Unravelling Relationships; Pythagoras Reconsidered

download Unravelling Relationships; Pythagoras Reconsidered

of 33

Transcript of Unravelling Relationships; Pythagoras Reconsidered

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    1/33

    1

    UNRAVEllING

    RElATIONSHIPS

    PYTHAGORAS RECONSIdEREd

    Matthew Kaser

    Published by Scribd

    A

    B

    C 2012 Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    2/33

    2

    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

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    3/33

    Unravelling Relationships

    Pythagoras Reconsidered

    Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    4/33

    1

    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.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    5/33

    2

    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.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    6/33

    3

    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.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    7/33

    4

    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

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    8/33

    5

    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 used in statistical analyses (for example, K-means clustering 9);

    and

    (H) Lanchesters Square Law (PCG/ G210).

    7 Note that the magnitude of distance between two points using the Lorentz Transformation is in the form (1 v2/c2); see Einstein, A. (1920) Relativity, 12, 13.8

    I might suggest that to be consistent with the equation, E = mc2, the remaining part of the kinetic energy ( Ek )

    equation could be Eb = 1/2mv2, where Eb may represent the nuclear and internuclear binding energy of the mass;

    the complete relationship is therefore considered to be Ek+b = mv2. Therefore a mass subject to Newtonian

    mechanics only gives up its half its total energy when brought to a halt; a mass subject to quantum mechanics willgive up all its energy when brought to a halt, for example a photon colliding with a photoreceptor molecule in theeye. This implies that a Newtonian mass m moving at c would therefore suffer considerable mass attrition due tocollisions with smaller masses, such as cosmic dust, since c cannot change.9 MacQueen (1967) "Some Methods for classification and Analysis of Multivariate Observations" 1. Proceedings of5th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. pp. 281297;Alon et al. (1999) Proc. Natl. Acad. Sci. 96: 6745-6750.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    9/33

    6

    (I) Datsko and Kopylov note that SEMWs (Surface Electromagnetic Waves) ... are of

    practical interest because their energy decreases in inverse proportion to the distance from a

    pointlike source while the energy of bulk electromagnetic waves (BEMWs) decreases in inverse

    proportion to the distance squared to the source. 11 .

    It should also be noted that in the field of ecology, Botkin, Janak, and Wallis 12 developed a set

    of algorithms that model tree and forest growth, all of which comprise square-law parameters of

    distance (see also Liu and Ashton, 1995 13 for other similar examples). Furthermore, this

    concept may also be applied anew to other physical relationships: for example, an alternative

    resolution to Maxwell's Equation comprising a three-dimensional matrix has been proposed

    (Earle Jennings, personal communication, 1999). I consider it likely that it should be possible to

    convert Jennings matrix to a function proportional to the square of the distance as posited here.

    Let us review at this stage the various fundamental equations that have been determined both

    empirically and theoretically for a considerable number of mathematical and physical concepts

    that have enable us to understand how the Universe and Nature function.

    Table 1.1 lists a few exemplary equations as well as their dimension value (where symbols 1d to

    4d represent the first to the fourth dimensions).

    10 Lanchesters Square Law (1916) states that the combat power PC of a given group G is not the sum of its groupsize, but group size squared, G2.11 Datsko and Kopylov (2008) On Surface Electromagnetic Waves Physics-Uspekki vol. 51(1), pp. 101-102. (Seealso their reference 3 attributed to Wyle).12 Botkin, Janak, and Wallis (1972a) Some ecological consequences of a computer model of forest growth J. Ecol.60: 849-872; ibid(1972b) Rationale, limitations, and assumptions of Northeastern forest growth simulator IBM J.Res. Dev., 16: 101-116.13 Liu and Ashton (1995) Individual-based simulation models for forest succession and management Forest Ecol.Management 73: 157-175.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    10/33

    7

    Table 1.1

    Name Function Square Parameter Dimension Value Notes

    Principal of massand energy

    E = mc2 Velocity 4d

    Kinetic energy Ek = 1/2mv2 Velocity 4dElectrical resistance P = I2 R Velocity 4d

    (LorentzTransformation)2

    1 v2/c2 Velocity 4d

    Coulombs Law Fe = K/d2 Distance 2d

    (Orbitalvelocity)2

    (2/r 1/a) Velocity 4d

    Gravitationalattraction

    Fg = Gm1m2/d2 Distance 2d

    Lanchesters SquareLaw

    PCG/ G2 Volume 3d e.g., one person 2

    x 0.4 x 0.25 m = 0.2m3 ( 0.4 m3 )

    K-means Distance 2d Distance from amean point

    R2 analysis Distance 2d Measure ofvolatility, goodnessof fit

    The established explanation for the inverse square law is that when some force, energy, or other

    conserved quantity is radiated outward radially from a point source. Since the surface area of asphere (which is 4r2) is proportional to the square of the radius, as the emitted radiation gets

    farther from the source, it must spread out over an area that is proportional to the square of the

    distance from the source. Hence, the radiation passing through any unit area is inversely

    proportional to the square of the distance from the pointsource 14.

