Toward the Unification of Physics and Number Theory

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Klee Irwin
Science
QGR Research papers
Toward the Uni¯cation of Physics and Number Theory
Klee Irwin
Quantum Gravity Research
Los Angeles, California, USA
Received 9 September 2019
Accepted 3 November 2019
Published 28 November 2019
This paper introduces the notion of simplex-integers and shows how, in contrast to digital
numbers, they are the most powerful numerical symbols that implicitly express the information
of an integer and its set theoretic substructure. A geometric analogue to the primality test is
introduced: when p is prime, it divides
p
k
 
for all 0 < k < p. The geometric form provokes a
novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of
the Riemann hypothesis. Speci¯cally, if a geometric algorithm predicting the number of prime
simplexes within any bound n-simplex or associated An lattice is discovered, a deep under-
standing of the error factor of the prime number theorem would be realized  the error factor
corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious
link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson,
Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests
how quantum gravity and particle physicists might bene¯t from a simplex-integer-based qua-
sicrystal code formalism. An argument is put forth that the unifying idea between number
theory and physics is code theory, where reality is information theoretic and 3-simplex integers
form physically realistic aperiodic dynamic patterns from which space, time and particles
emerge from the evolution of the code syntax.
Keywords: Physics; number theory; geometry.
1. Introductory Overview
The value of this paper lies in the following questions the following author hopes are
provoked in the mind of the reader:
1. Is there a geometric algorithm that predicts the exact number of prime-simplexes
embedded within any n-simplex?
2. If Max Tegmark is correct and the geometry of nature is made of numbers, would
they be geometric numbers like simplex-integers?
This is an Open Access article published by World Scienti¯c Publishing Company. It is distributed under
the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and
reproduction in any medium, provided the original work is properly cited.
Reports in Advances of Physical Sciences
Vol. 3, No. 1 (2019) 1950003 (79 pages)
#.c The Author(s)
DOI: 10.1142/S2424942419500038
1950003-1
Opinion Paper
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3. Would this explain the correspondence between number theory and physics and
support the conjectures of Freeman Dyson2 and Michel Lapidus,3 who posit the
existence of a missing link between number theory and fundamental physics?
4. If nature is made of geometric numbers, how would it compute itself into existence
and is the principle of least action an indication it is concerned with e±ciency?
5. Are quasicrystal codes maximally e±cient?
6. If nature is a symbolic language — a code operating at the Planck scale, can it
exist without choosing or \measuring" entities at that scale, as is required to put
in action the syntactically free steps in all codes?
Ontologically, it seems clear that the fundamental elements of reality are made of
information. We de¯ne information here as \meaning in the form of symbolic
language". And we de¯ne language as \a symbolic code consisting of (a) a¯nite set of
symbols, (b) construction rules and (c) syntactical degrees of freedom. This is in
contrast to deterministic algorithms which also use a¯nite set of symbols and rules
but have no degrees of freedom. We de¯ne symbol here as \an object that represents
itself or something else". And,¯nally, we de¯ne an object as \anything which can be
thought of".
Fundamental particles have distinct geometries and are, in some sense, geometric
symbols. For example, at each energy state, an orbiting electron forms a¯nite set of
shape-symbols  p-orbital geometries composed of the probability distribution of
the wave-function in 3-space. There are strict rules on how these fundamental
physical geometric objects can relate, but there is also freedom within the rules, such
that various con¯gurations are allowed. If we speculate that particles are patterns in
a Planck scale geometric quantum gravity code in 3D of quantized space and time,
we can wonder what the most e±cient symbols would be.
Simplexes are e±cient symbols for integers and their set theoretic substructure.
The prime simplexes within any bound of simplexes, n-simplex to m-simplex, are
ordered according to purely geometric reasons. That is, the set theoretic and number
theoretic explanation is incidental to the geometric one. Accordingly, speculations by
Freeman Dyson,2 Michel Lapidus3 and others on a hidden connection between
fundamental physics and number theory are less enigmatic when considering shape-
numbers, such as simplex-integers. This view brings fundamental physics and
number theory squarely into the same regime  geometry  a regime where
physics already resides.
As symbolic information, simplexes are virtually non-subjective and maximally
e±cient when used to express the information of an integer and its set theoretic
substructure. A quasicrystalline symbolic code made of simplexes possesses
construction rules and syntactical freedom de¯ned solely by the¯rst principles of
projective geometry.
All particles and forces, other than gravity, are gauge symmetry uni¯ed according
to the E6 Lie algebra, which corresponds to the E6 lattice and can be constructed
entirely with 3-simplexes. That is, it can be understood as a packing of 6-simplexes,
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Each simplex can each be constructed from 3-simplexes. More speci¯cally, all par-
ticles and forces other than gravity are uni¯ed according to the standard model SU
(3)  SU(2)  U(1) Lie algebra. The single gauge groups that contain this algebra
include SU(5) in the form of Georgi–Glashow Grand Uni¯ed Theory (GUT), SO(10)4
and E6.
5 All three are related by the complex octonion projective plane (C  O)P2
which is E6 divided by SO(10)  U(1) and by the 20-dimensional set of complex
structures of 10-dimensional real space R10, which is SO(10) divided by SU(5). These
algebraic objects are isomorphic to their Euclidean geometric analogues, which are
simple higher dimensional lattices constructed as packings of simplexes. So E6
embeds SO(10), which embeds SU(5), which embeds SU(3)  SU(2)  U(1). E6
embeds in E8. In 1985, David Gross, Je®rey Harvey, Emil Martinec and Ryan Rohm
introduced Hetronic string theory using two copies of E8 to unify gravity with the
standard model in an attempt to create a full uni¯cation theory. E8 can be con-
structed entirely by arranging, in 8D space, any choice of n-simplexes of equal or
lesser dimension than the 8-simplex. Other approaches using E8 for uni¯cation
physics include those of Lisi6 and Smith and Aschheim.7
Cutþ projecting a slice of the E8 lattice to 4D along the irrational hyper vector
prescribed by Elser and Sloan generates a quasicrystal. This quasicrystal can be
understood as a packing of 3-simplexes in 4D forming super-clusters 600-cells, which
intersect in seven ways and kiss in one way to form the overall 4D quasicrystal.
Because this 3-simplex-based object derived from E8 encodes the E6 subspace under
projective transformation, it also encodes the gauge symmetry uni¯cation physics of the
standard model along with the less well accepted uni¯cation of general relativity via E8.
A projection encodes the projection angle and original geometry of a pre-projected
object. Consider a copy of a unit length line [yellow] and its projection rotated by 60
[blue] as shown in Fig. 1.
Of course, this is a line living in 2D and rotated by 60 relative to a 1D projective
space, where it contracts to a length of 12 (as shown in Fig. 1). Similarly, a projection
of a cube (as shown in Fig. 2) rotated by some angle relative to a projective plane is a
Fig. 1. A unit length line [yellow] & its projection rotated by 60 [blue].
Fig. 2. A cube projected to a plane.
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pattern of contractions of the cube's edges, as encoded in the projection, which
join to form angles with one another. The total projection is a map encoding the
higher dimensional shape plus the rotation of the projector relative to the cube
and plane.
One can decode the projection itself to induce the pre-projected cube and the
possible projection angles. To form a quasicrystal, more than one cell is projected. A
slice of the higher dimensional crystal, called a cut window, is projected to the lower
dimensional space. The coordinate of the cut window can change to generate addi-
tional quasicrystalline projections that form animations. These changes can be made
by translating and or rotating the cut window through the lattice. The coordinates of
the various vertex types of the projection will change as the coordinate of the cut
window changes. The ways these changes can occur are called the phason rules and
degrees of freedom. This¯nite set of geometric angles and lengths and the rules and
freedom are collectively called the code or language of the quasicrystal. Refer to`Free
Lunch Principles-Forces' in Sec. 5.15 for more on the Cut-and-Project method.
The aforementioned E8 crystal can be built entirely of regular 3D tetrahedra 
3-simplexes. When it is projected to 4D, the tetrahedral edges contract, but do so
equally so that the tetrahedra shrink under projection but remain regular, generating
a quasicrystal made entirely of 3-simplexes. As will be discussed later, we then
generate a representation of this 4D quasicrystal in 3D.
A quasicrystal is an object with an aperiodic pure point spectrum where the
positions of the sharp di®raction peaks are part of a vector module with¯nite rank.
This means the di®raction wave vectors are of the form
k ¼
X
n
i1
hia

i ;
ðinteger hiÞ:
ð1Þ
The basis vectors a i are independent over the rational numbers. In other words,
when a linear combination of them with rational coe±cients is zero, all coe±cients
are zero. The minimum number of basis vectors is the rank of the vector module. If
the rank is larger than the spatial dimension, the structure is a quasicrystal.8 And
every aperiodic pure point spectrum in any dimension correlates to some quantity of
irrational cutþ projections of higher dimensional lattices.
There is an intriguing connection between quasicrystals, prime number theory
and fundamental physics. Both the non-trivial zeros of the Riemann zeta function
and Eugene Wigner's universality signature, found in all complex correlated systems
in nature, are pure point spectrums and therefore quasicrystals.9 However, with 1D
quasicrystals that possess many nearest neighbor point to point distances, as op-
posed to the two lengths in the simple Fibonacci chain quasicrystal, it is very di±cult
to know what \mother" lattices and projection vectors generate them. In other
words, we can know it is a quasicrystal because it is an aperiodic pure point spec-
trum. But we would have no deep understanding of its phason syntax rules or
information about the higher dimensional crystals and angles that generated it.
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Andrew Odlyzko showed that the Fourier transform of the zeta-function zeros has
a sharp discontinuity at every logarithm of a prime or prime-power number and
nowhere else.10 That is, the distribution is an aperiodic pure point spectrum 
a quasicrystal.2 Disorderly or chaotic non-periodic ordering will not generate an
aperiodic pure point spectrum.
Similarly, for any span of n-simplexes through m-simplexes, the density distri-
bution of simplexes with a prime number of vertices (prime-simplexes) is aperiodic
and non-random. Its prime density pattern and scaling algorithm exists for purely
geometric reasons. For example, one may consider the 99-simplex. It contains 25
prime-simplexes that have an ordering scheme that drops in density as the series
approaches the bound at the 99-simplex. The distribution of the 25 prime-simplexes
within this hyper-dimensional Platonic solid is based purely on geometric¯rst
principles and is not fundamentally related to probability theory. Of course, it can be
predicted using probability theory. The distribution of prime-simplexes, as shape-
numbers, within any bound is trivially isomorphic to the distribution of digital
integers within the same bound. The distribution pattern of digital primes is fun-
damentally non-probabilistic because the identical geometric distribution pattern of
prime-simplexes is not probabilistic.
Interestingly, Wigner's ubiquitous universality signature describes the quasipe-
riodic pattern of the zeta zero distribution.11 The pattern occurs in all strongly
correlated systems in nature. In fact, it shows up in the energy spectra of single
atoms. Indeed, nearly all systems are strongly quantum correlated. Terrence Tao and
Van Vu demonstrated universality in a broad class of correlated systems.12 Later,
we will discuss more about this pattern, which Van Vu said appears to be a yet
unexplained law of nature (see Sec. 4.9).
What could possibly correlate the distribution of primes in number theory to
something as ubiquitous as the universality signature? As Dyson recognized, the
distribution of prime numbers is a 1D quasicrystal and the universality signature
ubiquitous in physics is too.
Each prime-simplex integer is associated with a crystal lattice as a subspace of the
in¯nite A-lattice. For example, the 2-simplex is associated with the A2 lattice (see
Fig. 3), which is associated with a crystal made of equilateral triangles and is a
subspace of all A2þn lattices.
Fig. 3. A triangular lattice.
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If we cut þ project a slice of this crystal to 1D with the golden ratio-based angle of
about 52.24, we generate a quasicrystal code with three \letters" of the lengths 1, ’
and 1=’. As we go up to higher dimensional A-lattice crystals associated with a given
simplex-integer and project to 1D, the number of lengths or \letters" of the quasi-
crystalline code increases. Within any irrational projection of a prime-An lattice-
based crystal, there exists the projections of all A lattice crystals less than n,
including a distribution of prime-A-lattices. For example, the projection of the
crystal built upon the A99 lattice built of the simplex-integer corresponding to the
number 100, encodes the distribution of 25 prime-An lattice crystals.
