic World. In the Note 1830 Ostberger makesit clear that he regards the Bohr radius as a connective constant between our world of distances and the grav-electromagnetic 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:

a0Rµ =1

4πα

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in whicha0 isBohr’s radius,Rµ is Rydbergs constant andα is Sommerfeld’s constant. Rydberg’s constant belongsto Hydrogen. Bohr’s constant belongs to us and the product of the two belongs to Sommerfeld whose constant isthe geometric constant derived from Ostberger’sδ value inChapter 12. The constant1/4π joins area with linearmeasure.

A.4. Another theoremDo as the rope-maker did in the second rope trick. He had a spare metre of rope left in his workshop. So he twistedit into a circle which was 1 metre in circumference.

Figure A-9. One extra unit of circumference internally

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

Figure A-10. One extra unit of circumference externally

He wondered what the relationship was between the areas of the rope with a twist and the one without. He hadanothersurprise for he found that area produced by un-twisting the unit of rope was proportional to the originalradius of curvature and that he thought was impossible since it is known that the area is proportional to the squareof the radius. Something funny was going on. Somehow a two dimensional area had got mixed up with a onedimensional radius. But when he spoke to his friend, who was an engineer, he was told about something calledGreen’s Theorems which do just that. They relate ann dimensional domain to ann− 1 dimensional domain5.

The rope-maker remembered in his first rope trick the constant1/4π was a multiplicative constant. He was curiousto find that the constant1/4π had become an additive constant in this rope trick.

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The rope-maker decided to leave that to the mathematicians. But he was curious enough to want to discover moreaboutthis Green guy and so he continued his rope tricks.

Telle A Ostberger

Notes1. I have deliberately labelled the distanceh/2π as~. In Quantum Mechanics the Dirac~ is used solely for

the quantum of energy per cycle1.055 x 1034 joules per cycle per second. But there is no reason why thisprinciple should not be generalised so long as its special nature is preserved. It is special for at least thefollowing reasons:

1. ~ is linear and belongs to measurements near infinity. It is non-local.

2. It is imaginary. I cannot see that it can be divorced from its partneri becauseit is derived from anextravariant geometry.

3. The relationshiph = 2π~ has a unique geometric shape. The connection is circular. However we mustbe vigilant because the linearity of~ is only guaranteed for the first quantum. It too lives in a curvedspace.

2. Notes 4 and 124 and Note 2014 et.seq. These become axioms of the process.

3. One example of this problem is the teaching of multiplication. We tell the children theAxB = BxA andtheyare left with the impression that this is the rule. In fact we know that the truth is completely the reverse.AxB

is neverBxA except in the one special case; whereB andA are pure numbers. I fear for the future if we donot teach our children the truths we already know.

4. In mathematics we have not passed the stage where we believe that a vector can have its directions separatefrom its magnitudes. We teach only vectors whose magnitudes are parallel to their directions e.g. a car veloc-ity. But in my notes I have shown otherwise. Vectors can be found that have their direction orthogonal andantiparallel to their associated magnitudes. Indeed the vectors come in sets of four. I refer to those which havetheir magnitude and directions separated by0, π/2, π and3π/2.

5. The basis of Integration by parts.

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Appendix B. Geometric representations

TAO

The following partition the processes of creating geometric representations in my experience.

B.1. Presentation stages

1. A yin representation of direction only. A generalised geometry which is much like a guess. An undirectedgeometry.

2. A representation of stage 1 with the yang magnitudes on it. The shape or form which is expected. Anundirectedgeometry.

3. A representation which extends stage 2 to include the yin of the magnitude. That is the directions that areattachedto the line lengths which, at this stage are not defined in magnitude. A directed geometry.

4. A complete representation which includes all the above stages and the yang magnitudes which finalise it. Asetof scale factors relate the whole to our measurement of reality. A fully directed geometry with sufficientmagnitudes to permit a unique representation.

B.2. Generation

1. Absorption.

2. Reciprocity or inversion.

3. Real/Imaginary conversion.

4. Intravariant to extravariant condensation.

B.3. OrthogonalityRepresentations having their Magnitudes and associated Directions:

1. 0 radiansapart.

2. π/2 radiansapart.

3. π radiansapart.

4. 3π/2 radians apart

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Appendix B. Geometric representations

B.4. Density

1. No directions in the line. (e.g. plain geometry, position vectors.)

2. One direction in the line. (e.g. ordinary vectors.)

3. two directions in the line. (e.g. the tensors of rank 2 in the Law Fields.)

4. Multiple directions of greater density. (e.g. the Ricci and Weyl Tensors and the geometric Worlds of the firstandsecond Kind.)

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Appendix C. The case of two zeros

C.1. Something that actually happened in a companyBoth accountants and engineers who work with computer systems are well versed in handling two different zerosin 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 makethe systems work by saying what must happen to the money and the goods. When goods enter a company a“GoodInward 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 forhim 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 theseentries.

Now a tool which was critical to production was sent to a subcontractor for heat treatment. The tool was mislaidand a search for it began. The marketing department pointed out that advertising worth hundreds of thousands ofpounds 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 veryhigh 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 thefollowing 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 beensentto, 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 errorsWhen an accountant reconciles his balances and finds an difference of say, 4, he knows that this could representan error of millions because the double entry will allow any constant to be added to the difference. So he needsnumerical“tricks” to help him with his accuracy. The absorption matrix (Note 100) is a great help. It explainshow the numbers can be found even when there are errors.