    In my mind this explanation can best be described as an imaginative conjecture as first

    envisioned by Peter Medewar in the mid-1960s 15. He explained such conjectures thus:

    Hypotheses and other imaginative exploits are the initial stage of scientific enquiry. It is the

    imaginative conjecture of what might be true that provides the incentive to seek the truth and a

    clue as to where we might find it.

    14 http://en.wikipedia.org/wiki/Inverse-square_law15 Medewar, P. B. The Art of the Soluble, London: Methuen 1967.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    11/33

    8

    In the past, many other authors have left the nuts and bolts describing the gist of their new theory

    to the final sections in a book, using the main body to develop arguments and evidence from the

    work of others. As a reader, I become frustrated with this approach and yearn to discover the

    new idea at once. I will not disappoint you here.

    Whilst this explanation may be true it does not discount the fact that the Pythagorean Theory also

    bases a relationship between two points upon the square of their distance.

    I therefore posit that the instantaneous relationship is equal to the area of space-time between

    the two points; being instantaneous, the spacetime area can be in a different plane at any one

    moment however, the net result (mean area) for two stationary points is the same.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    12/33

    9

    Chapter 2

    Fifteen seconds of inspiration

    In Xanadu did Kubla Khan

    A stately pleasure-dome decree:

    Where Alph, the sacred river, ran

    Through caverns measureless to man

    Down to a sunless sea.

    Kubla Khan, A Vision Samuel Taylor Coleridge(1797, 1816)

    The image of the triangle and its walls wafted into my consciousness. For a few seconds, I

    probed the image, letting my mind drift around the shape. With a start, I understood what I was

    seeing. I realized that I must get this down on paper otherwise it might be lost forever. Quietly

    getting out of bed, not disturbing my sleeping spouse, I crept downstairs to our den and sat down

    at the computer. I wrote down the Pythagorean Theorem.

    Then I got out a scrap of paper and drew this:

    Figure 2.1 Figure 2.2

    Lying in bed, I had begun to conceive of the classical Pythagorean triangle with its attendant

    squares lined up along each edge. I then imagined those squares hinging up along the line of

    each side of the triangle creating virtual two-dimensional walls in place of the lines connecting

    A

    B

    C

    A

    B

    C

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    13/33

    10

    the points. I had suddenly I realized that if one could imagine traveling between each of the

    points (A to C or A to C via B) you would effectively be traversing within the same space of the

    two-dimensional wall, whichever path you took. Imagine that the lines AB and BC are the two

    vectors that define the line AC; conventionally they are perpendicular to each other. I imagined

    that they neednt be perpendicular but could be at any angle to one another, as shown in Figure

    2.3. It all fell into place during those ten to fifteen seconds: whichever pathway one took,

    irrespective of the vector, one would ALWAYS pass through the same effective volume of the

    wall(s) connecting A to C (Figure 2.4).

    a b c d e

    Figure 2.3

    Figure 2.4I had imagined that the Phythagorean triangle consisted of walls built perpendicular to and upon

    each line, each equivalent to its square value in dimensions. Through these walls was the

    A

    B

    C 2012 Matthew Kaser

    C

    AA AA

    B

    B

    B

    B

    CCC

    C

    A

    B

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    14/33

    11

    connection between points A and C and between A, B, and C: they were equally similarly

    connected to one another, having the same area; passing through each wall(s), either way (from

    A to C), was no different that passing through the other (A to C via B). By passing through, I do

    not intend to mean that an object or particle actually travels along a particular pathway of

    through a particular area, but merely that the walls represent the potential interaction or

    interactivity between the points A and C. Essentially, the walls having a dimension equal to the

    distance squared link points A and C either directly or through point B.

    I drew a few more triangles, where the angle at point B (B, ) was not a right-angle and ran

    through a few attempts to show that the concept probably applied to all B.

