Unravelling Relationships; Pythagoras Reconsidered

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    1

    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