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Appendix D. Uses of the word “Dimension”

D.1. Note 137 Summary page

1. Dimensions in units of 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 areMLT−2.

2. Dimension of a drawing.

Theordinary length measurement which scales the various attributes of a drawing. Particularly an engineeringdrawing but also in geometry.

3. Dimensions of a Riemannian kind.

Thenumber of orthogonalities required to analytically define a set of elements which associate at any numberof points in space. In ann dimensional space there will ben+1 points. I refer to thesen dimensions as being“magnitudinally separated”because there must be some size, distance or numerical value separating eachpair 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 theyare all mutually orthogonal despite being spatially separated. Likewise each of the 40 elements of a geometryof the second kind also constitute a dimension.

In three dimensional curved space there are three orthogonalities. The size of the space is determined bythe number of orthogonal correspondences there are between the points1 on the three orthogonal lines. Ifwe 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 Riemanniandimensions. And so on. A World of the first kind has ten. But these are generic dimensions when the LawFields are applied. Each law field restricts the correspondences of theπ 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 toall other points on its neighbouring line thus making it an highly overdetermined. Only in a curved spacecan we sort out this overdetermination of the spaces. Furthermore we may rank the spaces by the Law Fieldtechnique. 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 Ostbergerwork.

4. Dimension of the Einsteinian kind.

The number of orthogonalities required to define a set of elements at one point in space. This seems tobe 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“directionallyseparated”.

However any number of groups of four can be generated into higher spaces. In the World geometry of thefirst 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 thenth kind has4n groups of four. Such geometries would be extremelydense representations since the geometry of the second kind is sufficient not only to describe the whole of

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Appendix D. Uses of the word “Dimension”

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

Notes1. Points do not exist in the curved spaces of the Worlds, only new worlds. But what other language shall I use?

Thepoints of two orthogonal lines from a new world or quasi-world which we recognise as a new particle inphysics. Hence the Eight Fold Way.

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Appendix E. Connectedness anddisconnectednessSciencehas a great fondness for the speed of light. Much more so than the other constants of physics that areequally as interesting but less often brandished in the public eye. The Sommerfeld fine-structure constant1 forexample is fascinating because it is derived from a geometry, a pure shape in our universe. This shape is a discon-nected one. It is a quantised geometry and yet there is a connection to the set of Delta values2 that ensue from it.That connection is the particular rule used to draw the geometry in the first place. The rule is the connection andnot the geometry.

The idea of connectedness is very important because it underlies the whole structure of our universe. But it isnot just connectedness, it is disconnectedness too that drives the structure of form. The two vie with one and theother. Chaos bursts into the world and becomes a part of some new order. Wars bring about the human need forpeace. The genes swap about to produce rational and irrational humans or males and females. Asymmetry vieswith Symmetry to produce the rules of quantum mechanics. Even the interpretational arguments in science vieextra 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 notmatter, 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 Lorentztransformations apply equally to the“Quantum Mechanical view”as they do to the“classical”. In notes 1110and 1111 the geometry of the Lorentz transformation shows the connection between the relative velocity of twoobservers. The geometry shows the well known changes in the measurements of mass, length and time. As thespeed 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 thereis a directional effect as well as amagnitudinalone. The Magnitudinal one is the formulae for calculating theobservational 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 knowwhich curvature we are on in order to return home. Just as the intrepid explorers of the mid-millennium needed toknow 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 difficultyembracing 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 andothers who are intravariant by nature. Our energy source is from matter and we have a constant force motor some20 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 gravitationalfields that will directionally accelerate us without our motors. We are observing the Lorentz effects of the matteraround us and plotting its curvatures. We are calculating the geodesics along which we need to travel. There aredepartments 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’smotion. The question is, What happens when we reach the speed of light? The speed of light is about 300,000kilometres per second. Let us stand on the control deck and calculate how long it will take us to reach the speedof light.

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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 livewith our heads pointing forward. After one year we will be travelling at 315,619,2003 metres per second, well inexcess of the speed of light. On the 348th day (earth rotations) we have a party because we are at the speed oflight. Its a“disconnection”party.

Figure E-1. Finding the new universe

We are now disconnected from the folks back on Earth just as was Columbus. There is no grav-electromagneticenergy connection and we really are worlds apart. The Lorentz connection no longer applies because we cannotobserve each other by electromagnetic means. We aremacroscopically disconnected. We have become part of thefabric of the universe. We are like being in a fluid4.

The new explorers will not be back for many years and may even colonise new territory. They must be selfsustaining 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 positionthe craft in both the forward and the backward directions of all its curvatures of travel. The accountant and thephysicist 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 starsarrives on curves. The curves come from different Directions. Each curve is part of a 4-dimensional set belongingto the Laws of the universe. We have to sort them out in just the same way that we sort out the directions of theatomic 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 spaceforever is as arrogant as constructing a map of the Earth as if it were flat. It is a part of our present-day arrogance thatbelieves that we have some kind of power over nature.