    Figure 2.5

    It seemed correct even if the angle was not a right-angle, the total volume of the squares (i.e.,

    the sum of the square of the distances) would always equal the square of the distance between

    points A and C. This meant, I realized, that a relationship between any two points in the

    Universe could be defined a being proportional to the square of the distance between them. This

    is illustrated by the Pythagorean equation:

    z2 = x2 + y2 (1)

    It was likely that this was a fundamental concept of all points on a plane. The realization that I

    seemed to have unlocked one of the keys to understanding the Universe was profound. I needed

    to tell everybody. No, but wait; they would think I was a loony unless I had prepared the

    A

    B C

    2012 Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    15/33

    12

    discussion and lead-in properly. I knew that I had to develop the concept in greater detail and so

    spent the next twenty-four hours of my free time reviewing geometry texts and running through

    different numerical values to see if they gave the correct answer. These twenty-four hours were

    actually spread out over a period of about four years in total and was spent using brief periods

    late at night when I had time to cogitate on the matter. During that time I explored the concept

    that include all angles at B and realized that the fundamental trigonometric equation

    z2 = x2 + y2 -2xycos (2)

    (where is B) was the one to apply. I was also side-tracked for a year or so (i.e., an hour or

    two of real-time work) by making the incorrect assumption that if applied to three dimensions

    the general equation would be

    z3 = w3 + x3 + y3 (3)

    Even my car-pool buddies to and from Palo Alto were unconvinced when I asked for their

    comments: What an absolute load of old rubbish!! exclaimed my nuclear physicist car-mate

    from the back of the car. He was right. I realized how I had goofed even before I got the

    rejection card from Nature in February 199916 . By that time I had realized that the solution for a

    three-dimensional format was merely creating two perpendicular triangles, the base of one

    triangle being the hypotenuse of the other, thereby preserving the original Pythagorean theorem,

    as shown in the next drawing.

    Figure 2.6

    16 My recollection is 1999 but the postmark also suggests Feb 9 98.

    B

    A

    C

    D

    2012 Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    16/33

    13

    It should be now obvious to the reader that the concept can also be applied to the next dimension,

    that of time, as illustrated in the next sketch.

    Figure 2.7

    In this case, the dotted lines can represent time and the sketch shows the relationship between an

    entity at point A at time TA (e.g., at t = 0) compared with the same entity at point E (A')17 at

    time TE (e.g., t = 0 + n). The parameter of line AE (AA') is therefore velocity and it can be

    deduced from the other metrics in the other three dimensions (1d, 2d, 3d) and the units of time in

    the fourth dimension (4d). This sketch shows that any of the parameters of the dimensions up to

    at least four are each dependent upon those of the immediately previous dimension, i.e., the

    hypotenuse of a triangle in d is one of the bases of the triangle in d+1. The conjecture predicts

    that additional dimensions might be similarly dependent, but these are not discussed here.

    These manifestations into the third and fourth dimensions has been known for the Pythagorean

    Theorem for several centuries having first been described by Clairaut and probably well known

    to Descartes and Fermat

    18

    . However, as far as I am aware, those concepts had not beenconceived as a systematic and invariant series of relationships between the various points.

    17 Here the point letter prime (A') represents the same point A at time t, when point A was at t = 0.18 Clairaut, A.C., (1731) Recherches sur les Coubres Double Courbure; quoted in Maor, E. The PythagoreanTheorem (2007) Princeton University Press, Princeton NJ, pp.134 and 139.

    B

    A

    C

    D

    A' or E

    1d

    4d

    2d

    3d

    2012 Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    17/33

    14

    As an aside, remember that the path of an orbiting body around the central body over time tis

    always equal to the same area between the two bodies over that time. Keplers second law states

    that a line joining an orbiting body with the orbited body sweeps out equal areas during equal

    intervals of time 19; Keplers third law states that the square of the orbital period of an orbiting

    body is directly proportional to the cube of the semi-major axis of the orbit 20.