This connection can be summarized thus: (a) The distribution of non-trivial zeta
zeros and the distribution of prime numbers is a 1D quasicrystal. (b) All 1D quasi-
crystals can be derived by irrationally projecting hyper lattice slices. A likely can-
didate lattice is the one corresponding to simplex-integers. (c) Nature is deeply
related to mathematics. (d) The foundation of all mathematics, even set theory, is
number theory. The appearance of the universality signature in all complex systems
and the distribution of primes may relate to the in¯nite-simplex. That is, the crystal
associated with the in¯nite A-lattice and its projective representation in the lowest
dimension capable of encoding information, 1D.
We propose that the missing link between fundamental uni¯cation physics
and number theory is the study of simplex-integer-based lattices transformed under
irrational projection; quasicrystalline code theory.
In Sec. 5, we mention that a general feature of non-arbitrarily generated quasi-
crystals is the golden ratio. Speci¯cally, any irrational projection of a lattice slice will
generate a quasicrystal. However, only golden ratio-based angles generate quasi-
crystals with codes possessing the least number of symbols or edge lengths. We will
discuss how black hole theory, solid state materials science as well as quantum
mechanical experiments indicate there may be a golden ratio-based code related to
the sought after quantum gravity and particle uni¯cation theory. As preparation for
that, it is helpful to understand how deeply the golden ratio ties into simplexes.
Of course, the simplest dimension where an angle can exist is 2D. And the simplest
object in 2D is the 2-simplex. Dividing the height of a circumscribed 2-simplex with a
line creates one long and two short line segments (see Fig. 4). The ratio of the long to
the short segments is the golden ratio.
In fact, this is the simplest object in which the golden ratio exists implicitly,
since simply dividing a line by the golden ratio is arbitrary and not implied by
the line itself. So there is a fundamental relationship between  and ’ in the
circumscribed 2-simplex.
Simplexes are the equidistant relationships between an integer quantity of points
and, in their lattice form as packings of tetrahedra, they correspond to periodic
maximum sphere packings. For example, the maximum sphere packing in 3D
includes the FCC lattice, which is a packing of 3-simplexes. The points where the
spheres meet generate the lattice associated with the 3-simplex is called the A3
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lattice. Similarly, the E8 lattice is a packing of simplexes, and its points are the
kissing points of the maximum packing of 8-spheres.
2. Proof by Deductive Argument that Simplex-Integers are the Most
E±cient Number Symbols for Integers and their Set Theoretic
Substructure
2.1. Introduction
A number is a symbol used to measure or label. Generally, a symbol is an object that
represents itself or another object. For example, an equilateral triangle (as the delta
symbol) often represents the object called \change" or \di®erence".
However, an equilateral triangle (or any object) can also serve as a symbol to
represent itself, the equilateral triangle. Symbols can be self-referential and partici-
pate in self-referential codes or languages. An example is a quasicrystal, such as the
Penrose tiling (see Fig. 5) derived by projecting a slice of the 5-dimensional cube
Fig. 4. A triangle in a circle showing the golden ratio as the ratio between the red and the blue edges.
Fig. 5. The Penrose tiling quasicrystal, derived via cut þ projection of a slice of the Z5 lattice, is a code
because it contains a¯nite set of geometric symbols (two rhombs), matching rules and degrees of freedom.
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lattice, Z5, to the plane.
13 It is a language14 because it possesses (a) a¯nite set of
symbols, (b) construction rules and (c) degrees of freedom  called phason degrees
of freedom.
It has two \letters" which are two rhomboid shapes. The rhombus symbols can
only be arranged according to speci¯c assembly rules to form seven di®erent vertex
geometries that can be thought of as the \words" formed by the two building block
geometric \letters". But within the syntax constraints, there are also degrees of
freedom, called phason degrees of freedom. The quasicrystalline code is a language in
every sense of the word, conveying the meaning of geometric form such as dynamical
quasiparticle waves and positions. Yet, the geometric symbols, the building blocks of
this code, represent themselves, as opposed to ordinary symbols which represent
other objects.
It has been shown how 3-simplexes15 can be the only shape in non-space-¯lling
quasicrystals in 3D or 4D. All quasicrystals are languages, not just the Penrose tiling.
And the 3-simplexes, i.e., simplex-integers, in these special quasicrystalline symbolic
codes represent themselves.
2.2. Ultra-low symbolic subjectivity
Generally, symbols are highly subjective, where its meaning lies at the whim of the
language users. However, simplexes, as geometric numbers and set theoretic symbols
which represent themselves, have virtually no subjectivity. That is, their numeric, set
theoretic and geometric meaning is implied via¯rst principles. If space, time and
particles are pixelated as geometric code, low subjectivity geometric symbols with
code theoretic dynamism such as these could serve as quanta of spacetime.
2.3. Simplexes as integers
The geometric structure of a simplex encodes numerical and set theoretic meaning in
a non-arbitrary and virtually non-subjective manner. For example, the digital
symbol \3" does not intrinsically encode information about the quantity of three
objects. In fact, any object can serve as a symbol for a number. So, we introduce the
notion of simplex-integers as virtually non-subjective symbols for integers.
2.3.1. Symbolic function 1: Counting
The number of 0-simplexes in a given n-simplex indexes to an integer. For example,
the 2-simplex has three points or 0-simplexes corresponding to the digital symbol \3".
The most basic information of an integer is its counting function. The series of simplex-
integers counts by adding 0-simplexes to each previous simplex-integer symbol.
2.3.2. Symbolic function 2: Set theoretic meaning
Inherent to an integer is its set theoretic substructure. A simplex-integer is a number
symbol that encodes both the counting function and set theoretic substructure of an
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integer. For example, the quantity of four objects can be communicated by the
symbols, 4 or IV. However, when we use a 3-simplex to represent the counting
function of the number 4 we also encode its set theoretic substructure:
. Four sets of one
. Six sets of two
. Four sets of three
. One set of four.
This is geometrically encoded in the 3-simplex as four 0-simplexes (points), six
1-simplexes (edges), four 2-simplexes (faces) and one 3-simplex (tetrahedron).
2.3.3. Symbolic function 3: Binomial expansion
An n-simplex encodes the binomial coe±cient corresponding to a row of Pascal's
triangle (see Fig. 6).
The coe±cients are given by the expression
n!
k!ðnkÞ!.
Pascal's triangle is the arrangement into rows of successive values of n. The k
ranges from 0 to n generate the array of numbers. It is a table of all the binomial
expansion coe±cients.
2.3.4. Symbolic function 4: Sierpinski triangle fractal
Because each simplex is a higher dimensional map of the 2D Pascal's triangle table,
it too encodes the same Sierpinski triangle fractal when the positions of the table
are coded in a binary fashion to draw out the odd and even number pattern, with
fractal dimension log(3)/log(2) (as shown in Fig. 7). This same fractal can be a
cellular automaton generated by Rule 90,16 the simplest non-trivial cellular au-
tomaton.17 Speci¯cally, it is generated by random iterations of the time steps of
Rule 90.
2.3.5. Symbolic function 5: Golden ratio in the in¯nite-simplex
Pascal's triangle is analogous to a matrix representation of the sub-simplex sums
within any n-simplex. Diagonal cuts through Pascal's triangle generate sums that are
successive Fibonacci numbers (as shown in Fig. 8). Any two sequential Fibonacci
Fig. 6. The Pascal triangle until the¯fth row.
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numbers are a close approximation of the golden ratio. The series of ratios converges
to the golden ratio.
2.4. Symbolic power of simplex integers
In graph theory, one can use a graph drawing as a numerical symbol to count
quantities of objects and explore their set theoretic relationships (connections). This
makes a graph diagram analogous to the counting function and set theoretic sub-
structure function of a simplex-integer.18 For example, the complete and undirected
graph of three objects expresses the set theoretic substructure implied by the integer
3. The graph drawing symbol is usually the 2-simplex (as shown in Fig. 9).
A key aspect of this symbol is the equidistance between its points  its con-
nections. The complete and undirected network (graph) of three objects has no
magnitudes in its connections. To geometrically represent the complete graph of
three objects, equidistance symbolizes the notion of equal magnitude of the graph
Fig. 7. Sierpinski triangle showing Pascal triangle values.
Fig. 8. Fibonacci numbers as diagonal sums of a Pascal triangle.
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connections. So the 2-simplex is an e±cient or waste-free symbol for the complete
and undirected graph of three objects.
The diagrammatic symbol graph theorists typically use for the complete and
undirected graph of four objects is given in Fig. 10. It encodes the full set theoretic
substructure of the integer 4.
But here, we see a breakdown in e±ciency because we have wasted information in
the drawing or symbol. It does not inherently represent the notion of equal con-
nection magnitude because four connections have one length and two have a longer
length. This super°uous information in the symbol must be ignored by the graph
theorist. It is wasted.
The only way to have equidistance between four points in a geometric symbol is to
extrude an additional spatial dimension  to go to 3D. In this case, the tetrahedron
can symbolize in a non-subjective and waste-free manner, the equidistant relation-
ship of four points and their full set theoretic substructure. This is the case for all
simplexes where each encodes a positive integer and its full set theoretic substructure
in the most e±cient manner possible without wasted symbolism and where all
connections are of the same length or magnitude.
Of course, we cannot make symbols in spatial dimensions greater than three.
However, we can work with higher dimensional simplex symbols in the form of their
associated geometric algebras.a
Next, let us sketch out a proof that simplex-integers are the most powerful
numbers to express counting function and set theoretic substructure.
Fig. 10. The three-simplex, a tetrahedron, a complete graph of four elements projected to a square.
aEach simplex is associated with a Lie group of the series An and its Lie algebra an, which corresponds to
geometry because a Lie group encodes geometric operations (mirror re°ections and rotations). Each
simplex is associated to a Cli®ord algebra (commonly named geometric algebra), while each sub-simplex is
a basis element of the same dimension.
Fig. 9. The two-simplex, a triangle.
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2.5. A method for ranking symbolic power
Here, the term symbolic power shall be synonymous with symbolic e±ciency. Our
discussion is concerned only with the e±ciency ranking of symbols that can represent
the meaning of (a) integers and (b) their set theoretic substructure. The rank of
meaning or information content of an integer and its set theoretic substructure
increases with size.
The challenge is to logically rank the magnitude of inherent information of a given
symbol, then we can consider the set of all symbols which might encode the numeric
and set theoretic meaning of integers to see if there is one type with maximal
e±ciency.
Symbolic e±ciency here is concerned with the ratio of:
1. Irreducible sub-symbols
2. Meaning
Because the numeric and set theoretic meaning is established, we are seeking to
understand what symbols are the most minimalistic or elegant  the least
complex for this purpose. Accordingly, let us discuss the magnitude of a symbol's
complexity.
If a symbol can be reduced to irreducible sub-symbols, it is a composition of some
quantity of simpler symbols. We will use that quantity as the magnitude rank of
symbolic complexity. For example, the sentence \The dog ran fast" is itself a symbol.
But it can be decomposed into clauses, words and letters. Indeed, the letters them-
selves can be decomposed into simpler subparts as points and connections or lines.
Again, symbols are objects that represent themselves or another object. In the
universe of all objects, the empty set and 0-simplex are equally and minimally simple.
There can be no simpler object. These are the only two to possess the quality of being
non-decomposable into simpler objects. That is, all other symbols are composites of
other objects/symbols.
It is di±cult to conceive building composite symbols and a symbolic language out
of empty sets. Points (0-simplexes), on the other hand, can be arrayed in spaces to
form familiar symbols or can be connected graph theoretically without spaces to form
non-geometric symbols.
The simplest object in n-dimensions is the n-simplex.19 And every n-simplex is a
composite of irreducible 0-simplexes or vertices fv1,v2; ... ;vng, where every subset
in the structure is a simplex of n-m dimensions. Subsets with one element are points,
subsets with two elements are line segments, subsets with three are triangles, subsets
with four are tetrahedra and so on.
The reason a simplex is the simplest object in any spatial dimension is because it is
the least number of non-decomposable symbols (0-simplexes) needed to form a
convex hull occupying all sub-dimensions of a given spatial dimension. Of course, the
simplex need not be regular to possess this quality. However, regular simplexes have
only one edge length, one edge angle, one dihedral angle, etc. Irregular simplexes
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possess far more information, where members of a set must be distinct from one
another. Accordingly, irregular simplexes possess more information or sub-symbols.
For this reason, a regular n-simplex is the simplest object possible in any spatial
dimension.