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Appendix E. Connectedness and disconnectedness

Notes1. SeeChapter 12

2. SeeAppendix J.

3. One day is 60 x 60 x 24 x 365.3 seconds = 31,561,920 seconds. The speed is 31,561,920 x 10ms−2

4. Ostberger makes notes on the possible effects of being in a fluid of the universe. The Laws of fluidics applyandso communication may be possible through those laws. Because we have traversed a geodesic bend andare 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.

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Appendix F. Newtonian cases of Magnitudeswith DirectionsA brief look at the six cases mentioned in the text. Each of these six cases have components in Euclidean 3-space. There are two case each of theDirection, the Momentum and the Force. The first case is having thedirections parallel and the second is having them orthogonal. A further two cases of each of these phenomena inthe extravariant World will also exist. These are in the microscopic World of Quantum Mechanics.

F.1. Direction Zg: Directional case, Potentialsorthogonal to ForceThe gravitational Force is equal to the Gradient of the gravitational potential.

Fg = gradVg = iδV/δx+ jδV/δy + kδV/δz

The potentialsV are the orthogonal case of the use ofZg the Direction belonging to the Newton Field. Herewe have the potentialsV orthogonal to the gravitational forceF . The gradient is the measure of the slope ofthe potential in the region considered. The greater the slope the greater the gravitational pull. It is the same on acontour 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 field

F.2. Direction Zg: Magnitudinal case, Directions parallelto VelocityThe 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-static plane

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

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

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Momentum parallel to direction

Figure F-3. The rocket is an 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 changeof 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’slaws. We may say that,“The rate of change of momentum of a body is proportional to the force. When the changein the motion is directional the force acts at right angles and when the change in the motion is magnitudinal theforce acts parallel.”

This leads to the invention of abest trajectorytake off for momentum exchange devices to clear the Earth’sescape velocity. By combining both of these laws together a launching device can be proposed which maximisesthe benefits of the momentum laws (note No 930).

F.5. Force: Directional case - the gyro, Forceorthogonal to precession/directionThe attributes of the gyro1 were explained briefly inSection 4.4(see notes 1010-4). It epitomises all rotatingdevices that are in the field of an external torque such as a wheel fixed on one side only, a bicycle wheel that canturn in a headstock, a train turning a corner2, a car accelerating round a bend and so on. The internal torque is theone that drives the flywheel. A torque is acurving forceand in the case of the gyro sets up a miniature law fieldset on a spacial surface (Section 4.4).

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

F.6. Force: Magnitudinal case, Force parallel todirectionThe internal torque drives the gyro’s flywheel. Most of the twentieth century transport is driven this way withsome extremely inefficient designs of engines.

We are lead to the conclusion that,“to every momentum there is an equal contra momentum”and we can see thisas the counterpart to the law which says that,“to every force there is an equal and opposite reaction”and therewill 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 un-derstanding these laws.

These laws come from the intravariant World representing the macroscopic laws. The corresponding set of sixin the extravariant World case represent the microscopic laws. These are the cases in which the Magnitudes areantiparallel and anti-orthogonal (270 degrees) to the Directions.

Notes1. The gyro represents a quite general case. A torque applied to rotate a body with a second moment of inertia

possessesa rotational inertia. Associated with this are all the attributes of the gyro even if the resulting effectsare small or resisted naturally.

2. The original British Rail high speed train, theHS150, was intended to give the passengers a comfortable rideby defying the laws of the gyro on the bends. It failed! The braking system, on the same train, consisted ofexternal 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 slowspeeds can such forces be arrested. At higher speeds the mathematics tells us that any approaching objecthaving a frictional effect will be thrown off abruptly. The brake shoes would judder uncontrollably.

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Appendix G. Grav-electromagnetic relationsIt is not my intention that this should become a mathematics book but I feel that it is necessary to introduce someof the connections that exist in the spaces. Here are some of the relationships that exist in the three fields of theGrav-electromagnetic World.

Below in Table G-1are the effects of the rotations (Curls) that take place in the fields. The additional magnetictermmarkedδD/dt comes from the rate of change in the field. It was how Maxwell explained the radio receptionin an Edison earpiece. The top four are Maxwell’s equations in the electric and magnetic field. The lower one iseffectively the equation of a syphon in the gravitational field. The flow in the syphon is zero if the two ends havethe same potential. The path between them has no bearing on the force that creates the flow except that frictionplays a part where a pipe is involved. However where motion in space is concerned a body accelerates from aplace of higher potential to one of lower potential. Friction is almost absent and plays a much lesser role. Whilstthe 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 Vector notation

Magnetic∫C

Hs ds = I +dϕdt

Curl H = J +δD

dt

Electric∫C

Es ds = −dφdt

Curl E = −δBdt

Gravitic∫C

Fs ds = 0 or − ka

Curl F = 0 or k

Notes:a. 0 in a Schwarzchild field,−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 happensin a curved gravitational field, that is, one in which the force bends round an arc. These say that we cannot havesuch a force as it would imply that an acceleration could take place in the path of the arc which would lead to asingularity of infinite velocity.

What these formula are about is the rotation of the forces as the World enlarges. The Curls are the internal rotationswhich produce an effect in the field.