    19 Kepler, J. (1621) and (1995) Epitome astronomiae Copernicanae (Epitome of Copernican Astronomy) Book. V(translation by Charles Glenn Wallis) Prometheus Books, Amherst NY pp.138-143 & 153.20 Kepler, J. (1619) and (1995) Harmonice Mundi (Harmonies of the World) Book V (translation by CharlesGlenn Wallis) Prometheus Books, Amherst NY pp.181-183, 213-215, & 235.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    18/33

    15

    Chapter 3

    The Universal Square Law Conjecture Exemplified

    The Book of Nature is written in the language

    of mathematics.Il Saggiatore, Galileo Galilei (1623)

    The Pythagorean Theorem can be expressed as

    x2 + y2 = z2 (1)

    This ancient equation forms the basis of the following conjecture The relationship between two

    points in space is proportional to the square of the distance between them. As shown in the

    previous chapters, in my mind it is extremely probable that this conjecture can be applied to any

    relationship between two entities, no matter which parameters are used, as long as those

    parameters are a measure of distance, virtual or real.

    Of note, Edward O. Wilson has promoted the concept of consilience for some years (reviewed,

    for example, in Edward O. Wilson, Consilience, ibid). Wilson has suggested that there are some

    fundamental functions which would describe interrelationships between different entities and

    that these functions would be translatable across disciplines. It is my contention that the analyses

    presented here confirms Wilson's original concept of consilience and that equation (1) represents

    at least part of this interrelationship.

    What now needs to be done is to uncover the relationships between as yet unknown entities,

    using the parameters we can now recognize as being relevant to the relationship. Without

    knowing the multiplier factor we should be able to determine a mathematical relationship

    between two points (for example, datapoints from an experiment) and, having identified a metricthat can be used to plot or model the dataset, we can therefore very easily compute the multiplier

    factor (or constant) based solely upon the square of the distance between the two data points.

    The majority of these examples have been used for centuries in some case as representing the

    fundamental laws of the Universe. It is quite likely that as we proceed further up the hierarchy

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    19/33

    16

    of organization, namely that for (1) the relationships between subatomic particles (quarks and

    gluons (also neutrons, protons, electrons), etc., and their antimatter counterparts) based upon

    type and distance (strong force), (2) the relationships between all particles based upon mass and

    distance (gravity; gravitons), (3) relationship between sub-nuclear particles, radiological decay

    (weak force), (3) the relationships between atoms for molecular interactions (electromagnetic

    forces; electron shell; electrons; photons), (5) the relationship between molecules through charge

    interactions (electrostatics), (4) the relationships between different molecules (for example, Van

    der Waals forces and electronic forces and distance) 21 , (5) the relationship between temporal

    variation between different molecules (exemplified by expression profiles of genes over time

    using clustering algorithms), (6) the relationship between temporal variation between different

    molecular pathways (hormonal control of enzyme activation and/or gene expression;

    transmission of nerve impulses over distance and consequent activation of (a) neurotransmitters,(b) ion movement, (c) gene expression, (d) protein synthesis, (e) protein modification), (7) the

    relationship between different compartments within the same organ or tissue (intracellular

    signals, brain/memory, pancreas/insulin, liver regeneration, kidney/diuresis, gut/enzyme

    secretion, etc.), (8) the relationship between different organs or tissues (intercellular signals,

    hormonal regulation at a distance, interactions between hormone, receptor, signal transduction

    molecules, etc.), (9) the relationship between different organisms (physical relationship, e.g.,

    birds flying in strict formation at a set distance apart); ecological relationships; psychological

    relationships, relationships between groups or populations, economics, etc.); (10) the relationship

    between an organism with an artificial intelligence ( la Kurzweil 22), (11) the relationship

    between an organism and a digital entity (e.g., listening to a recording of anothers voice or a

    synthetic voice 23), (12) the relationship between different nations (temporal, distance,

    psychological, economic, etc.), (13) the relationship between the terrestrial organisms and the

    underlying structures in the context of the atmosphere (soil sciences, geology, climate, humidity,

    ambient temperature, etc.), (14) the relationships between the geology/atmosphere and terrestrial

    21 See also the recent paper from Israelachvilis group (Donaldson, S.H. et al. (2011) PNAS 108(38): 15699-15704;

    10.1073/pnas.1112411108). Their equation for predicting the molecular forces between molecules is E(D) = -2i (a

    a0)eD/D

    0 , where E is energy, D is distance, a is area-per-molecule, and I is initial interfacial tension. One isled to speculate whether substituting D

    2at all places might prove a better predictor.

    22 Kurzweil, R. (2005) The Singularity is Near, Penguin, New York, NY.23 Think of Mozarts Requiem (von Karajeim, 198X) or Allegris Miserere (ed. Willcocks, 1963).