It is clear that the irreducible 0-simplex is the simplest object that can form com-
posite symbols. It is clear that compositing a set of them to form a series of symbols
called simplex-integers non-arbitrarily and inherently encoding the numeric and set
theoretic meaning, we are concerned with e±ciency and non-arbitrarily symbolizing.
Can we allow, for example, an irregular equilateral triangle to encode the meaning
we are interested in and still call it equally simple because it uses the same number of
points? It is true that the irregular triangle encodes the numeric and set theoretic
meaning. It is also true that a graph diagram in 2D for, say,¯ve objects encodes the
same information we need with the name number of points (as seen in Fig. 11).
However, as a symbol, it inherently possesses two di®erent connection lengths. It
has additional information not required or needed in our attempt to symbolize the
number¯ve and its set theoretic substructure corresponding to equal magnitude of
connections. One must ignore that additional inherent meaning of two connection
magnitudes is implied by the symbol. We may consider this as an equation, where the
right side is the meaning of the quantity of¯ve objects and their set theoretic
substructure. The left side is a package of inherent geometric information in the
symbol we are considering to equal the right side of the equation. We start by
counting the quantity of irreducible 0-simplexes on left side  the symbol side.
Counting them gives the value 5. If we allow any irregular 5-simplex, it leaves us with
an in¯nite number of geometric con¯gurations of¯ve points with more than 1 con-
nection magnitude. We wish to minimize the left side further to¯nd the one con¯g-
uration that is simpler or generates the least amount of unneeded inherent geometric
symbolism/information. Speci¯cally, we need all connections between points to be
equal. Otherwise, additional information accumulates on the left side of the equation
 the inherent information of the symbol itself. Only an n-simplex can achieve this
task for any quantity of points. Via this logic, simplex-integers are the most powerful
numbers to encode the numeric and set theoretic meaning of the integers.
Fig. 11. The four-simplex, a complete graph of¯ve elements projected to a pentagram.
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2.6. Section 2. Conclusion
The simplest object in n-dimension is the n-simplex. The simplest object in any
dimension and the only non-decomposable or non-composite symbol is the 0-simplex.
The simplest set of composite symbols is the n-simplex series, which adds one
0-simplex to each successive member of the set. The count of 0-simplexes serves as a
number  a geometric symbol representing the counting function of an integer and
its inherent set theoretic substructure.
Adding more points in some lower dimension, such as 2D, can also serve as a
simple counting symbol that also encodes the set theoretic substructure in the form
of the connections on a complete graph drawing. However, this symbol deviates from
the pure implied meaning in the simplex-integer series because, without extruding an
additional spatial dimension for each added 0-simplex, the connections of the graph
drawing take on di®erent length values. The implied information and unnecessary
complexity of the symbol breaks down with this more mathematically complex object,
where the equal set theoretic \connections" are no longer intrinsically implied with
virtual non-subjectivity. One must ignore this extra information and subjectively and
arbitrarily interpret the various connection lengths as having equal magnitude.
We prove through this deductive argument that the n-simplex series is the most
powerful set of symbols to represent the integers and their set theoretic substructure.
A given simplex embeds the full set of theoretic and numeric information of all
simplexes within it, including the distribution of isomorphic prime simplexes to the
distribution of prime numbers on the ordinary number line. Accordingly, the in¯nite
simplex is the most powerful representation of the integers, prime distribution and
the set theoretic substructure of each integer.
3. A Geometric Primality Test and the Prime-Simplex Distribution
Hypothesis
3.1. Introduction
It is generally believed that if an exact expression were found which determines the
number of primes within any bound of numbers, it would lead directly to a proof or
disproof of the Riemann hypothesis. This is because the non-trivial zeros that fall on
the critical line on the complex plane in the Riemann hypothesis correspond to the
error terms created by using inexact expressions to estimate the number of primes
within a span of integers. Inexact expressions are all that have been discovered so far
in prime number theory. An exact expression is still outstanding and would lead
directly to a proof of the Riemann hypothesis if discovered.
There are 25 primes in the bound of numbers 2–100. However, the prime number
theorem expression of Carl Friedrich Gauss of the form ðxÞ  x= logx20 incorrectly
predicts 21.7 primes in the same bound. The expression of Peter Gustav Lejeune
Dirichlet of the form Li(x) - (x)21 incorrectly predicts 30.1 primes.
The delta between these inexact approximations and Riemann's exact function
result in error terms that can be written in terms of the non-trivial zeros of the
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Riemann zeta function. Number theorists have not proven the Riemann hypothesis
because they do not deeply understand these non-trivial zeros. That is, they do not
understand how to exactly predict the distribution of primes within a bound of
integers.
0ðxÞ ¼ RðxÞ 
X

RðxÞ  1
lnx
þ 1

tan1

lnx
:
ð2Þ
Because an analytical expression for the second term in Equation (1) does not exist,
this term quanti¯es the error in prime counting functions. It is a sum over the non-
trivial zeros of the Riemann zeta function () on the critical line. The Riemann
hypothesis states that all non-trivial zeros lie on the critical line. Riemann's formula
is exact if and only if the Riemann hypothesis is true. Again, mathematicians have
not proven the Riemann hypothesis because they do not deeply understand the non-
trivial zeros. If an exact form for 0ðxÞ is found that does not depend on the zeros, it
could be used to shed light on their nature and should lead to a solution of the
Riemann hypothesis.
Mathematics is like an upside down pyramid, where sophisticated math is built
upon a foundation of simple math, and base math is the counting numbers. For
example, before one can think about set theory, one must possess a notion of
counting numbers. A huge collection of proofs of potential new theorem exist in the
literature. Essentially, they state: \We prove that if the Riemann hypothesis is true,
then this theorem is true."22
With the current state of prime number theory, we have somehow missed
something deep. It is trivially true that within the geometric structure of the in¯nite-
simplex and its irrational projection to 1D, the distribution of prime-simplex and
prime digital numbers is encoded. Accordingly, this \deep" aspect of prime number
theory, and therefore all mathematics and mathematical physics which we have
missed is geometric number theory.
3.2. Geometric primality test
We introduce a geometric analogue to the primality test that when p is prime, it
divides,
p
k
 
¼ p  ðp 1Þ; ... ; ðp kþ 1Þ
k  ðk 1Þ; ... ; 1
for all 0 < k < p:
ð3Þ
Our geometric form provokes the prime-simplex distribution hypothesis that, if
solved, leads to a proof of the Riemann hypothesis.
Claim. If and only if the quantity of vertices of an n-simplex is evenly divisible into
each quantity of its sub-simplexes is that simplex a prime-simplex and associated with
a prime A-lattice.
p is prime () ð8k 2 N; k < p ) pj#fSk  Sp1gÞ;
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p is not prime () ð9k 2 N; k < p; ð#fSk  Sp1gÞ½p 6¼ 0Þ:
Note : #fSk  Sp1g is the binomial coefficient
p
k
 
¼
p!
k!ðp kÞ! :
We want to prove that p is prime if and only if p divides into C, where C is given by
Eq. (X). C ¼
p!
k!ðpkÞ! for any k between 2 and p 1. C satis¯es this equation: p! ¼
C k! ðp kÞ!
First, we demonstrate that, if p is prime, p divides C.
Conjecture 1. If p is prime, p divides C.
Proof. Because p divides p!, p also divides one of the three factors on the right side:
C or k! or ðp kÞ!
k < p and k! is a product of numbers smaller than p:Therefore, p does not divide
k!. If k is greater than 1, ðp kÞ! is a product of numbers smaller than p. Therefore, p
does not divide ðp kÞ!. So, necessarily, p divides C.
Conjecture 2. If p is composite, let p ¼ a b, where a
6¼ 1, b
6¼ 1 then at least one of
the coe±cients is not divisible by p.
Next, we demonstrate that, if p is composite, let p=ab, where a
6¼1, b
6¼1 then at
least one of the coe±cients is not divisible by p:
Take
k ¼ a : C ¼
ða bÞ!
a!ðaðb 1ÞÞ! :
ð4Þ
We can rewrite as
C ¼ bða b 1Þða b 2Þ; ... ; ða b aþ 1Þ=ða 1Þ!
ð5Þ
C is not divisible by a b, because none of the factors (a b 1), (a b – 2); ... ;
(a b aþ 1) is divisible by a, and b is not divisible by a b.
[Credit goes to Raymond Aschheim for assistance with the above equations.]
3.3. Prime-simplex distribution hypothesis
When studied as simplex-integers instead of digital integers, there is a
simple formula that separates prime numbers from composite numbers.
That is, there is a non-constant polynomial that takes in only prime values.
There is no known formula that separates primes from composite numbers. However,
there exists a purely geometric reason why a given simplex is prime or why there are,
for example, 25 prime simplexes embedded in the 99-simplex. The reason is not
directly number or set theoretic. Number and set theoretic aspects are merely inci-
dental or secondary to the geometric structure. The reason is based solely on the¯rst
principles of Euclidean geometry.
An extension can be made to A-lattices, which correspond to simplexes. For
example, the 4-simplex corresponds to the A4 lattice, which embeds the A2 and A3
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lattices. Using the 99-simplex example again, the A99 lattice, built as a packing of 99-
simplexes, embeds 25 prime-An lattices and describes their distribution exactly
without probability theory based approximations.
The unknown formula expressing the drop in prime-simplex and prime-A lattice
density within some bound is also purely geometric. In our future work, we intend to
focus on this problem. However, we can state with certainty that the algorithm can
be expressed with quasicrystalline formalisms when studied via the irrational pro-
jective transformation of a slice of the in¯nite-A-lattice.
The vertices of a prime-simplex are evenly divisible (without a remainder) into
each sum of its sub-simplexes. When one considers what this means in terms of shape
analyses, such as symmetry or topology, it becomes clear that there must be special
shape qualities present in prime-simplexes that are not evident in non-prime-
simplexes. For example, the 3-simplex is the¯rst to fail this geometric primality test.
Its sub-simplex quantities are:
4
0-simplexes,
6
1-simplexes,
4
2-simplexes.
Its four vertices does not evenly divide into its six edges. By contrast, when we
look at the simplex-integer associated with the prime number 5, we see sub-simplex
sets of
5
0-simplexes,
10
1-simplexes,
10
2-simplexes,
5
3-simplexes.
For lack of a better term, there is a division symmetry in this simplex with respect to
its geometric parts. The \beauty" of¯ve vertices evenly dividing into the sums of
each sub-simplex inspires the curiosity about what special volumetric, topological or
symmetry qualities this shape possesses.
3.4. Prime number distribution and fundamental physics
As far as impacting science is concerned, the discovery of the actual algorithm pre-
dicting the distribution of prime simplexes within an n-simplex may have important
implications for fundamental physics, shedding light on an equally monumental
outstanding problem: the theory of everything that uni¯es the theory of space and
time (general relativity) with the theory of the quantum world (quantum mechanics).
It is certainly true that nature is deeply mathematical, which means its founda-
tion is built upon counting numbers. But nature is deeply geometric as well. So
geometric counting numbers, like simplex-integers, are interesting  there is a
mysterious connection between physics and the distribution of prime numbers.
Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis,
we have studied23 the Schroedinger QM equation involving a highly non-trivial
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potential whose self-adjoint Hamiltonian operator energy spectrum approaches the
imaginary parts of the zeta zeros only in the critical line.
Sn ¼
1
2
þ in:
ð6Þ
This is consistent with the validity of the Bohr–Sommerfeld semi-classical quanti-
zation condition. We showed how one may modify the parameters which de¯ne the
potential,¯ne tuning its values, such that the energy spectrum of the (modi¯ed)
Hamiltonian matches all zeros. This highly non-trivial functional form of the
potential is found via the Bohr–Sommerfeld quantization formula using the full-
°edged Riemann–von Mangoldt counting formula for the number N(E) of zeroes in
the critical strip with imaginary part greater than 0 and less than or equal to E.
Our result shows a deep connection between the most foundational model we have
for reality, quantum mechanics, and prime number theory.