There are three“fluxes” involvedV ,B andD, the velocity (density)1, the Electric flux density and the Magneticflux density. The“click” causedδD/dt by is what would come from a circuit when it is suddenly switched off. Itis the radio signal part of the equation.

Table G-2. The magnetic, electric and gravitic inverse square laws

Magnetic Electric Gravitic

Fm =1

4πηp1p2

r2Fe =

14πε0

q1q2

r2Fg =

14πζ

m1m2

r2

Theinverse square laws are shown inTable 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. (SeeAppendixF and notes).

141

Appendix G. Grav-electromagnetic relations

Another set of equations that relate these three fields is shown inTable G-3. The gradient is like going up ahillside.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

H = −gradVma

E = −gradVe Fg = gradVg

Notes:a. The steepest slope on a hill-side shown as contours on a map (gradV = iδV/δx+ jδV/δy + kδV/δz).

We see here that the potentials, which are a part of the Direction belonging to the Newton Law field are relatedto the forces that exist with with them in a simple way. The steeper the gradient of the potential (Direction) thegreater the force. These three fields are hill sides in the space, albeit a three dimensional hill. What the Law fieldexpresses is that the surface of the hill can be represented by the the isoclinic contour lines and their orthogonalrisers which describe the rate of incline. What we have done is to describe the shape of the hill. We might easilybe 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 mea-surement. The Magnetic and Electric fields are viewed from the outside looking in whereas the gravitational fieldis viewed from the inside looking out. To relate all three, two must be of opposite sign. Another way of expressingthis is to realise that the negative sign means that the first two are asymmetric and will combine with separatefields in a symmetric way whereas the gravitic field is symmetric and combines asymmetrical.

Notes1. We live at a particular velocity. Our actual motion consist of the components of rotation of the Earth, around

the Sun, around the galaxy, etc. So, all velocities are relative and we may see this as the relativedensityofthe velocity at which we live. As we increase in velocity thisdensitychanges and we become aware of newobservations. Velocity is the flux of the gravitational field.

142

Appendix H. Spherical surface or sphere

Figure H-1. Elements of a sphere’s surface

Figure H-2. Elements of a spherical surface

The difference between a sphere and a spherical surface is shown inFigure H-1andFigure H-2. The line elementsthat can be described over the surface of a sphere make an angle at the surface which is the same as the angleexpressing latitude. This is not so in the case of a spherical surface where the angle at the surface can be any whichis self consistent with a representation. In the case of the Law World the angle at the surface is always orthogonalto the line element.

143

Appendix I. Linear algebra relationsThesubject of Linear algebra is too deep to be discussed in any way other than a mathematical text book. HoweverI consider it possible that we can glean the essence of the subject from a glance at some of the important theoremsof the subject.

Table I-1. Algebraic Structure

TheRing The Absorptive element of theLaw fieldforms aring underthe definition of analgebraic ringR. If a world surface is formed in which the ring is the great circlethen the small circles are thesub-rings.

TheField A ring with a unit element is called afield if every non-zero element ofR has amultiplicative inverse. The reciprocity element of theLaw fieldforms a field. Afield is necessarily anintegral domain.

TheGroup The First Law Field forms agroup.

Moduloor Clock Numbers Part of the thirdness in the Law Field.

The table below shows the relationship the first Law Fields and the various classes of complex numbers andoperators.

Table I-2. Comparing Law Fields with Linear Algebra

Law Fieldelement

Class ofcomple xnumbers

Behaviourunderconjugation

Class of operators in afinite dimensionalinner product space

Behaviourunder theadjoint map

Law

Reciprocalelementof the:

upperfield

Unit circle(mod iz = 1)

z = 1/z Unitary operators, complex T* = T−1 Inver-sion

lowerfield

(mod z = 1) Orthogonal operators, real

TheReal/Imaginaryelementof the:

upperfield

Real axis z = z Symmetric operators,complex

T* = T Conver-sion

lowerfield

Symmetric operators, real

The Ab-sorptionelementof the:

upperfield

Imaginary axis z = −z Skew symmetric operators,complex

T* = −T Absorp-tion

lowerfield

Skew symmetric operators,real

144

Appendix J. Table of delta valuesThetable shows the first fourq values of these constants. The particular value that belongs to the Sommerfeld fine-structure constant is the value whenp = 1 andq = 1. A comprehensive list is contained in the Note 1850. Thereare proposals for calculating the scalars for atoms heavier than hydrogen. The value used for the fine-structureconstant is that produced by the−δ1

1 term (0.085786438).