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    20/33

    17

    radiation and solar radiation, (15) the relationships between the terrestrial system and the other

    bodies in the Solar System; should I go on? 24

    I believe that if the conjecture is found to be true, then I expect that it might have an almost

    universal application in determining how interactions between two points in space may be

    calculated. These two points might be two subatomic particles, two people, two corporations, or

    even two galaxies. The following table (Table 3.1) posits a number of different disciplines in

    which the conjecture may be used. Note that these are merely examples of how the conjecture

    may be used and that many others may be considered for testing.

    Table 3.1

    The above list is clearly incomplete and that many more interactions are likely to be deduced in

    the future. In addition, although the concept appears to be, at face, a reductionist conjecture, I

    would prefer that it be treated a simple conjecture that may also apply to non-reductionist

    modeling, such as for example, how an individual suddenly conceives of a novel idea, based not

    24 Steven Jay Gould made a case that, in terms of upon which natural selection operates, there are a number ofhierarchical levels, for example, genes, cells, organisms, demes, species, and clades. In the present work, although Ihave listed them in levels of increasing complexity and organization, I would actually prefer to not place suchlevels in discrete units. Rather, I would consider that each level of organization actually blends into the nextlevel as a continuous interacting structure and that it is only human nature that seeks to compartmentalize them andattempt to identify which factors influence different elements at those artificial compartments/levels. (See, forexample, Gould, S.J. (2002) The Structure of Evolutionary Theory, Harvard University Press, Cambridge, MA,p. 681).

    Field Model Method Likely metric

    Artificial Intelligence Game Theory Linear distancePolitical Science Schema Theory

    Conflict Theory GameTheory

    Rhetoricdistance

    Ecology Game Theory Linear distance;Proteomics Pathway Analysis Spectral

    MappingMolecularinteractions

    Genomics Expression Analysis ExpressionProfiling

    Co-expressionparameters

    Metabolomics MetabolicProfiling

    Substratereaction rates

    Computationalquantum chemistry

    Electronic structure ofmolecules

    Molecularinteractions at

    atomic scaleCombinatorialchemistry

    Electrostatic structureof molecules

    Chemical space Molecularinteractions atatomic scale

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    21/33

    18

    upon an apparent sequence of logical deductions, but as an almost instantaneous gelling (? re-

    equilibrating ?) of past experiences, leading the brain to suddenly make a new model of the

    environment. This general concept has been expanded elsewhere 25.

    Figure 3.1 illustrates the concept of how the different levels of organization might be conceived,

    drawn as a column. I have labeled it as a Biosphere Column but of course this is only part of

    the column that encompasses the Universe, itself part of a similar columnar Cosmos.

    25 Kaser, M. The Brain as a Constantly Iterating Organ, in preparation.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    22/33

    19

    Figure 3.1

    One may at first consider the hierarchy as a form of Mandelbrot pattern; however this might notbe a true representation as a Mandelbrot pattern is constructed using repeats of a single form of

    an equation, whereas in the current case, despite the fact that each set of relationships is based

    upon the distance between each point, the hierarchy is based upon a multiplicity of different

    algorithms, and therefore constructing a real-time, instantaneous model would therefore be

    complex.

    Sub-atomic Particle

    Atom

    Sim le Molecule

    Molecular Complexes

    Cell

    Tissue

    Organism

    Group

    Complex Molecule

    Fundamental Particle

    Cellular Or anelle

    Multi-Cellular Complexes

    Organ

    Famil

    Communit

    S ecies

    Bios here

    A BIOSPHERE

    COLUMN

    Point of Energy/Mass

    Symbionts

    Ecos stem

    Environment (incl. eolo )

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    23/33

    20

    Chapter 4

    An Infinite Universe

    Hypotheses and other imaginative exploits are

    the initial stage of scientific enquiry. It is theimaginative conjecture of what might be true

    that provides the incentive to seek the truth and

    a clue as to where we might find it.

    The Art of the Soluble Peter B. Medawar,London: Methuen (1967)

    The Conjecture also has an additional prediction that appears to answer a question regarding the

    structure of the Universe. We can extrapolate Fig. 2.3e so that 0, see Fig. 4.1a. From this

    it seems to me evident that when 0 (e.g., Fig. 4.1b), lines AB and BC would be close to

    infinite in length, where one of the two would be marginally longer to account for the difference

    squared equaling line AC2. The Universe thus is most likely infinite.