Patterns in nature over time or space can only be of three fundamental species:
1. Periodically ordered
2. Aperiodically ordered
3. Random.
There is no solid evidence for randomness in nature. In fact, demonstrating it is
impossible because one cannot write it down, as can be done with periodic and
aperiodic patterns. An experimentalist can only concede she has not been able to¯nd
periodic or aperiodic order. The lack of¯nding order is not good experimental evi-
dence for the theory of randomness. What does have good supporting experimental
evidence is the theory of non-determinism, which¯ts our code theoretic axiom.24 For
example, in 1984, Dan Shechtman reported his observation of code-theoretic aperi-
odic order in a material known by the scienti¯c community to be disorderly-randomly
structured (amorphous).25 The consensus belief was built upon a bedrock of crys-
tallographic mathematical axioms and decades of failure to observe order in this type
of material. And yet, Shechtman observed the signature of aperiodic order in the ma-
terial. A good portion of the scienti¯c community, led by Nobel laureate Linus Pauling,
rejected his¯ndings due in part to the popular theory that randomness is real.26
Similarly, number theorists have no idea why or how the quasiperiodic spectrum
of the zeta zeros possess the universality spectral pattern. Some mathematicians
think there may be an unknown matrix underlying the Riemann zeta function that
generates the universal pattern. Paul Bourgade, a mathematician at Harvard, said,
\Discovering such a matrix would have big implications for¯nally understanding the
distribution of the primes".27
So why would proving the Riemann hypothesis help in the search for a theory of
everything? Because there is a unifying principle in the form of (1) simplex-integers
and (2) quasicrystal codes based on simplexes. The notion of randomness in physics
would become an old paradigm giving way to the new ideas of aperiodic geometric-
language based physics and the principle of e±cient language.24 Non-deterministic
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syntactical choice would replace randomness as the ontological explanation for non-
determinism. Discreteness would replace the older notion of smooth space and time.
Number and geometry would be uni¯ed via the mathematical philosophy of shape-
numbers, where nature is numerical  simplex-integers forming the substance of
reality  geometry, all within a logically consistent self-actualized code-theoretic
universe.
3.5. Are digital numbers a dead-end approach to prime number theory?
Clearly, the universality aspect of complex physical systems is deeply rooted in the
geometry of particles and forces acting in 3-space. Prime and zeta zero distribution
display the same quasicrystalline pattern.
Non-geometric methods, such as probability theory and brute force computa-
tional methods, are typical tools for modern number theorists working on prime
number problems. If the 2300 years of stubbornness of this prime distribution
problem is a deep geometric challenge, then we have been using the wrong tools for a
long time.
As mentioned, within, for example, the 99-simplex, there are 25 prime-simplexes,
which are simplexes with a prime quantity of vertices. The reasons for why this
bound of simplexes 2–99 has a density of prime simplexes of 25 is a purely geometric
problem, even though the solution is exactly the same as the unknown algorithm
determining the exact quantity of prime digital numbers in the same bound. In other
words, the algorithm determining the distribution of prime-simplexes in some bound
is the missing and correct algorithm that encodes the error term, i.e., it equals the
error term plus the incorrect result of solutions using the prime number theory
algorithm or others.
A fresh and little-focused-on approach is to move prime number theory problems
from digital number theory into the domain of simplex-integer number theory 
into the realm of pure hyper-dimensional Euclidean geometry and its associated
geometric algebras and moduli spaces.
The purely geometric algorithm that determines the number of prime-simplexes
within an n-simplex is knowable. It would give the exact number of prime numbers
within any span of integers. But what is so deeply di®erent about the digital versus
simplex-integer approaches? Two core things: (1) non-transcendental convergence
and (2) non-homogeneity of sequential number deltas.
3.6. Non-transcendental convergence
Digital integers do not converge when you add them as a series. And divergences are
unhelpful because they tell you little  i.e., they don't give you a number because
they explode toward in¯nity. In order to convert digital integers into a convergent
series, one applies the zeta function. For example, we put a power on each integer and
then invert it. We do this with the next integer and add to the previous solution. We
repeat this with all integers to transform the integers into a convergent additive
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series that tells us something deep about the integers and their fundamental skeleton,
the primes:
1=ð12Þ þ 1=ð22Þ þ 1=ð32Þ; ... ;¼ 2=6:
ð7Þ
What is remarkable is that  is a deeply geometric number even though integers do
not appear to be associated with geometry. It is generally believed that  is tran-
scendental, although this is debated.28 The two most famous transcendental num-
bers are  and the basis of the natural logarithm, e.
Both e and  and are deeply geometric. The exponentiation is the fundamental
operation to transform an angle  into a complex number Exp(i ) which, multiplied
by a vector, also expressed as a complex number, operates the rotation of this vector
from this angle. The constant e is de¯ned by the choice of radian as a unit for the
angle, which sets  to measure a half circle rotation by Expði Þ ¼ ei, or e ¼ Expð1Þ.
This also involves I ¼ p  1.
Both e and  fundamentally relate to the Riemann hypothesis but only when
explored via digital numbers.
Speci¯cally, e is a part of the false error generating algorithms that imperfectly
predict the number of primes in a bound  thereby generating the error term that
maps to the zeta zeros and perhaps preventing a proof of the Riemann hypothesis.
And,  is related by the convergence of the zeta function itself to 2=6. The zeta
function process is how we plot the zero solutions related to the errors onto the
complex number plane.
It may be helpful to inquire, \If the error term generating method using digital
numbers relates to the transcendental numbers  and e, is the inverse true, where in
some sense we can say the use of  and e generate the error term?" Although this
question is confusing, it cuts deep. In other words, there is little choice when using
digital integers  the zeta function using digital integers convergesb to . And e is
deeply related to  by similar reasoning associated with the choice to use digital
integers  as opposed to simplex-integers.
It is reasonable to realize, though, that  and e are deep aspects of the error term.
And the error term is the problem. It is simply the delta between the imprecise
temporary \placeholder" prime density prediction algorithm and the currently
unknown imprecise ones.
So is it as simple as that? Can we avoid the error term by avoiding digital numbers
and, by extension,  and e?
We will not see  and e when we attack the problem via simplex-integers. The true
algorithm determining the number of primes within a bound is geometric and related
to an algebraic number. Speci¯cally,
p
2. Interestingly, we do see a relationship to 2 in
current prime number theory based on digital integers. The non-trivial zeros are all on
a coordinate at 12 the length of the strip bounded by  12 on the left and 12 on the right.
bMore technically, zeta(x) converges to 0 when x goes to in¯nite, but zeta(2n) for any positive even integer
is expressed as a rational fraction multiplied by  at the power 2n.
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As mentioned, the zeta function is a method to uncover something deep about
the primes by expressing them as a convergent additive series. The additive series
of the simplex-integers is naturally convergent without need to invert values. Con-
sider the circumradius of each n-simplex and index each successive one to a cir-
cumradius generated by an n-simplex with a number of vertices equal to that prime
number. The circumradius of the 1-simplex is 12. And the circumradius of the in¯nite-
simplex is 1=
p
2. The circumradii of all simplexes can be related as a series of con-
centric circles, each two with a di®erent distance between them than the distance
between any other two. The distance between each two corresponds uniquely to a
certain prime or non-prime integer such that we may call each delta between se-
quential circumradii a unique integer. And the sum of all deltas is 1=
p
2. Within any
span of such rings, there is a subset that are prime based, wherein the pattern of radii
are neither periodically nor randomly arrayed. They are arrayed as a quasicrystal.
Here, we see the¯rst example of a radically di®erent form of convergence in
simplex-integers, where the convergence value is an algebraic number instead of a
transcendental number like 2=6.
3.7. Non-homogeneity of sequential number deltas
A key di®erence between digital and simplex-integers is the information encoded in
the delta between successive numbers. For example, a few of the geometric deltas
between two successive simplexes are:
. Dihedral angles (series ranges from ArcCos½ to ArcCos(0).
. Circumradii
. Hyper-volumes
Again, the deltas index to integers. The delta between the circumradius of the
1-simplex and 2-simplex would index to the integer 3 because the 2-simplex corre-
sponds to 3 vertices or 0-simplexes.
The salient point is that the delta between two successive simplex-integer geo-
metric indexes is unique and di®erent than the delta between any other successive
pair of simplex-integers.
By contrast, the di®erence between any two successive digital integers is always 1.
Accordingly, it is homogeneous and therefore gives absolutely no number theoretic
information. This rich extra information of simplex-integers provides a wealth of
geometric clues to fuel a new approach to search for the exact scaling algorithm for
density distribution of geometric primes as prime-simplexes  the deep reason for
why a prime-simplex shows up every so often in a given series of simplexes, and this
answer is trivially isomorphic to the distribution of prime digital integers.
3.8. Extensions to lattices and geometric algebras
Consider a 2-simplex, the equilateral triangle. Around it, there can be an in¯nite
2-simplex lattice, called the A2 lattice.
29 This is a tiling of the plane with equilateral
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triangles. The lattice associated with the simplex-integer 4 is the lattice described by
a maximum density packing of spheres in 3D  the way oranges are stacked in the
supermarket. This is called the A3 lattice and is composed by rotating A2 lattices
from one another by ArcCos(1=n), where n is the integer corresponding to the A2
lattice (in this case, 3). This continues ad in¯nitum, where, for example, the lattice
associated with the integer 100, the 99-simplex lattice called A99, is a stack of irra-
tionally rotated parallel lattices A2 through A98.
We can extend the idea of prime-simplex distribution to prime-simplex-A-lattice
distribution. Each n-simplex can pack to form a crystal of n-simplexes 
compositing to the An-lattice for a given dimension. We can then algebraically
explore the reasons for why prime A-lattices appear where they do in a given span
or stack.
A given A-lattice is associated with various geometric algebras, such as Lie and
Cli®ord algebras. The geometric algebra of a given A-lattice contains an algebraic
stack of sub-algebras associated with each sub-A-lattice. This algebraic space cor-
responds to a point array and is called a moduli space.30 These geometric algebra
tools can be used to work on the geometric problem of¯nding the actual and precise
scaling algorithm for the density distribution of prime-simplex associated A-lattice
geometric algebras within a larger stack of algebras  again, an algorithm identical
to the unknown precise density algorithm for the distribution of prime digital
numbers, which would immediately lead to the proof (or disproof) of the Riemann
hypothesis.
The \writing on the wall" seems clear. The 2300 years of searching for the correct
prime distribution algorithm via digital numbers and the 157 years of mathemati-
cians trying to prove the Riemann hypothesis via digital numbers are impressive.
This apparent roadblock supports the argument that the solution can only be found
within the realm of geometry. In addition, the geometric physical connections and
the geometry of  and emake it even more reasonable to surmise that digital number
theory and stochastic approaches will not lead to an answer.
Because only prime simplex vertices divide evenly into all quantities of sub-simplex,
there is an aspect of these special prime shapes that is di®erent than non-prime sim-
plexes. This geometric di®erence has additional aspects other than the pure set the-
oretic qualities of the divisibility of vertices. It is analogous to the fact that an
equilateral triangle contains all the set theoretic information of the complete and
undirected graph of three objects. However, it also contains additional shap-related
geometric information that goes beyond the set theoretic information of the graph.
Prime-simplex-integers possess special group theoretic qualities, topological qualities
and various geometric qualities that set them apart from non-prime-simplexes.
3.9. Graph-drawings
As discussed, one may gain intuition by understanding simplexes and A-lattices via a
growth algorithm that generates a graph drawing, where the quality of the connection
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magnitudes are lengths, making the object a graph drawing because the connection
types are geometric line segments. Abstractly, this can be true without admitting a
smooth spatial substrate, such as R3. In other words, the space is discretized such
that there exists no information or space between the line segments of the network.
This minimalistic graph-drawing space is in some sense a quantized subspace of a
continuous space.
Picture a line segment and rotate a copy on one end by 60 into another¯nite
1D spatial dimension to generate the three points of a 2-simplex and its associated
A-lattice. We can play with semantics by saying we have rotated a copy of the¯nite
length 1D universe into another¯nite 1D universe. The objective of the visualization
is to disassociate ourselves from the idea of a smooth Euclidean space in a network
of¯nite 1-simplexes, where length is real but restricted to the connections of the
network  a graph-drawing made of 1-simplexes.
Or we can index the length value to an abstract graph theoretic magnitude.
In any case, the connection relationship between the three points (from the two lines
sharing a point) is isomorphic to a 2-simplex (as shown in Fig. 12).
Next, we rotate a copy of the second line (note that we do not need the green line
in the diagram in order to generate the 3 points of the 3-simplex) into the third
spatial dimension by 60 from the previous line to generate the 4 points of the
3-simplex and so on. Again, we reject the assumption of a smooth 3-space in favor of
an approach that is graph-theoretic.