Table J-1. Definition of the delta functions

+δqp = p+ q/2 +√p(p+ q) −δqp = p+ q/2−

√p(p+ q)

Table J-2. The delta values forq = 1

p q p+ q/2√p(p+ q) +δqp

−δqp 1/+δqp 1/−δqp0 1 0.5 0.000000000 0.500000000 0.500000000 2.000000000 2.000000000

0.5 1 1 0.866025404 1.866025404 0.133974596 0.535898385 7.464101615

1 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

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 1 7.5 7.483314774 14.98331477 0.016685226 0.066740906 59.93325909

8 1 8.5 8.485281374 16.98528137 0.014718626 0.058874503 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



−δqp 1/+δqp 1/−δqp0 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

145

Appendix J. Table of delta values



−δqp 1/+δqp 1/−δqp0 3 1.5 0.000000000 1.500000000 1.500000000 0.666666667 0.666666667

0.5 3 2 1.322875656 3.322875656 0.677124344 0.300944153 1.476833625

1 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



−δqp 1/+δqp 1/−δqp0 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 K. The magnitudinal conicsThereare at least two historical references to the magnitudinal representation of conic sections. There isBasicPhysics of Atoms and Moleculesby U. Fano and L. Fano1 and D Goodstein and J Goodstein inFeynman’s LostLecture2. Both provide examples of the use of such a technique. The former use the method to derive RutherfordScattering 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 ReneDescartes. I have chosen a different method because it seems to produce more useful results and is easier for themind to assimilate.

In the Cartesian method we describe shapes by algebraic equations and draw them on pre-set axes. The equationsare the magnitudes and the drawings are the directions. We have historically attached a greater significance tothe magnitudes than to the directions. However, there is no foundation for believing that one should fare moresignificant that the other. There are many ways of representing our reasoning processes. The trick is to find theone that is most revealing and gives consistent and contiguous results. Results that can be connected indefinitelyand 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 shownhow to separate the Magnitudinal form of our ideas from their Directional form. Here is an example of this processwhich is applicable to the conic sections. The table inFigure K-1shows the two forms. There is more than onemethodof 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 circleto 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 and Directional form

Notes1. [Fano59]

2. [Goodstein99]

148

Bibliography

[Carse87] Finite and Infinite Games (http://www.randomhouse.com/BB/catalog/display.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 history of time (from the big bang to black holes) (http://www.amazon.co.uk/exec/obidos/ASIN/0553175211/qid=1016635382/sr=2-2/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 formx cosα+ y sinα = 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 angle2θ”, 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 TrustCompany.

[Note612] “Squaring the circle and quantising space (blue geometry)”, Telle A Ostberger, The Educational TrustCompany.

[Note665] “Sommerfeld’s fine-structure constant from the blue and red geometries”, Telle A Ostberger, TheEducational Trust Company.

[Note666] “Sommerfeld’s fine-structure constant from the first delta value”, Telle A Ostberger, The EducationalTrust 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 Com-pany.

[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.

[Note1210] “Extending the half angle geometry to all values”, Telle A Ostberger, The Educational Trust Company.

[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-payingpublic”, 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 EducationalTrust 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”, 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

AAbsorption

The process in which a quantity or a vector direction disappear by virtue of have equal and opposite propertiesof exactly the same character by definition. Positive and negative rational numbers of the same magnitudeabsorb. Irrational numbers, ones that have not cut-off or ending cannot be said to absorb.

151

Glossary

Absorption matrix

A two dimensional array of numbers or quantities in which the row and column vectors absorb. The deter-minant 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 inthe Krebb cycle of the digestive system, the process by which we extract energy from food. During the stagesthe Hydrogen atom is split into its proton and electron for recombination.

Agenerate

The concept of building up a process or system towards its state ofCondensation, condensing it and repeatingthe process again at its next level ad infinitum. This is a negative entropic process. The complementaryprocess isdegeneracy, 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 directionallyπ 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 la-belled E, P, N and back-E, back-P and back-N.

152

Glossary

ATP

AdeninineTri-phosphate. SeeADP.

BBack-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 reversedcharacter 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 thedirections 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.

CCircel

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

153

Glossary

Colour nomenclature

Thenomenclature adopted by Ostberger that uses blue for the vectors of formative or firstness character, redfor 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 ofcomplex variables. It does not imply continuity necessarilyfor 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 uponits surrounding.

Contra momentum

Any momentum which gyrates about some centre which its co momentum into equilibrium. Our body mo-tions 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 contrapostingor entryin a book of accounts. It is negative by convention and often shown red. Liabilitiesare primarily credit numbers as are Sales.

Curl

Another term for Rotation in vector analysis. It used to be calledRot. The term includes any kind of rotationand is mathematically well defined.

Curvature

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

Curved vectors

Why not? Satellites have elliptical ones.

DDebit

Thepostingor entryin a book of accounts. It is positive by convention and often shown black or blue. Assetsare primarily debit numbers, as are inventories.

Delta geometry

The special geometry which Ostberger used to represent the energy changes in atomic phenomena.

Delta values

The greek letterδ was used by Ostberger to define his table of constants which lead to the discovery of theSommerfeld fine-structure constant, a natural number.

Descartes, Rene

Rene was the man who set us on the path of analysing the wrong kind of spaces.

155

Glossary

Dirac, Paul A M

The Lucasian Professor of Mathematics who, like Steven Hawkin, could see through the curtain of theundiscovered. He gave us a new vector mathematics with Bra and kets which led him to the vector solutionto the Hydrogen atom.

Direction

Any phenomena can have a directional aspect. The term is used by Ostberger with an upper caseD to relateto a phenomena of nature having a directional character.

Directional character

In the nature of yin.

Div

A vector term which is well defined in mathematics to identify the components of an the expansion orenlargement of a spacial surface.