    Figure 4.1

    a

    b

    B

    B

    C

    C

    A

    A

    2011 Matthew Kaser

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    24/33

    21

    Chapter 5

    A Universe Beneath

    I have shown that the method can be applied to more than two dimensions by simply using thehypotenuse of the base triangle as one of the vector lines in the next dimension, as shown in

    Figure 5.1. It therefore follows that we could assume that one of the base vector lines of the

    original triangle we envisioned, e.g., Figure 2.6, could also be the hypotenuse of a lower sub-

    dimension, made up of non-Cartesian dimension(s), resulting in triangle AB.

    Figure 5.1

    Of course, the reader will have also remembered that string theory also has predicted multiple

    dimensions, particularly up to 26 for bosonic strings, eleven for M-theory, and ten for the related

    superstring 26. Understanding how string theory might be relevant to this new interpretation of

    the Pythagorean theorem might then resolve problems in the physics and philosophy

    communities regarding the Anthropic Principle 27. Since all of Nature may be explained by a

    26 Susskind, L. (1993) String theory and the principle of black hole complementarity Phys. Rev. Lett. 71, 2368;Hellerman, S. and Swanson, I (2006): Dimension-changing exact solutions of string theory J. High EnergyPhysics 709: 096; Aharony, O. and Silverstein, E. (2006) Supercritical stability, transitions and (pseudo)tachyonsPhys. Rev.D75: 46003; Duff, M.J., Liu, J.T. and Minasian, R. (1995) Eleven Dimensional Origin of String/StringDuality: A One Loop Test Nucl. Phys. B452: 261-282; and Polchinski, J. (1998) String Theory, CambridgeUniversity Press, Cambridge, UK.27 Carter, B. (1974) Large Number Coincidences and the Anthropic Principle in Cosmology IAU Symposium 63:Confrontation of Cosmological Theories with Observational Data. Dordrecht: Reidel. pp. 291298; Barrow, J. D.;

    B

    A C

    D

    First sub-dimension

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    25/33

    22

    simple relationship concept and that all matter and energetic interactions have a common

    underpinning, as I have posited in this book, it should be unnecessary to recruit arguments that

    the Universe is fine-tuned for particular natural manifestations and connections in order for life

    to have come about. In my opinion, life is a sustained series of interacting chemical reactions

    resulting in continuous near-equilibrium, the reactions fueled by an external energy source. This

    would happen no matter how the interacting components behave, as they would find a suitable

    balance for the relationship anyway. This balance could be at the quantum level, the atomic

    level, the electronic and molecular levels, etc., etc. Hence, the Anthropic Principle is no more.

    Tipler, F. J. (1988) The Anthropic Cosmological Principle Oxford: Oxford University Press; see also Dicke, R. H.(1957) Gravitation without a Principle of Equivalence. Reviews of Modern Physics 29(3): 363376 and Dicke, R.H. (1961) Dirac's Cosmology and Mach's Principle. Nature 192(4801): 440441.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    26/33

    23

    Chapter 6

    Spheres

    There is nothing special about spheres. Contemporary cosmology and mathematical theorypostulates that space-time can be curved 28. It has also long been understood that the

    Pythagorean Theorem cannot be applied to a curved surface, such as the surface of the Earth.

    Figure 6.1 illustrates this conundrum.

    Figure 6.1

    However, remember that a triangle illustrating the Pythagorean Theorem is not limited to having

    a right-angle (Chapter 2, equation (2) and Chapter 4); the hypotenuse always has the property

    that its square is equal to the sum of the two squares of the opposite sides minus the correcting

    factor dependent upon the cosine of the angle B ():

    z2 = x2 + y2 -2xycos (2)

    28 For example, see Einstein, A. (1920) Relativity, 31.

    A

    B

    C

    Not a right-angle

    Not a right-angle

    Not a right-angle

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    27/33

    24

    Therefore, as the relationship between the points still hold even if the angle B is not a right-

    angle, then is surely must be applicable between points upon a curved surface. Thus it should be

    fairly simple to calculate such relationships between points upon the surface for the Earth, for

    example, whether it is using straightforward metric distance, or perhaps a virtual distance, such

    as when communicating electronically.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    28/33

    25

    Chapter 7

    Quantum Field Theory and Quantum Mechanics

    To date, theoretical physics and cosmology have not been able to reconcile Euclidian

    space/Newtonian physics (and of course, general relativity) with quantum field theory 29.

