# Unravelling Relationships; Pythagoras Reconsidered

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UNRAVEllING

RElATIONSHIPS

PYTHAGORAS RECONSIdEREd

Matthew Kaser

Published by Scribd

A

B

C 2012 Matthew Kaser

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Matthew Kaser58 West Portal Avenue #121

San FranciscoCA 94127

USA

Copyright 2013 Matthew R. Kaser

All rights reservedincluding the right of reproductionin whole or in part

Published by Matthew Kaser at Scribd

Includes bibliographic references

First edition

ISBN 978-0-9891749-0-9ISTC-A02-2013-00000214-3

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Unravelling Relationships

Pythagoras Reconsidered

Matthew Kaser

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Introduction

Consilience, the term conceived by William Whewell in 1840 1 and as used more broadly by E.O.

Wilson in his popular book of the same name, embodies an unknown algorithm, or perhaps a set

of common functions or parameters, that are the essence of the Universe2

. Wilson conceivedthat defining or uncovering the consilient truths will lead us to understand how the Universe is

organized. The only thing missing from his excellent synthesis was any hint of what the nature

of this truth might be; he believed it would not be resolved in his lifetime and that it would most

likely be rather a rather complex mathematical function in form. He urged scholars of all

disciplines to collaborate and isolate a functional form or set of parameters that could be used

irrespective of the applicable circumstance. Fortunately, I think we now have the answer.

Remarkably, that answer has been the cornerstone of mathematics and many principals of

geometry for millennia. We just were not looking for it there, probably due to its simplicity and

apparent universality.

Richard Dawkins, in the introductory chapter of a book that changed the way we biologists

viewed the reach of the gene in a fundamental way, noted that it is possible for a theoretical

book to be worth reading even if it does not advance testable hypotheses but seeks, instead, to

change the way we see 3. I am hoping that this book will change the way that all of us see.

I would like to present to you, the reader, the simple, well-known mathematical theorem,

Pythagoras Square Theorem, which can probably be used in most instances to define a

relationship between any two points in space, from subatomic particles to supra-galactic

structures and beyond. By relationship, I mean the manner in which the elements at those two

points in space interact with each other and how the degree of that interaction may be further

reflected at higher levels of complexity.

1 Whewell, W. (1840) The Philosophy of the Inductive Sciences, Founded upon their History. Longmans, Green, andCompany, London.2 Wilson, E.O. (1998) Consilience. The Unity of Knowledge. Vintage Books (Random House Inc.) New York NY,paperback edition.3 Dawkins, R. (1982, 1999) The Extended Phenotype. The Long Reach of the Gene OUP Oxford New York p. 2.

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I would predict that this proposed conjecture may be used to further generate mathematical

relationships on progressively higher scales, such as atoms, molecules, biological cells,

organisms, planets, solar systems, galaxies, supra-galactic structures, and, more particularly, the

cosmos 4. Perhaps this may be the consilience we have all been looking for? I shall leave the

reader to decide.

4 Presented as an abstract published at the first Conference of the World Knowledge Dialogue (2006) held inSwitzerland. E.O. Wilson was the Plenary Session speaker.

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Chapter 1

The Square (or Inverse-Square) Law is Ubiquitous

It is nevertheless remarkable how this ancient

theorem still plays its fundamental part now atthe infinitesimal level

The Road To Reality Roger Penrose,London: Jonathan Cape (2006)

To paraphrase the introduction in Richard Dawkins book The Extended Phonotype 5 this work

seeks . to change the way we see and also to inform and educate the reader in ways not yet

having being expounded upon. The aim of this book is to lay a foundation for future research

and technologies and it should provide a framework in which many disciplines will advance their

fields, and ultimately will be used as a model for all interactions.

When I was about 11, my parents allowed me to walk to and from school, which was about a

couple of miles away. Unlike the usual grid pattern of streets in the US, the roads in Oxford

were a mixture of straight sections, curves, staggered cross-junctions, &c., probably due to the

fact that north Oxford had been laid out over fields and meadows. I particularly remember that I

began to be beguiled by triangles, inasmuch that instead of treating a simple crossing of the street

as taking the shortest distance between the two sides, I would think ahead to where I wanted to

be after I had crossed the street. Therefore I would pick a spot further down on the other side

and head to it, crossing diagonally. In Britain, even in those areas of suburbia, there was usually

very little traffic, so I could presume that I would be fairly safe. I began by thinking that the

most efficient way across would be a straight diagonal, i.e., at 45. But then I realized that I

could save more time by taking a longer route across, the hypotenuse being relatively shorter and

shorter compared with the sum of the two sides, and so began to extend the hypotenuse of the

triangle more and more. Walking for a longer time in the road also exposed me at greater risk to

traffic (even bicycles) so I ended up taking as long a diagonal as I could, given local traffic

conditions, the cost/benefit ratio in north Oxford being clearly fairly low.

5 Dawkins, R. 1982, 1999 The Extended Phenotype. Oxford University Press, Oxford, England, p.2.

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Vectors

A vector, or more accurately, a Euclidian vector, is a geometric object that has magnitude and

direction. In general, one vector in one plane (or dimension) indicates the virtual movement of a

point in space from one position to another. As shown in Figure VV, the movement (or

transformation) of an object from point A in space to another point C in two dimensions can be

described by a combination of two transformations, one in a first dimension (red, x units; A

B), the other (blue, y units; B C) in a perpendicular dimension, the resulting movement (or

transformation) being mathematically described as (x, y). This will be familiar to those readers

who have studied matrices and geometry. As shown in Figure 1.1, the vector combination and

effective movement of the object from point A to point C can be described by a third vector,

which to all intents and purposes, is essentially the hypotenuse of the triangle ABC. Vectors

may be used to illustrate and describe movement of an object from one point to another in more

than two dimensions, of course. In addition, they are used to describe magnitudes and directions

of other parameters, such as force, electric and magnetic fields, gravitational fields, momentum,

etc. in both Euclidian and pseudo-Euclidian space (e.g., Minkowski space-time 6). Interested

readers are encouraged to go to other sources for a more in-depth understanding of vectors and

vector fields.

Figure 1.1

6 For a through explanation of Minowski space-time, see Penrose, Roger (2010) Cycle of Time Alfred A. Knopf,Random House, Inc. New York NY pp. 80-95, 108-109. See also rescaling of identical structures in Minowskispace time: g 2 g . ibid, p. 89.

x

y

(x, y)

A B

C

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Squared distance parameters are fundamental mathematical and physical concepts

A plethora of examples from different disciplines of science suggest that the concept of

square/inverse-square laws as likely to be ubiquitous holds true. Many of these reflect

relationships between mass and energy or between energy and velocity. They include:

(A) general relativity field equations (+/- c2d2); force of gravitational attraction

between two bodies proportional to the inverse square of the distance between them (Fg =

Gm1m2/d2);

(B) for a mass uniformly accelerating on a friction-compensated slope (inclined plane),

the distance travelled is proportional to square of the time, in this case time t is a distance in

four dimensions (d t2);

(C) the relationship between rest energy and rest mass of a body at different points in

space-time as shown in Einsteins principal of the equivalence of mass and energy (E = mc 2)

and special relativity: that is, bodies in rectilinear and non-rotational motion relative to each

other 7;

(D) conservation of energy (attrib. Leibniz), e = mv2 for a set of particles; kinetic

energy (Ek = 1/2mv2 )8;

(E) the relationship between power, resistance, and current or voltage (P = I2 R; P =

V2/R);

(F) Coulombs Law (Fe = K/d2);

(G) clustering algorithms us