Double entry

I trade with you. You give me one guinea. You are one guinea less AND I am one guinea more. (Unless wecheat!) There is only one coin.

EEinstein, Albert

Evidently did not understand double entry but did a sterling job. Thanks.

Einsteinian dimension

Any of the Four Directions that coincide at one point orthogonally. SeeAppendix D.

Electrics

The observable effects of all phenomena of Physics which have electrical characteristics. These includeelectric force, electric intensity, charge, current, resistance, capacitance, inductance, ion exchange, dielectricproperties, potentials, discharges, voltages, XXX.

156

Glossary

Electron

Thehalf unit spin particle which is the sole cause of all chemical study. Its Fermi particle.

Element line

A general term for looking at the lines in a space.

Entropy

The state of dis-organisation of a thermodynamic system.

Euclid

Did a great job starting us all on the road to the Ostberger studies. We can’t do without him.

Extravariant

A new term used by Ostberger to describe a space in which the imaginary regions are wholly contained by afour dimensional set of elements.

FFermions

Spin half particles and multiples thereof.

Fine Structure Constant

Labelledα it is, according to Ostberger, the natural scalar with which we can measure the Hydrogen atom.

Firstness

A term used by the mathematician and philosopher Charles Saunders Pierce to describe his philosophicalthinking. Ostberger used it in his honour.

Fleming’s R and L H rule

The rules used to determine the motion, current and magnetic flux in an electric motor or generator. It worksin transformers as well. More importantly it works anywhere in the universe.

157

Glossary

Fluidic

Of the nature of materials whether solid, liquid or gaseous.

Force

The phenomena that has the potential to cause a flux to act. There is an electric one, magnetic one and agravitic one.

Four dimensions

Of two kinds. One magnitudinal in character and the other directional in character. In the former there aren-dimensions in a group but in the latter there are only four in each group. SeeAppendix D.

Frequenc y

Cycles per second or Hertz. Ostberger measures them as circular functions in a circle.

Fully occupied states

Nature, says Ostberger, has the special property that she is fully occupied. This implies that we cannot addinto her structure anything that is not a part of her structure. Distance is a man made concept; nature doesnot recognise it. The implications of this is far reaching. The suggestion is that even in the human bodythis principle continues. When a space occurs it is only recognised as something else that we call a disease.The question that arises is“is a virus simply a manifestation of a space in the biology of the body?”Theexample that Ostberger gives is that of a plan view of traffic in New York. Some Aliens are studying themotions of the little metal boxes whilst others are studying the gray ones in between. The gray ones have theremarkable property that they replicate easily and quickly but tend to disappear at the junctions. The theoryis that the junctions are holes to another underground system of traffic. Many experts have put their namesto this theory. But one has not. He says they are just spaces between the metal boxes and they are calledvehicles!

GGrad

The vector definition for the slope in a region of 3-dimensional space.

Grav-electromagnetic

The combined World of Magnetics, Electrics and Gravitics and their combinations. This includes magneto-electric effects (real), electro-magnetic effects (imaginary), electro-gravitics and gravito-electrics (see XXXEisen), Gravito-magnetics and magneto-gravitics (see note XXX).

158

Glossary

Gravitic

Theterm used by Ostberger to refer to the physics of gravitational phenomena. These include velocity (theproperty of transferring mass from one coordinate set to another), gravitic direction (gravitational potential),Newton force, torque, precession, spin, rotation, XXX.

Great circle

The circle on the surface of a sphere whose plane passes through the centre of the sphere. The Ostbergergeometry of curvatures has no centre and therefore no great circles. Instead they are called geodesics.

Groups

The Law field meets the mathematical definition of a Group.

Gyro

SeeChapter 4. Any mass that has a second moment of inertia and rotates follows the laws of the gyro.

Hh-bar

A special character first used by Dirac to linearise Planck’s constant. SeeChapter 11.

Half angle formulae

A particular set of formula in mathematics. The geometries show the properties of the tangent of the halfangle.

Hawkin, Steven

The Author of“A brief History of Time” who is the Lucasian professor of Mathematics. His foresight pre-dicted Ostberger’s work.

Heisenberg

The creator of matrix mechanics which will be even more useful in the future.

Helium

The second element of the Periodic Table. It has two electrons, two protons and two neutrons. It is almostsymmetric. Its nucleus is the alpha particle.

159

Glossary

Hilbert space

Thespace of a symmetric World geometry without the phenomena.

Hydrogen

The first element of the Periodic Table. It has one electron, one proton and no neutron. It is happy in pairs asa diatomic molecule.

IImaginary

A general term. Different terms are used in other subjects to the same concept. Notional in accounts. Spiritualin human activity. Complex in mathematics for both real and imaginary. The square root of -1 is the basis ofall mathematics that uses imaginary numbers. Imaginary things are quite tractable. If this were not so naturewould not survive.

Imaginary geometry

Not yet studied by mathematicians in the form of directions in space. Ostberger examines some imaginarygeometries and how they work.

Imaginary time

As Steven Hawkin says, if imaginary time is used in place of real time we can join classical physics withquantum physics because the fourth dimension becomes a Direction. Ostberger suggests that some peopleare genetically predisposed to thinking in terms of imaginary time and others in terms of real time. Thisaccords with Professors Carse’s book[Carse87].