    However, if the square of the distance conjecture holds true for Euclidian space/Newtonian

    physics, then perhaps it may also be applied to quantum field theory, if the conjecture is

    universal.

    Indeed, the inconsistencies reached when attempts to reconcile general relativity with quantum

    field theory has been partially answered with the development of string theory 30. Interestingly,

    string theory predicts that there are multiple lower dimensions of space; this is compatible withthe predictions made in Chapter 5.

    Regarding quantum mechanics, more specifically relating to the Heisenberg Uncertainty, it

    might be possible to define the position of a Heisenberg object at two moments in time, and

    approximate the mean distance between those positions, thereby establishing at least one value

    for d2. The Heisenberg matrix coefficients might then be applied to the d2, the time measured,

    and then a better value for momentum (mv) might then be calculated.

    Furthermore, as I suggested in Chapter 1, in regards to Maxwells equation, the matrix

    mechanics derived by Heisenberg, Bron, and Jordan 31, might also be resolved by treating the

    matrices as encoding the position of objects in dimensional space, the objects thereby being

    resolved having dimensional distance between them.

    29 Messiah, Albert (1999), Quantum Mechanics, Dover Publications, ISBN 0-486-40924-4; Carlip, Steven (2001),"Quantum Gravity: a Progress Report", Rept. Prog. Phys. 64 (8): 885942, arXiv:gr-qc/0108040, Bibcode2001RPPh...64..885C, doi:10.1088/0034-4885/64/8/301; Rovelli, Carlo (2000). "Notes for a brief history of

    quantum gravity". arXiv:gr-qc/0006061 [gr-qc]; Rovelli, Carlo (1998), "Loop Quantum Gravity", Living Rev.Relativity 1, http://www.livingreviews.org/lrr-1998-1.30 Nielsen, H.B. (1973) Local field theory of the dual string Nucl. Phys. B57: 367-380DOI: 10.1016/0550-3213(73)90107-7; Nambu, Y. (1974) Strings, Monopoles and Gauge Fields Phys. Rev. D10:4262-4281; and Susskind, L., et al. (1987) Strings in space In Santiago 1987, Proceedings, Quantum Mechanics ofFundamental Systems 2, pp. 153-165.31W. Heisenberg, W. (1925) ber quantentheoretische Umdeutung kinematischer und mechanischerBeziehungen Zeitschrift fr Physik, 33: 879-893; Born, M. and Jordan, P. (1925) Zur QuantenmechanikZeitschrift fr Physik, 34: 858-888,; M. Born, M., Heisenberg, W., and Jordan, P. (1925) Zur QuantenmechanikII Zeitschrift fr Physik, 35: 557-615.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    29/33

    26

    Chapter 8

    Conclusions

    Perhaps theres even some quite simple rule,

    some simple program, for our Universe?Stephen Wolfram, TED, Long Beach,California, February 2010

    The relationship between two points in space is proportional to the square of the distance

    between them. This appears to be a universal law that forms the foundation of many previously

    identified universal laws that were the results of much empirical evidence and that forms the

    basis of my conjecture.

    Richard Feynman once said: It isnt that a particle takes the path of least action but that it smells

    all the paths in the neighborhood and chooses the one that has the least action.... 32 . He said

    this in regards to properties of photons, but of course it should equally apply to any interaction

    between two points (objects) in space and would apply to electromagnetic radiation, gravity, as

    well as interactions at the atomic level.

    I had imagined that the Pythagorean triangle consisted of walls built upon each line, each

    equivalent to its square value. Through these walls was the connection between points A and Cand between A, B, and C: they were equally similarly connected to one another and passing

    through each wall(s), either way, was no different that passing through the other. This further

    suggested that there is a substantial connection between all organisms and matter and that those

    connections and relationships are contiguous throughout the environment. Using the knowledge

    regarding the relationship between any two points, it will be relatively simple for anyone to use

    the measured metrics to make a powerful predictive function or algorithm regarding the

    interaction and/or activity between those two points.

    32 Feynman, R.P. (1964, 2006, 2010) The Principle of Least Action In The Feynman Lectures on Physics: MainlyElectromagnetism and Matter: The New Millennium Edition, Feynman, R.P., Leighton, R.B., Sands, M. (Eds.),California Institute of Technology, California, pp. 19-1 to 19-14 (p. 19-9).

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    30/33

    27

    With the recent finding that Ohms Law is consistent at atomic scale 33, I would predict that it

    will soon be determined that other electrical, gravitational, and relativistic laws are also in effect

    at such scales, thereby consistent with my thesis.