Each of these 60 rotations from the previous edge is equal to rotating the edge by
ArcCos½ðn 1Þ=ð2nÞ from the total simplex construct below it, where n is the
simplex number.
As mentioned, this iterative process results in the stack of simplexes converging to
a circumsphere with a diameter of
p
2 at the in¯nite-simplex. When the construction
of any simplex, such as the 99-simplex, is visualized with 60 rotations extruding
successive spatial dimensions, one realizes that the lines form a hyper-dimensional
discretized non-Archimedean spiral.
In other words, the circumradii of the simplex series starts at 12 for the 1-simplex
and converges at 1=
p
2 at the in¯nite-simplex. This is in stark contrast to the cubic
series of circumradii, which is divergent, converging at an in¯nite radius for the
in¯nite-cube. So from the simplex-integer 2 (1-simplex) to the in¯nite-simplex, the
increasing radii values must be distributed over a distance of 1=
p
2 12 	 0:207.
Fig. 12. A blue and a red one-simplex rotated by 60 degrees make a two-simplex.
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But in just the distance from the circumradius of the 1-simplex to the 2-simplex, we
cover about 0.077 or more than 37% of the total 0.207 distance to convergence 
a distance that must be distributed over an in¯nite number of simplexes.
Of course, this rapid convergence is massively exponential.
The distribution of prime numbers on the digital number line drops with distance.
For years, number theorists have used the prime number theory algorithm (and
improved versions) related to the natural logarithm number e to predict the number
of primes within a given bound. This correlates to a non-Archimedean spiral called
the logarithmic spiral because the distances between turnings increase in geometric
progression as opposed to an Archimedean spiral.
Is it the natural logarithm number e 	 2:718 that corresponds to the actual
algorithm for prime distribution  the one that does not generate the error term?
As mentioned, e is an artifact of the exploration of the problem via digital numbers
and is fundamentally part of the error term. That is, the algorithm for predicting
prime density that is related to e simply does not work. It is the chosen formalism of
the approximation itself that generates the error term corresponding to the non-
trivial zeros. The logarithmic spiral correlated to the distribution of prime-simplexes
should logically relate to a hyper-spiral corresponding algebraic irrational
p
2. So,
just as the prime number theorem algorithm corresponds to the non-algebraic
transcendental number e, the simplex-integer prime number algorithm for prime
distribution would correspond to the algebraic number
p
2 (and to the golden ratio
by arguments beyond the scope of this paper).
3.10. Creation of a prime number quasicrystal
There is an interesting approach we will explore in a subsequent publication. We will
cut + project various A-lattices to lower dimensional quasicrystals and do spectral
analyses on the Fourier transform of each. We predict prime-A-lattice associated
quasicrystals will possess distinct spectral signatures. If so, a spectral analysis of the
superposition of a span of simplex-integer associated A-lattices projected to lower
dimensional quasicrystals is expected to reveal the signature of a prime A-lattice
distribution scaling algorithm. Such an algorithm would predict the exact prime-
simplex density distribution for any bound of projected A-lattices.
3.11. Section 3. Conclusion
We have established a hypothesis, which should be true by trivial deduction;
an exact algorithm for the distribution of primes exists in the realm of pure
geometry.
4. Simplex-Integer Uni¯cation Physics
If nature were a self-organized simulation, it would be a simplex-integer based
quasicrystalline code derived from E8.
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4.1. Introduction
The digital physics31 view is the idea that reality is numerical at its core [See work by
Ed Fredkin,32 To®oli,33 Wolfram,34 and Wheeler35]. But the numbers need not be
digital. They can be shape-numbers, such as simplex-integers. Because reality is
geometric and has three spatial dimensions, one could surmise the following: If na-
ture were built of 3D bits of information that are also numbers, the most powerful
candidate for a 3D geometric number is the 3-simplex. Power in this context is
synonymous with e±ciency in the manner explained in Part 2.
Higher-dimensional lattices, such as E6 and E8 that are associated with uni¯ca-
tion physics, can be constructed entirely from 3-simplexes. Certain projective
transformations result in the 3-simplexes remaining regular but being ordered into a
quasicrystalline code that encodes the higher-dimensional lattices and associated
gauge symmetry physics.
A hallmark and general characteristic of quasicrystals is the golden ratio.
For example, the simplest quasicrystal possible is the two length Fibonacci chain, as
1 and 1/. Virtually all 3D quasicrystals found in nature are golden ratio based on
icosahedral symmetry.
A self-organizing code on an abstract quasicrystalline substrate is in some sense
like a computer but better described as a neural network. Computer theory is con-
cerned with the e±ciency of creating information in the form of solutions to pro-
blems. Information theory is concerned with the e±ciency of information transfer.
Neural network theory 36 is concerned with both the e±cient creation and transfer of
information in a network. Neural networks operate via codes, i.e., non-deterministic
algorithmic processes  languages. Neural networks in nature are spatial (geometric)
arrays of nodes with connections, such as particles connected nonlocally by quantum
entanglement or forces. Clearly nature, like a neural network, accomplishes the dual
task of (a) creating new information (computation) and (b) transferring information.
So the universe as a whole is a neural network in the strictest sense of the term.
A special quality of neural networks that sets them apart from computers is that
they are non-deterministic. If one subscribes to the theory of randomness and does
not require a theory to explain the generator of randomness, one can decide that the
free choices in a neural network code are random. On the other hand, there is a
special cases in Physics where human free will emerges in a biological neural network,
which itself emerges from fundamental particle physics and presumably some un-
known quantum gravity theory. In this case, the free will can act on the syntactical
choices in the code-theoretic neural network, providing an explanation for the syn-
tactical choices that might be more explanatory than stopping at randomness as the
unprovable axiom. It is generally known that physical reality is: (a) non-determin-
istic and (b) that it creates the emergence of non-random free will, at least in the case
of humans. As this is not a philosophical paper, we will simply say that whatever free
will is, it is non-deterministic. It behaves similar to the concept of randomness insofar
as being non-deterministic. The di®erence is that choices, as the actions of free will,
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are made with a blend of subjective meaning, perceptions or opinions combined with
logic and choices of strategy. So symbolic language and meaning are deep principles
embedded in the theory of free will. Conway and Kochen proved that if free will is
real, fundamental particles have some form of non-random free will.37
Another fundamental feature of nature is that, as a network, it is concerned with
e±ciency in the form of the principle of least action and similar laws.38 In fact,
e±ciency may be the most fundamental behavior of reality  leading directly to
Noether's second theorem about conservation and symmetries in nature,39 conser-
vation laws and from there to the modern gauge symmetries uni¯cation physics, such
as seen in the standard model of particle physics.
4.2. E8 in nature
The most foundational symmetry of nature uni¯es all fundamental particles and
forces. It can be described as:
All fundamental particles and forces, including gravity, are uniquely
uni¯ed according to the gauge symmetry transformations encoded by
the relationships between vertices of the root vector polytope of the E8
lattice – the Gosset polytope.6
We have also shown cosmological correlations to E8 in Heterotic Supergravity with
Internal Almost-Kahler Con¯gurations and Gauge SO,32 or E8 x E8, Instantons.
40
However, our general approach is to exploit projective geometry as the symmetry
breaking mechanism in a quantum gravity plus particle physics approach, which
recovers particle gauge symmetry uni¯cation.
The simplest polytope in eight dimensions is the 8-simplex. The E8 lattice is the
union of three 8-simplex based lattices called A8. This lattice corresponds to
the largest exceptional Lie algebra, E8. That is, the simplest 8D building block of the
Gosset polytope and E8 lattice is the 8-simplex  the most powerful number
that inherently encodes the counting function of 32 = 9 and its full set theoretic
substructure. Nine, incidentally, is the¯rst odd number that is not prime.
4.3. E8 derived quasicrystal code
In Starobinsky In°ation and Dark Energy and Dark Matter E®ects from Quasicrystal
Like Spacetime Structures41 and Anamorphic Quasiperiodic Universes in Modi¯ed
and Einstein Gravity with Loop Quantum Gravity Corrections,42 we show how
quasicrystalline codes can relate to quantum gravity frameworks.
A 4D quasicrystal can be created by projecting a slice of the E8 crystal.
43 This 4D
quasicrystal is made entirely of 600-cells, which are each made of 600 regular
tetrahedra. So the fundamental 3D letters or symbols of this quasicrystal are
3-simplexes. The allowable ways these geometric number symbols can spatially relate
to one another is governed by cut þ project-based geometry,44 which generates the
syntax of this non-arbitrary quasicrystalline code. The term code or language applies
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because the syntax allows various legal con¯gurations that are determined by the
size, shape and position of the cut-window in the hyper-lattice from which the
Quasicrystal is cut and projected from. A language or code must have degrees of
freedom within the rules and a¯nite set of symbols that must be arranged by a code
user in order to create meaning. Geometric codes, such as quasicrystals, generate
geometric meaning, such as waveform and quasiparticle position. The code user may
emerge from the evolutionary complexity of the system itself and can be as sophis-
ticated as a human consciousness and beyond. Or it can be simple, like the guiding
tendency of a tornado to preserve and grow its dynamical pattern for as long as
possible in a new physical ontology based on code theory instead of randomness.
Furthermore, how two or more syntactically legal quasicrystals can be ordered
in a dynamic pattern or animation has a separate syntax scheme based on how the
cut-window moves through the hyper-lattice. That is, all behaviors and rules are
part of a code based solely on geometric¯rst principles with no arbitrary ad hoc
mathematical contrivances.
The caveat is that a free will chooser of some form must execute the free choices in
the code. This is the case with all codes, whether that be a computer language or
song-language of birds. A general quality of codes is that meaning is not maximized
and breaks down when strategic choices in the syntax are replaced by, for example, a
pseudo random number generator.
Discovering a fundamental uni¯cation model of all particles and forces based on
such a geometric¯rst principles code is a worthy but formidable challenge. It would
be a microscopic¯rst principles theory of everything. Currently, there exists no¯rst
principles explanation for the¯ne structure constant, the speed of light, Planck's
constant or the gravitational constant. In other words, there is no known¯rst
principles uni¯cation theory. In fact, the fundamental constants h and G and c are
only known to about four places after the decimal. The CODATA values, which go
out to a few more places after 4, are an averaging of six established experimental
methods that all disagree at the 5th place after the decimal.
Syntactically legal con¯gurations of these quasicrystal-based simplex-letters form
simplex-based words and sentences. In other words, groups of simplexes have
emergent geometric meaning  shape and dynamism. Sets of these inherently
nonlocal quasicrystalline simplex sentence frames can be ordered into animated
sequences and interpreted as the physical dynamic geometric patterns of space that
have both wave and particle like qualities, a well understood dualistic quality of
phason quasiparticles in quasicrystal codes.45
While this part of the discussion is a mix of fact and conjecture, the reader may
agree that, if nature is a code or simulation based on maximally e±cient symbolism,
the following three ideas may be at play:
(1) Nature must be an e±cient symbolic code capable of simulating a 3D reality.
Accordingly, the most powerful symbol in 3D— the 3-simplex—may be exploited
because it is the simplest and most e±cient 3D quantum of information.
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(2) Spacetime would be discrete. While in quantum mechanics the spin is quantized
(the quantum of action is the Planck's constant), energy is quantized (the photon
is the quantum of energy), and here we also propose spacetime which itself is
quantized. Nature must have an e±cient geometric \pixel" or foundational
symbol, just as a dynamic image on a high-resolution video monitor is composed
of invisible microscopic building-block pixels or just as a binary computer code is
made of irreducible elements symbols, where bytes can be further decomposed
into bits but no further. In our proposed framework, the simplex-integer is the
irreducible and non-transformable \pixel" in the simulation that composes our
3D reality. It is the fundamental shape-symbol in a geometric code/language. In
short, simplex-integers \switch-hit" as both numbers and spatial building blocks.
(3) It must be a symbolic code derived from E8, which encodes the gauge symmetry
uni¯cation of all fundamental particles and forces. We have generated a 3D
quasicrystal language of 3-simplexes derived from E8ð15Þ.
The causal dynamical triangulation program of Amjorn and Loll46 is encouraging
evidence that fundamental physics can be modeled with aperiodic con¯gurations of
3-simplexes as the only building-block element (see Fig. 13). They have generated
very close approximations of Einstein's¯eld equations.47
4.4. Quasicrystals as maximally e±cient codes
Just as simplex-integers are the most powerful numbers to express counting function
and set theoretic information, quasicrystals are the most e±cient codes possible in
the universe of all codes.