Integ ers

Whole numbers. They have no decimal parts or fractions.

Intravariant

The contra World to extravariant. In systems, physics and mathematics this is the unstable set of laws. Theyare continuous in character and pass over the World surface as spirals.

Inversion

Of a reciprocal nature. The property of turning upside down.

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Glossary

Irrational number

Numberthat don’t have endings. Better defined mathematically.

KKind

Expresses an entire generic development of a geometry. Also used in tensor calculus to describe the highersets of vectors in place of the wordrank.

LLaw field

See Also:Law fields.

Law fields

SeeChapter 8. A generic set of three infinite elements (straight lines) which meet the origin at the values1, 0 and singularity (or infinity) orthogonally. The elements have the property of absorption, inversion andconversion.

Linear

Contained by a regular matrix of scales.

Lorentz

The physicist who identified mathematically the transformations that take place in objects that motion to-wards the speed of light. The Fitzgerald contraction in the direction of motion, and the time dilation. Os-tberger says that the direction of the motion has been ignored and that the traveller takes a curved pathand does not dilate or contract. However the observational information is correct and is a measure of thetraveller’s velocity. However, he will return. SeeAppendix E.

MMacroscopic disconnection

After 348 days of travel under 1g acceleration we reach the speed of light. We become electromagneticallydisconnected from Earth. We can no longer see our home. Like Columbus going over the horizon.

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Glossary

Magnetics

The observable effects of all phenomena of Physics which have magnetic characteristics. These includemagnetic force, intensity, direction, potential, field, strength, pole, flux, hysteresis, XXX.

Magnitude

The result of reasoning with numbers. Ostberger uses the term more generally than in mathematics, andreserves the termmodulusfor an unsigned number. Used with an upper caseM it relates to a phenomena ofnature having a magnitudinal character.

Magnitudinal character

Yang-like. The character of a geometry which makes it tend to be controlled by numbers rather than direction.

Matrices

Arrays of numbers, algebraic formula, vectors, differential quantities etc.

Minkowski

The mathematician who conceived of aWorld but not in geometric form.

NNecessary conditions

The conditions that are essential for a solution or formula to exist. It also applies to geometries.

Ness

See Pierce, Charles Saunders.

Neutron

The Fermi particle in the atomic structure which conveys mass and magnetic spin properties to the atom.

New rules

A vector was considered to have a magnitude and two possible, opposite directions. Ostberger says that wemust double the number of possibilities to included negative magnitudes.

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Glossary

Newton Law Field

TheLaw Field that describes Newton’s laws of motion.

next-level-up

In planning an engineering product a structure is created which breaks down into generations or families eachlevel of de-construction of the product. The reverse process of constructing the product requires reference tothe next-level-up generation.

Notebook

Telle Ostberger kept a notebook. It is voluminous. See the Summary Notebook index.

Notional

Used in accounting for money that is forecast. Also Risk money and potential emoluments.

OOccupied

SeeFully occupied states.

Operator

A mathematical symbol that says a whole process of operations must be applied to the formula or characterswhich follow.

Orthogonal

The property of being one quarter of a rotation apart. Mathematically it can be measured in the plane by thesine of the angle being zero. If the plane is not flat another process is required.

Ostberger

Telle.

Ownership

The living process in which humans retain their omnipotence over assets assigned to them eternally. A changeonly takes place when and if ever theownerrelinquishes his omnipotence of their own volition and in theirown time. All monetary gains from the use of transaction of the assets are added assets of the owner.

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Glossary

PParallel

Being directionally zero radians apart indefinitely. This applies to lines, surfaces and volumes.

Parallel principle

All vectors having the same direction and magnitude which are parallel are the same vectors.

Parametric equations

A particular method of expressing algebraic forms. SeeFigure 6-14.

Pen stand note

Note 103. SeeChapter 2.

Penrose, Roger

Wrote“The Emperor’s New Clothes”which contains useful directional hints.

Planck, Max

The mathematician who formulated the ideas of his time into a quantum of energy. Ostberger identifiesPlanck’s constant as the circumference of the smallest circle whose area is the energy per unit frequency.

Polar

In Cartesian thinking, a set of ordinates which uses a radial vector and an angle to define the space from acentral point. The Pen Stand Note shows it to be under subscribed in coordinates.

Precession

The peculiar and interesting property found in a gyro which makes it rotate under natural laws. Ostbergeridentifies this with the Directional property of Newton’s Laws.

Proton

The Fermi particle that conveys positive electric charge and mass to the atomic elements.

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Glossary

QQM

QuantumMechanics.

RRational number

Numbers that end.

Real

Tangible things and ideas that can be directly seen or measured.

Reason

Jesus said,“It will be said that they killed me without Reason”. Reason is the tool of our survival. If we donot reason with one another we we become extinct.

Reciprocity

The property of turning upside down, exchanging by inversion. For everyupconcept in a set there is anotherset with a correspondingdownconcept.

Red geometry

The colloquial name given to a particular 600 series geometry which is useful for the representation ofmagnitudes in the atomic structure.