    I wish to advance this conceptual conjecture in order to establish a framework upon which others

    may build. It would be arrogant of me if I felt that I should set forth arguments in fields in which

    I have little or no experience.

    I readily admit that time has not permitted me to read every reference in detail and that some

    before me may have suggested or predicted these ideas in earlier publications. It is not my place

    to announce or define a new Universal Law; that would not only be presumptuous but also

    unethical, as I am not a mathematician by profession. Others will need to perform theexperiments and the calculations to determine whether all other systems and levels of

    organization possess this property. Until that time, I can only feel justified in labeling this a

    conjecture, and hope that it may eventually develop (via a theorem) to a true Universal Law.

    I think that we are now of the verge of a second period of Enlightenment and I would seriously

    encourage students of all disciplines to consider this conjecture and proceed to utilize its

    concepts in their research. I expect that someday Stephen Wolframs company, Mathematica,

    will be able to come up with a simple algorithm that one could just plug into ones model and

    which could then be used to test whether the square of the distance (d2) concept for the test

    system is correct.

    One additional problem I have is using the term relationship. This may to some readers be

    rather to anthropomorphic in essence, so I have come up with a number of different, but possibly

    equivalent terms that could substitute for that concept.

    33 Weber, B. et al. 2012 Ohms Law Survives to the Atomic Scale Science 335(6064): 64-67 DOI:10.1126/science.1214319

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    31/33

    28

    Table 8.1

    AssociationAffinityAffiliate

    LinkEndolinkCis-linkage

    How may this concept be defined in the normal world you might ask yourself? Well,

    philosophers and mathematicians since writing began have independently discovered the

    geometric principle that is colloquially known as Pythagoras Theorem. The Theorem

    implicitly contains the concept of a universal parameter that can be used in probably any

    circumstances where the scholar is attempting to define a relationship between two or more

    events. An extremely full and enjoyable narrative of the history of the Theorem is to be found

    in the first chapters of Geometry Civilized: History, Culture, and Technique by J.L. Heilbron

    (Oxford University Press, Oxford, UK & New York, NY, 1998) in which he describes how the

    principle was deduced seemingly independently by Chinese, Veddan, and Egyptian priests and

    philosophers.

    Edward O. Wilson wrote: A united system of knowledge is the surest means of identifying the

    still unexplored domains of reality. It provides a clear map of what is known, and it frames themost productive questions for inquiry. Historians of science often observe that asking the right

    question, even when insoluable in exact form, is a guide to major discovery. 34

    I think we may be there.

    34 Edward O. Wilson Consilience. The Unity of Knowledge. 1998 Vintage Books (Random House Inc.) New YorkNY, paperback edition p. 326.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    32/33

    29

    Epilogue

    I agree that theorizing is to be approved,

    provided that it is based upon facts, and

    systematically makes deductions from

    what is observed

    Hippocrates, Precepts

    I have spent many hours over the past seventeen years or so tweaking the story here and there; I

    believe that I have presented enough facts to enable anyone to derive their own conclusions

    about this apparent phenomenon and to apply it to their own area of expertise.

    My hope that by publishing this freely available online, that this will alleviate ignorance

    throughout the world.

    Dedication

    This work is dedicated to the late Professor George Varley, the much-respected Hope Professor

    of Entomology at Oxford, who, over a Sunday lunch in 1977, opened my eyes to the multiplicity

    of relationships in biology, big and small.

  • 7/28/2019 Unravelling Relationships; Pythagoras Reconsidered

    33/33

    About the Author

    The author was born in Geneva, Switzerland, and grew up in Oxford, England. He graduated

    with a B.Sc. (General) degree from U.C.N.W., Bangor, then joined the Biochemistry Department

    at the University of Oxford. After gaining an M.Sc. (Biochemistry) from the University of

    Wales, Cardiff, he then undertook basic research in growth and development under the

    mentorship of Drs. Margery Ord and Lloyd Stocken at Oxford and obtained his doctorate in 1988.

    After several postdoctoral positions in the U.K., Texas, and California, he was appointed as an

    Assistant Research Biologist at the University of California, San Francisco, where he studied

    lung surfactant proteins until 1997 with Dr. William H. Taeusch. He then went into the private

    sector and passed the US Patent Bar. He is now with a small patent law firm in San Francisco

    specializing in the life sciences.