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UNRAVEllING
RElATIONSHIPS
PYTHAGORAS RECONSIdEREd
Matthew Kaser
Published by Scribd
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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
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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 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.
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(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.
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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.
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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.
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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
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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
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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
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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.
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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
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. 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.
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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.
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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
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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).
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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
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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.
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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 )
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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
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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
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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.
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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
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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.
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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.
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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).
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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
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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.
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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.
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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.