Red Shift

The relativistic effect on the frequency of light arriving from distant objects which is thought to be a measurethe velocity of the object away from the Earth. It is interpreted by using the idea of the Doppler effect.Ostberger says that this is not a measure of the velocity of the objects but rather a measure of the amount ofrotation of the light from its object along the geodesic.

Representation

A model in any stage of development.

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Glossary

Riemann, George Bernhardt

The Father of the mathematics of Relativity, tensor calculus and 4-dimensional geometry in the algebraicform.

Riemannian Dimension

SeeAppendix D. The many magnitudinal dimensions in a space. Riemannian dimensions are separated byorthogonality.

Rope-maker

A character used by Ostberger to set the scene of theorems which were directional in character.

Rope tricks

Ostberger’s way of depicting line theorems.

Rotation

The motion in space about a place outside the line element is an external rotation. If the place is in or on theline it counts as an internal rotation.

SScalar

A number scale seemingly having no directional characteristic.

Schroedinger, Edwin

Creator of the differential calculus approach to the atomic structure particularly the hydrogen atom.

Secondness

A term used by Charles Saunders Pierce together with firstness and thirdness. Adopted by Ostberger toindicate the generic levels of the World constructions. Secondness isOperation.

Sommerfeld

Discovered the natural number that scales the Hydrogen atom known as Sommerfeld’s fine-structure con-stant. Ostberger identifies this as coming from a geometry which he describes and calculates.

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Glossary

Spin

Theattribute of particles represented by a vector that identifies its particular gravitic characteristic that ap-pears to be a rotation. Similar axial vectors are used by Ostberger to identify charge and magneton spin.

Spiritual

The directional side of human beings!

Stages of development

SeeAppendix B.

Standar d forms

Geometric representations can be categorised. They may be representing a unique set of conditions or aspecial space. They may be something less of a more general character. But first they have to be discovered.They can be placed in a library with a date and Author and updated by the next Author as they are improvedor perfected. This process is the library of Standard forms.

Sufficient conditions

No other conditions are required. A term used mathematically.

Surds

Unending numbers.

Symmetric

Capable of description using a mirror.

TTensor

A vector of rank higher than one. Because there are an infinite number of vectors in a line element thereare various geometric representations. They divide into series and parallel groups. They possess magnitudeand direction. The electromagnetic tensor, for example, describes the complete range of frequencies of allelectromagnetic phenomena that we observe.

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Glossary

Theorems directional

Mathematicaldeductions that can be made without representing numerical values, but instead, relying onlyon the geometric form.

Theorems magnitudinal

The usual mathematical theorem.

Thermodynamic

Belonging to that World.

Thirdness

A term used by Charles Saunders Pierce together with firstness and secondness. Adopted by Ostberger toindicate the generic levels of the World constructions. Thirdness isInterpretation.

Torque

A rotational force. Force applied about a lever. One of the two characteristics of the output of a heat engineused for mechanical drives. The other is speed or frequency.

Trigonometric

About the measure and mathematics of triangular forms, usually in a plane.

UUsership

The living process in which humans retain their omnipotence over assets assigned to them for a period oftime determined by their use of them. A change takes place according to usage (see Ostberger notes 2000).All monetary gains are part of a usage measurment of the user.

VVector

The representation of the magnitude and direction of a phenomena using a line element. The most commonstyle is to let the length represent the magnitude. Other methods are used by Ostberger.

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Glossary

Velocity

A vector quantity giving the speed and direction of a mass. Ostberger’s definition is“that phenomena whichtransforms gravitic energy from this set of coordinates to another”.

WWhite hole in hydrogen

Hydrogen is the only atom in the periodic table without a neutron. It is the first element in the table. Perhapsthe hole is a white one? In which case the Sun is a white hole energy source.

World of the first kind

See text. The 4 dimensional World which is either intravariant or extravariant.

Worlds of the second kind

A geometric model which contains one intravariant World superposed with six extravariant Worlds in sixlocations, each a different orientation.

Wunman

A one dimensional fictitious character that lives in a two dimensional world.

YYang

From Chinese Taoism. The male, magnitudinal, often positive, left thinking characteristic half of an idea orrepresentation. That part of the Tao that combines with the Yin to create the structure of nothingness fromwhich all things are derived.

Yin

From Chinese Taoism. The female, directional. often negative, right thinking characteristic half of an idea orrepresentation. That part of the Tao that combines with the Yang to create the structure of nothingness fromwhich all things are derived.

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ColophonPrior to this book the contents were previously copyrighted in 1971 and again in 1991 with the object of provingits existence in the face of a total lack of communication from any quarter. Ostberger sent out hundreds of com-munications to universities and potentially interested parties. He never received a single reply in thirty years. Hevisited universities and received a welcome that he describes as“worse than a hunter greeting an animal”. Hesays,“I greet my business competitor, he who wishes that I was not there, with a magnanimity which is a thousandtimes greater than these supposed intelligentsia, with whom I have no competition and to whom, I had come togive a gift. I am astonished at their inhumanity”.

He requests the following:“I request that Universities draw up a code of conduct which ensures that all personswho are members of the tax-paying public approaching them in humility are treated with humanity and respectand I have laid the foundation of such a code in my[Note1466]. ”

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Documents

FINITE AND INFINITE WAYS: The Ostberger Story by Bob Beaumont