This is a major claim. To understand it, we should¯rst establish the fact that an
n-dimensional quasicrystal is a network of quasicrystals in all dimensions lower than
it. For example, the Penrose tiling, a 2D quasicrystal, is a network 1D quasicrystals.
A 3D Penrose tiling, called Ammann tiling48 is a network of 2D quasicrystals, which
are each networks of 1D quasicrystals.
So the building block of all quasicrystals are 1D quasicrystals. Reducing further,
we should understand that there are an in¯nite set of 1D quasicrystals. The \letters"
of a 1D quasicrystal are lengths. A 1D quasicrystal can have any¯nite number
of letters. However, the minimum is two. The Fibonacci chain is the quintessential
1D quasicrystal. It possesses two lengths related as the golden ratio. In order for a
Fig. 13. Causal triangulation of a surface. The vertical dotted lines are the timelike edges.
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quasicrystal greater than 1D to have only two letters, the letters must be 1 and the
inverse of the golden ratio. Interestingly, this simple object has a deep relation to E8.
When a slice of E8 is projected to 4D according to a non-arbitrary golden ratio-based
irrational angle,43 the resulting quasicrystal is made entirely of 3-simplexes and is the
only way to project that lattice to 4D and retain H4 symmetry. The angle between
adjacent 3-simplexes is 60 þ ArcCosð½3’ 1=4Þ ¼ ArcCosð1=4Þ, where ’ is the
golden ratio. This quasicrystal, fully encoding gauge symmetry uni¯cation physics,
can be described as a network of Fibonacci chains. These are the most powerful 1D
quasicrystals for two reasons. As mentioned, the power of a code relates to how many
building block symbols it has. This de¯nition of power relates to the discussion
earlier, where we spoke of the left side of the equation as being the magnitude of
simplicity of the symbolic system. But a code cannot have fewer than two funda-
mental symbols for obvious reasons. This is what makes binary codes so powerful.
Secondly, Fibonacci chain quasicrystal codes are based on the Dirichlet integers 1
and 1=’, which possess remarkable e±ciency characteristics, such as error detection
and correction abilities and multiplicative and additive e±ciencies. For example,
they are closed under multiplication and division. As with all quasicrystals,
Fibonacci chains are fractal.49
The function of division stands out in the arsenal of powers that the golden ratio
possesses because it relates to measurement, which is the deepest and most enigmatic
aspect of quantum mechanics that is not yet fully understood.
Physics Nobel laureate, Frank Wilczek of MIT said,50 \The relevant literature [on
the meaning of quantum theory] is famously contentious and obscure. I believe it will
remain so until someone constructs, within the formalism of quantum mechanics, an
observer, that is, a model entity whose states correspond to a recognizable caricature
of conscious awareness."
Wilczek is speaking of the need to build measurement into a new quantum me-
chanics mathematical formalism that incorporates an operator capable of measuring
at the Planck scale. His motivation is on solid ground. Quantum mechanics indicates
that the ontology of physical reality must be based on measurement, where all that
exists is that which is measured. Certainly, it is extreme to postulate that physical
reality needs humans to measure it in order to be real. A physical formalism based on
the premise that reality is made of information in the form of a code would require
a quantum-scale mathematical operator capable of actualizing information via
measurement.
A measurement of any form is ultimately a spatial relationship between the
measurer and two additional points in space. This is the case with any detector,
such as a human eye or a Geiger counter. Waveforms are reducible to quantum
particles. All detectors are reducible to component particles that interact with signal
particles, such as photons, that are emitted from another particle being measured
at a distance. For example, a camera takes a photo of a tree by receiving rays of
photons that trace to the camera lens. The irreducible measurement, however, is the
relationship between a detecting particle and two other particles at two other
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coordinates. This forms a triangle, where the detector is one vertex and the two
measured coordinates are the other vertices. The fundamental information being
registered by the detector is a transformation via contraction of the edge length of
the triangle that is not connected to the detection particle. For example, you observe
two friends in the distance. We can conceptualize you and the two friends as three
points in space. There is an actual distance length between your two friends, which
we will call L. Because it is impossible for you to have a perfectly equal distance
between you and each of your friends, you observe or measure the information of L-l,
where l is some contraction value on L.
Your transformation of L gives you information about the relationship of your
two friends and their relationships to you. If they are standing one car length apart
but your angle relative to them is such that you perceive it as ½ a car length, then
you intuitively know how to decode that information to tell you their actual distance
as one car length plus your position relative to them. Similarly, when you look at the
complexity of, say, a tree, the massive package of information from that measure-
ment/observation is merely a composite of these individual length transformations
between pairs of points and the measuring detector, forming a transformation of the
pre-transformed triangle.
So L and l form a relationship in your mind as a ratio. The meaningful information
of your measurement is not l it is the ratio of L to l, which tells you information about
the relationship of the two measured points to one another (their actual length
relationship) and their length relationship to you from your vantage point.
Consider this set of three points that are equally spaced in a line in 3-space. If you
measure them with your eye from a golden ratio-based vantage point equal to a
rotation of the line of three points by a golden ratio angle, then you can divide the
total length into two parts as 1 and 1=’ (as shown in Fig. 14). This is true only in
perspective projection, never in orthogonal projection where the segment sizes would
stay equal.
To review, all measurements are divisions or ratios. And a choice of measurement
(observation) is necessary to actualize or make-real any information. If we consider
that reality is information theoretic or code-based, we must model a mathematical
measuring operator, as discussed. So why would the golden ratio obey the principle of
e±cient language better than any other ratio? Why would it be more powerful in
terms of the ratio of symbolism to meaning?
Fig. 14. Two segments of equal lengths rotated by an angle and seen through perspective projection. Blue
is nearest to the observer and looks bigger than the red segment. The ratio is the golden ratio (beware–that
will not be the case with the orthogonal projection).
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For golden ratio divisions, going down from long to short, the ratios between
successive pairs is the golden ratio. Going up, it is the inverse of the golden ratio. This
quality is known as in°ation and de°ation and the golden ratio achieves it with only
two symbols or numbers 1 and 1=’.
But for divisions other than the golden ratio, going down needs two ratios (3/2,
2/1 for example); going up also requires two ratios. Consider the idea that Planck
scale measurement operators in the quantum gravity code use abstract observation
actions in the E8 derived quasicrystalline point space to actualize compact symbolic
objects that are themselves ratios  simply ordered arrays of the two Dirichlet
integer values 1 and 1=’. This binary pair of values is maximally e±cient in terms of
the symbolism to meaning ratio.
For the ’ (golden ratio) division, there is only one ratio needed for encoding the
relationships of the consecutive segments going down in length (as shown in Fig. 15).
a
b
¼ b
c
¼ ’;
ð8Þ
Or going up in length
c
b
¼ b
a
¼ 1
’
:
ð9Þ
For example, any other division, such as dividing into thirds, requires two ratios and
therefore more symbolic information to express
a
b
¼ 3
2
;
b
c
¼ 2
1
or
c
b
¼ 1
2
;
b
a
¼ 2
3
:
ð10Þ
The other aspect of the golden ratio that is powerful and may be important for a
simulation code of reality is its fractal nature. In the last 37 years, fractal mathe-
matics has been found to be at play at all scales of the universe from cosmic to the
sub-atomic scales.49 Dividing a line by the golden ratio, if we take the short length
and place it on top of the long length, we are left with a section of the long length that
is left over. That length is even shorter than the short length of the¯rst division and
the ratio of this new short length to the original short length is the golden ratio. This
process can continue to in¯nity in the smaller direction with the ratio of the re-
mainder to the previous length always being the golden ratio. Furthermore, this
process can be applied in the other direction, where we add the long piece from the
original division to the undivided length. The ratio of the new combined length to the
long length from the¯rst division is the golden ratio. This also continues to in¯nity.
The golden ratio is the ultimate recursive fractal, generating the most information
for the least amount of symbolic symbolism and measurement action.
Fig. 15. The golden ratio: B is to A what C is to B.
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A phason °ip in a quasicrystal is a binary state change of a point, where it is
registered as being on or o®. If it is on, it is an active node with a connection to other
points in the quasicrystal. The syntax rules allow legal choices of whether a point can
be on or o®. One can call the total set of points the possibility space. The points that
are chosen to be on by the code user, are active in that frame of the dynamic
quasicrystal. Active or on points have connections and are syntactically legal se-
lection con¯gurations of the possibility space. For example, this is a projection of the
32 vertices of the 5-cube to the plane, where we see 31 total points with an overlapped
32nd point hidden in the middle. Note that the Penrose tiling is made by projecting a
slice of the 5-cube lattice to the plane. These 31 points are a small section of the
possibility space that the dynamical phason code of the Penrose tiling operates on
(see Fig. 16).
The Penrose tiling is a tiling of two types of selection patterns of 16 of the 31 point
decagonal possibility space. The two decagons can overlap other decagons in two
ways or kiss without overlapping. In Fig. 17, we highlight those two selection
Fig. 16. The projection to 2D of a¯ve dimensional cube with 32 vertices projected to 31 giving possibility
space of Penrose Tiles.
Fig. 17. Two types of selection pattern of 16 of the 31 point decagonal possibility space, as they exist in a
larger Penrose tiling Quasicrystal.
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patterns, as they exist in a larger Penrose tiling quasicrystal. You can easily visualize
how to select one of the two 16 point combinations by looking at the projection of the
full 31 point possibility space (shown in Fig. 16).
The empire of a given point consists of points that are forced to be on by the given
point. For example, look again at the 31 points in Fig. 16. Note that if you select the
center point to be on, you are forced to have a certain 15 additional points also be on
and another 15 to be o®.
Those 15 points that were forced to be on are the empire of the one point which
you, the language user, chose to be on in a phason °ip binary action.
Why is this interesting in terms of e±ciency? In computer theory, we try to
conserve binary actions. It costs electricity and time to open or close logic gates in a
computer. So e±ciency is important. We want codes that achieve maximal infor-
mation with as few actions as possible.
Changing a single point to be on or o® in a Fibonacci chain 1D quasicrystal forces
an in¯nite number of additional points throughout the possibility space of the 1D
chain to also change state. This global or nonlocal \spooky action at a distance" is
very powerful in terms of the ratio of action to meaning.
If we live in an information theoretic universe, then, abstractly, the action we are
trying to conserve is binary choice. Speculation of what the substance or entity or
action is that makes the choices in the code is °exible. Our frame work deals with the
math and behavior of the code, not so much who or what the operator of the code
must be. The principle of e±cient language requires the operation of the code to tend
toward maximal meaning for the least number of on–o® choices.
Resources are always used to make a choice in any physical model. For example,
in the neural network of a human brain, choices cost calories and time. In an arti¯cial
neural network, choices require time and electricity. So e±cient neural networks
generate as much meaning as possible with a given number of connection actions 
they generate maximal information for as few binary choices as possible.
Each Fibonacci chain is isomorphic to a Fibonacci word, which is a string of 0s
and 1s that encodes a unique integer.51 Of course, the larger the integer, the greater
the magnitude of the information. For example, a one millimeter Fibonacci chain
with Planck length tiles is isomorphic to a Fibonacci word with 1031 0s and 1s and
corresponds to an equally enormous integer. Changing one point on the Fibonacci
chain possibility space from o® to on, changes the state of points along the entire
chain, thereby changing the Fibonacci word to a di®erent integer. The principle of
empires in arti¯cial neural networks consisting of networks of Fibonacci chains
involves enormous e±ciency when one is interested in conserving binary actions or
choices. If nature is a computational language based on neural network theory and
globally distributed computation and connectivity, quasicrystal codes are the most
e±cient possible.
When a network of Fibonacci chains is formed in 2D, 3D or 4D, a single binary
state change at one node in the possibility space changes the Fibonacci chains
throughout the entire 1þ n dimensional network of chains.
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4.5. DNA and quantum computers as examples of 3D
neural networks
Manmade computer code symbols are, in some sense, minimally e±cient. That is, the
binary symbols encode an instruction to do one binary state action in a logic gate to
express only one bit of information. DNA is not quite a computer. It is a neural
network in the sense that it both computes information using its code rules and it
transfers information within its structure. Like a neural network, it achieves its com-
putations and information storage in a distributed manner within 3-space. A single
position in the DNA possibility space of coordinates where one of the four molecules in
the code can exist serves as information in more than one 1D string of code. For
example, an adenine molecule can exist at some location in the DNA coordinate space.
This then forces certain states of the 4-letter code for other positions in the string,
according to syntactical rules of the code. That string of code is wrapped around the
double helix and has an empire of forced coordinate identities from the four letter code.
But the empire is not just in 1D along that single string. Information relative to that
one molecule selection of adenine is also encoded into strings that run in-line with the
axis of the double helix and also diagonal to the axis. It is similar to the analogy of
the game Scrabble, where a choice of a single letter on the grid of the possibility space of
the game can encode information in more than one word. So the choice of the adenine
molecule at that coordinate achieves a great deal of e±ciency by (a) playing a role in
multiple 1D strings and (b) by forcing other syntactically controlled actions of coor-
dinates in the empire of that single registration of adenine in the DNA possibility space.
DNA has quasicrystalline structure. In fact, Erwin Schrodinger¯rst deduced that
DNA has a quasiperiodic structure in his book What is Life, published nine years
prior to Watson and Crick's discovery of DNA in 1953.52 His deduction is in-line with
the theme of this paper. Speci¯cally, crystalline structures are deterministic and have
no degrees of freedom in terms of their abstract construction. They are not inherently
languages because they are too rigid in their construction rules.
On the other hand, amorphous or disorderly materials do not have structural
rules and can have a virtually in¯nite number of microscopic geometric relationships
– geometric symbols. The lack of rules and lack of a¯nite set of geometric symbols
prevent a dynamic code from evolving within amorphous materials. The \sweet
spot" between order and disorder, where a language or code can emerge, is in qua-
sicrystalline order. Only within aperiodically-ordered structure is there a true code
with a¯nite set of geometric symbols, rules and syntactical freedom.
Themost powerful codes are based on the golden ratio because the ratio of symbolism
to geometric meaning output is maximal. For example, DNA ismade of two helices that
have pentagonal rotational symmetry, which is based on the golden ratio. The two
helices themselves are then o®set from one another by a golden ratio related value called
a Fibonacci ratio, which is a rational approximation of the irrational golden ratio.53
Quantum computers are another example of systems where one node serves
multiple roles in various relationships. 3D clusters of atoms, often with golden
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ratio-based icosahedral symmetry54 in a quantum correlated state interact with one
another in various combinations to process and create information as a group 
a spatial network of nodes very di®erent than the ordinary notion of a 1D
computational system.
4.6. Symbolic power of Fibonacci chain networks
It is well known that Fibonacci codes have unique and powerful properties in terms of
error correction and detection.51
For example, all sequences in a Fibonacci word end with \11". And that sequence
appears nowhere else in the data stream of that symbolic group object. Changing a
bit corrupts the sequence (the symbolic group object). However, within a few more
symbols, the pattern \11" will appear again, which indicates the end of the string or
group symbol.
The system or user can then simply resume coding with only those few symbols
felt to be incorrect. The power in this is that one bit can only corrupt up to three
symbols. No other code shares this property. Error detection is fast, and errors are
limited in how much damage they can do. Error correction is similarly powerful and
unique. Let us say that a 0 is erroneously changed to a 1 that is adjacent to a correct
1. A 1 that is part of the data stream gets changed to a 0. A 1 that is part of the
ending 11 gets changed to a 0 and so on.
When an error occurs in ordinary codes, it will exist uncorrected in the string
forever.
The power of 3D networks of Fibonacci chains relates to the spatial dimension of
the quasicrystal being able to host objects with icosahedral symmetry. For example,
the 4D analogue of the icosahedron is the 600-cell.55 The icosahedron is one of the
¯ve regular polytopes in 3D  the Platonic solids. Three of the solids correspond to
crystal symmetries because their combinations can tile space. These are the square,
octahedron and tetrahedron. The other two are correlated with quasicrystal sym-
metry, the 600-cell and the 120-cell. These correspond to the quasicrystal-based
Platonic solids called the icosahedron and dodecahedron, each possessing icosahedral
symmetry. Again, in 3D there are¯ve regular polytopes. In 4D, there are six. And in
all dimensions higher than 4D, there are only three  the analogues of the tetra-
hedron, octahedron and cube  the crystal-related polytopes. The quasicrystal-
related regular polytopes are exclusive to dimensions less than 5D. So the special
dimensions for Fibonacci chain-related quasicrystals are 1D, 2D, 3D and 4D. Of these
dimensions, 4D can host the quasicrystal with the densest network of Fibonacci
chains, where 60 Fibonacci chains share a single point at the center of the 600-cells in
the E8 to 4D quasicrystal discovered by Elser and Sloane.
43 In other words, a binary
state change in the possibility space for this quasicrystal changes the state of
many other Fibonacci chains associated with that point. And numerous other
points in the possibility space also change state, not just the ones in the Fibonacci
chains connected to the aforementioned point. All this binary state change  the
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empire  occurs due to the geometric¯rst principles via the state change of a single
node in the possibility space.
If the universe is a neural network interested in maximal e±ciency, this would use
a substrate like this. The fact that this quasicrystal and its 3D analogue discovered
by our group called the quasicrystalline spin network (QSN) encodes gauge sym-
metry uni¯cation physics may be evidence for the trueness of the conjecture. And
this would be more likely if the universe is a neural network code concerned with
expression of maximal meaning for the minimum number of binary state choices/
actions.
The principle of e±cient language guides the behavior of the code choices in this
framework, where binary actions in the code are chosen such that maximum infor-
mation or meaning is generated for the least number of binary choices.
Meaning comes in two categories:
(1) Physical or ultra-low subjectivity geometric information — the prototiles of the
quasicrystalline code, wherein all particles and forces can be simulated such that
the simulation are one and the same and are themselves in physical reality.
(2) Emergent or virtually transcendent and highly subjective information, such as
Mathematics and Humor. This form of information can never be separated from
the geometric physical information and quasicrystalline code. For example, the
abstract thought of \love" comes with a package of memories and associations
that trigger countless changes in the nonlocal waveform domain of quantum
mechanics, gravity and electromagnetism.
At a physical level, evidence for this tendency toward e±cient code use would exist in
the form of the principle of least action and similar principles and conservation laws.
At a non-physical level, evidence for this would exist in the form of the delayed choice
quantum eraser experiment56 and Bem's retro-causality experiments57 in addition to
well-known experiments of quantum entanglement over space and time. As engines
of abstract meaning generation and perception, humans would be a special case in a
universe obeying the principle of e±cient language, where our perceptions of
meaning and information far exceed the brute simple geometric meaning expressing
physical phenomena in the quasicrystalline code.
The degree 120 vertices of the E8 to 4D quasicrystal appear to be the maximum
possible density of Fibonacci chains in a network of any dimension and therefore the
most powerful possible possibility space for a neural network. 3D quasicrystals or-
dinarily have a maximum of degree 12 vertices with six shared Fibonacci chains.
Fang Fang of Quantum Gravity Research discovered how to create a 3D network of
Fibonacci chains with degree 60 vertices.15
This quasicrystal is made entirely of 3-simplexes, the simplest possible \pixel" of
information in 3D (see Fig. 18). It encodes E8 uni¯cation physics and is derived from
the aforementioned E8 to 4D quasicrystal. Refer to Fang Fang's et al. paper An
Icosahedral Quasicrystal as a Packing of Regular Tetrahedra103 regarding the
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construction of a dense, quasicrystalline packing of regular tetrahedra with icosa-
hedral symmetry.
4.7. Is the error correction code found in gauge symmetry physics a
clue that nature computes itself into existence?
James Gates, the John S. Toll Professor of Physics at the University of Maryland and
the Director of The Center for String and Particle Theory found the widely used
doubly-even self-dual linear binary error-correcting block code embedded in the
network of relationships of the gauge symmetry uni¯cation equations of fundamental
particles.58 These are the exact same codes used in web-browsers and peer-to-peer
network simulations to ensure the consistency of information transfer from client to
client. Furthermore, he found that the error correction codes relate speci¯cally to
geometric symbols, called adinkas,59 which encode the relationship of particle gauge
symmetry equations. This astounding¯nding is one of the most powerful pieces of
evidence in support of the digital physics view that is growing in popularity in aca-
demic circles the view that reality itself is a computation, essentially a simulation.60
Gates himself commented, \We have no idea what these things are doing there".61
4.8. Is there evidence for a golden ratio code in black hole equations,
quantum experiments and solid state matter?
Other compelling evidence used to support the digital physics view includes black
hole quantum gravity theory and an idea known as the holographic principle, which
is derived from the mathematical proof called the Maldacena conjecture.62 It states
that the total amount of binary information from all the mass and energy pulled into
a black hole is proportional to its surface area, where every four Planck areas of its
surface encodes the state of a fundamental particle that fell into it.
It is a distinctly binary code-based framework that comes directly from the¯rst
principles application of general relativity and quantum mechanics at the limit of
Fig. 18. A chiral quasicrystal derived from E8 – a tetrahedral decorated E8 to 4D then to 3D quasicrystal.
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both theories  the environment of a black hole. Black hole quantum gravity
equations are one of the best clues we have about what a theory of everything might
look like.
As stated, quasicrystals generally relate to¯ve-fold symmetry and the golden
ratio. The pentagon is the 2D analogue of the icosahedron and the quintessential 2D
quasicrystal is the Penrose tiling with its 5-fold symmetry related golden ratio
structure. Virtually all of the 3D quasicrystals discovered in nature have icosahedral
symmetry. That symmetry is possessed by any object having the combination of 2-
fold, 3-fold and 5-fold rotational symmetry. The E8 to 4D quasicrystal has these
symmetries and is fundamentally based on the golden ratio.
Black hole physics relates deeply to the golden ratio. It is the precise point where a
black hole's modi¯ed speci¯c heat changes from positive to negative.63
M4
J2
¼ :
ð11Þ
It is a part of the equation for the lower bound on black hole entropy.
e
8Sl 2
P
kA 
 :
ð12Þ
The golden ratio even relates the loop quantum gravity parameter to black hole
entropy.64
2 ¼ :
ð13Þ
In 1993, Lucien Hardy, of the Perimeter Institute for Theoretical Physics, discovered
that the probability of entanglement for two particles projected in tandem is65:
5:
ð14Þ
In 2010, a multinational team of scientists found an E8-based golden ratio signature
in solid state matter. Cobalt niobate was put into a quantum-critical state and tuned
to an optimal level by adjusting the magnetic¯elds around it. In describing the
process, the researchers used the analogy of tuning a guitar string. They found
the perfect tuning when the resonance to pitch is in a golden ratio-based value
speci¯cally related to the geometry of E8. The authors speculated that the result is
evidence in support of an E8-based theory of everything.66
Xu and Zhong's short paper,67 Golden Ratio in Quantum Mechanics, points out
the connections to the golden ratio in various works  linking it to particle physics
and quantum gravity (quantized spacetime). The short piece is worth reprinting
here, and we have included their citations in our bibliography.
The experimental discovery of the golden ratio in quantum magnetism68 is an
extremely important milestone in the quest for the understanding of quantum me-
chanics and E-in¯nity theory. We full-heartedly agree with the explanation and dis-
cussion given by Prof. A®leck69 ... ...For this reason, we would like to draw
attention to a general theory dealing with the noncommutativity and the¯ne structure
of spacetime which comes to similar conclusions and sweeping generalizations about
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the important role which the golden ratio must play in quantum and high energy
physics. Maybe the most elementary way to explain this point of view is as follows:
Magnetism is just one aspect of the¯ve fundamental forces of nature. In a uni¯ed
picture where all the¯ve forces melt into one, it is reasonable to suspect that the
golden ratio will play a fundamental role. This fact immediately follows from the work
of the French mathematician Alain Connes and the Egyptian engineering scientist
and theoretical physicist M.S. El Naschie. In Connes' noncommutative geometry, his
dimensional function is explicitly dependent on the golden mean. Similarly, the bi-
jection formula in the work of El Naschie is identical with this dimensional function
and implies the existence of random Cantor sets with golden mean Hausdor® di-
mension as the building blocks of a spacetime which is a Cantor set-like fractal in
in¯nite dimensional but hierarchal space. Invoking Albert Einstein's ideas connecting
spacetime to geometry with energy and matter, it is clear that these golden mean
ratios must appear again in the mass spectrum of elementary particles and other
constants of nature. There are several places where this work can be found.70–72
4.9. Wigner's universality
The universality pattern is another fundamental clue about what a theory of
everything should look like. It is aperiodic but ordered  liberally de¯ned as a
quasicrystal. It was¯rst discovered by Eugene Wigner in the 1950s in the energy
spectrum of the uranium nucleus.73
In 1972, number theorist Hugh Montgomery found it in the zeros of the Riemann
zeta function, so it deeply ties into the distribution of prime numbers.11 In 2000,
Krbalek and Šeba reported it in the complex data patterns of the Cuernavaca bus
system.74 It appears in the spectral measurements of materials such as sea ice75 and
human bones. In fact, it appears in all complex correlated system  virtually every
physical system. Wigner's hypothesis states that the universality signature exists in
all complex correlated systems.9 Van Vu of Yale University, who has proven with
coauthor Terence Tao that universality exists in a broad class of random matrices,
said, \It seems to be a law of nature".12
Why something as fundamental as the universality signature would relate to both
the distribution of primes and complex physical systems is a mystery  unless
somehow number theory and an unknown theory of everything are deeply related. Of
course, that is trivially true since the entire edi¯ce of mathematics is built upon the
counting numbers. And the foundational \skeleton" of the counting numbers are the
primes. Eugene Wigner famously said that nature is unreasonably mathematical.76
So the ultimate foundation of both complex mathematics and nature herself reside in
number theory.
Freeman Dyson de¯nes a quasicrystal as \a [aperiodic] pure point distribution
that has a pure point spectrum". He said, \If the Riemann hypothesis is true, then
the zeros of the zeta-function form a one-dimensional quasicrystal...".2 Andrew
Odlyzko published the Fourier transform of the zeta-function zeros. It showed sharp
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peaks at the logarithm of the primes and prime.10 This demonstrated that the dis-
tribution is not random but is aperiodically ordered. By the same de¯nition, the
universality signature is a quasicrystal. Quasicrystals in nature generally correspond
to the golden ratio. So how might the universality signature correspond to it?
Universality relates fundamentally to matrix math. It de¯nes the spacing between
the eigenvalues of large matrices¯lled with random numbers. This is interesting be-
cause the four-term two-by-two binary matrix is the most fundamental of all matrices.
14 of its 16 possible combinations of 1 and 0 have either trivial or simple eigenvalues as
0, 1 or 2. However, the remaining two eigenvalues are golden ratio based as
þ ¼  and  ¼ 
1

:
ð15Þ
Quantum systems, such as the hydrogen atom, are governed by matrix mathematics.
Freeman Dyson said, \Every quantum system is governed by a matrix representing
the total energy of the system, and the eigenvalues of the matrix are the energy levels
of the quantum system."80
Based on work done by Suresh and Koga in 2001,77 Heyrovska78 showed the
atomic radius of hydrogen in methane to be the Bohr radius over the golden ratio.
rH ¼
a0

:
ð16Þ
The random matrix correspondence to physics is not an indication that actual ran-
domness occurs. The matrices of some correlated systems, like a hydrogen atom, can
be worked out precisely. However, more complicated systems, such as a uranium atom,
are non-computable by current methods. The values of its unknown matrix become
super-imposed like the blur of voices in a crowded conference hall. Although, there is
no randomness in the conversations of the people in the crowd, the super-position of
soundwaves behaves exactly like the solutions to a matrix with random numbers.
Scientists are still trying to¯gure out why universality has the exact pattern that
it does. Vu said, \We only know it from calculations". Because this pattern also
matches perfectly to the distribution of the non-trivial zeros in the Riemann zeta
function, the distribution of primes must relate to a strongly-correlated matrix.
Dynamically, quasicrystals obey random matrix statistics.79 And they are strongly
correlated and nonlocal, due to the empire concept discussed above.
The distribution of prime numbers is encoded in the spectral pattern derived by
an irrational projection of a slice of an An lattice to 1D. The cell types of An lattices
are simplex-integers, n-simplexes, where each An lattice and its n-simplex cell type
embeds the stack of allAn lattices with dimensions lower than it. That is, the series of
simplex-integers, including prime-simplex-integers, are encoded in the projection of a
slice of an A-lattice to 1D.
The salient point for now is that the distribution of primes and, accordingly, the
zeta zeros corresponds to geometric-number theory  simplex-integers and their
associated An lattices. We conjecture that our quantum gravity framework based on
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a quasicrystal projected from the 8-simplex based E8 lattice will explain why the
quasiperiodic universality pattern appears both in nature and prime number theory.
That is, the matrix analogue of our quasicrystal may be the missing matrix corre-
lated to the universality signature.
Like all quasicrystals, the dynamical behavior of our E8 derived quasicrystal is
described by a complex matrix.79 Because its complex phason code is strongly and
nonlocally correlated, it will obey random matrix statistics and map to the univer-
sality signature. But the random matrix and universality pattern would be secondary.
We agree with Laszlo Erd€os of the University of Munich, who said \It may happen
that it is not a matrix that lies at the core of both Wigner's universality and the zeta
function, but some other, yet undiscovered, mathematical structure. Wigner matrices
and zeta functions may then just be di®erent representations of this structure".80
4.10. Section 4. Conclusion
This section began with the conjecture:
If nature was a self-organized simulation, it would be a simplex-integer based
quasicrystalline code derived from E8.
We have defended the reasonableness of the conjecture. Now it is up to our
institute and the scientists who work here continue to publish a series of theoretical
and experimental papers that transform the toy framework into a rigorous formalism
worthy of attracting a community of collaborators. The approach is certainly outside
the box. However, an outside the box approach may be what is needed. String theory
is now 50 years old and it has not made a successful prediction. We believe that a
fresh but rigorous new approach such as ours is overdue. It is possible there are
bridges to aspects of the string theory approach. In fact, the most foundational string
theory was¯rst introduced by David Gross et al. in 1985, heterotic string theory.
It exploits the power of E8.
105
However, our primary approach achieves symmetry breaking in an intuitive
manner via projective geometry to lower dimensions, where full recovery of hyper-
dimensional uni¯cation physics can be achieved. The resulting spacetime and par-
ticle code is a simulation, much more similar in form to loop quantum gravity, where
the code itself is the structure of dynamical spacetime.
Appendix A. Overview of Emergence Theory
Emergence theory, developed by our institute over the last eight years, exploits the
ideas discussed above. The program is at an intermediary stage of development.
A.1. Foundational papers
Fang Sadler et al. published the foundational tetrahedral golden ratio rotational
relationships and helical behavior in 2013.81 In 2012, Kovacs et al. introduced the
sum of squares law82 and in 2013, Castro-Perelman et al. proved the derivative
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sum of areas and volumes law.83 In 2014, Fang et al. derived the golden ratio rotation
from the¯rst principles approach of the icosagrid method. In 2016, she and coau-
thors published the construction rules of the 3-simplex based quasicrystalline pos-
sibility space and introduced the term golden matrix (\GM") to describe it along
with its E8 derived sub-spaces.
15
A.2. Conceptual overview
Our program is an Occam's razor approach to physics, where we aim to start with
irreducible¯rst principles and relentlessly question status quo assumptions. Because
nature seems to be governed by rules and beautiful math, it is safer to say that there
exists an analytical expression for the¯ne structure constant, the Planck constant,
the magnitude of the speed of light and the gravitational constant than it is to say
there is not. Put di®erently, either there exists a¯rst principle theory of everything
that explains these values or there is not. However, no such theory has been dis-
covered yet. All theories start with those values and then create equations relating
them and their composite objects.
It is helpful to understand the di®erence between a uni¯cation theory and a
simulation theory. A uni¯cation theory is a network of equations that show how
di®erent things transform into one another. A simulation theory uses geometric building
blocks as the mathematical operators that themselves are physical reality  the sim-
ulation  instead of merely describing it. Such a framework would spit-out the uni¯-
cation equations while also serving as the \pixels" or functional building blocks of reality.
We want to know what reality is, not just the equations that tell us how it behaves or
how it is uni¯ed. Loop quantum gravity is the most popular simulation theory.
Because reality appears to have three spatial dimensions, we start there and
inquire whether or not it is possible to simulate physics using the simplest building
block or pixel of 3D information, the 3-simplex. The idea is known as a background
independent model because it starts with spacetime building blocks and makes
particles the propagating patterns in that system. The second part of our basic idea is
that we use a quantized irreducible unit of measurement at the Planck scale sub-
structure of our model. We call this operator a quantum viewer. The building block
simplex-integers are made of information. But, they are ontologically real because
they are being actualized by quantized units of primitive measurement  the
quantum viewers.
We will now highlight a few of the key components of our framework.
A.3. Ontology and symbolic language  \All that exists is that
which is measured"
We would agree with Ilija Barukčić's statement:
\Roughly speaking, according to Bell's theorem, there is no reality
separate from its observation".84
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Classical physics indirectly de¯nes energy as information in the form of an abstract
quantity called the \potential for work". Spacetime is permeated with energy, where
di®erent energetic potentials within it, are equal to local densities of curvature.101
Einstein's mass-energy equivalency reduces matter to the notion of bound up energy.85
Quantum mechanics is more clearly information theoretic, dividing reality into the
abstract possibility space of the wave function and the actualized collapse into a
particle coordinate in the form of measurement data.86 J. A. Wheeler was one of the
¯rst to point out that reality is made of information.35 Max Tegmark and many other
modern physicists hold this view today. Information is real, so ontologically, there is
a division between the potential for information, which is not real, and information
as a product of measurement/observation, that is real.87 The measurement problem
associated with quantum mechanics relates in large part to the choice of ontological
interpretations of what the equations and experiments mean. It is a topic of hot
debate with no broad consensus. Einstein and many others have said that there is
something we are missing and that the formalism is incomplete.88 Some have taken
the bold position that humans or entities at our level must measure something to
actualize it into physical existence. Einstein was one of the¯rst to take issue with this
idea, saying, \I like to think that the Moon is still there even when I'm not looking
at it". So we take the more conservative position that there is some self-actualizing
measurement operator at the Planck scale, where the quantized pixels of reality exist.
We call this operator a \quantum viewer". Its function is to generate a trinary state
change in the 3-simplex quanta of space in a possibility space of such objects. The
possibility space is called the QSN.15 It is an E8 derived space of 3-simplexes, wherein
the trinary state selection actions create syntactically legal quasicrystalline sub-
spaces of the QSN that are physically real frames of space with particle patterns
embedded within it. The trinary quality state choices are: (1) on right, (2) on left and
(3) o®. For example, if a 3-simplex is in the \on right" state in one quasicrystalline
frame and is \on left" in the next frame of a dynamical sequence, the formal action is
a Cli®ord rotor or spin operation on the possibility space. However, there is an
ontological requirement to manifest these actions with an irreducible measurement/
observation operator  the quantum viewer action. To understand this, visualize
the idea of standing to the left of a friend and taking a photo. Next, walk to her right
and take a second photo. Each photo is a transformation-symbol. The ordered set of
two photos express the physical information of a discretized rotation of your friend
changing orientation relative to your camera if you are stationary and she rotated
between the two orientations. So as each quantum viewer performs its operation, it
captures symbols which are projective transformations that are equal to a state change
of a tetrahedron as either on-right, on-left or o®. The quantum viewers actualize, via
observation/measurement, the action of a Cli®ord rotor or spin operation on the QSN.
As mentioned in Sec. 4.4, the 3-simplex network can be decomposed as a network of
1D Fibonacci chains with line segments in the golden ratio proportion. The quantum
viewers generate either a right or left-handed rotation of a tetrahedron, which divides a
given edge by the golden ratio on one side or the other (as shown in Fig. A.1).
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Mathematically, the coordinates of the quantum viewers the camera positions
are the edge crossing point set of the QSN. A key geometry of the network can be
understood by taking 20 evenly spaced 3-simplexes that share a common vertex at
the center of the cluster. Rotating each either right or left on an axis running from
the outside face center through the shared inner vertex by the golden ratio based
angle 12 cos
1 1
4
  120

 creates a 20-group that is either twisted to the right or left
(see Fig. A.2). This is an absolute chirality not relative to one's vantage point.
In summary,
(1) The quantum viewers are the observation or measurement operators.
(2) They make projective transformations based on their position, just as a camera
transforms a 3D image to a 2D image, which is really a network of transforma-
tions of 1D actual lengths (lines) between pairs of points to contracted or
transformed lines. So the irreducible measurements are 1D phason °ips that divide
a line into the golden ratio with the long side on one side of the line or the other.
(3) The transformations are information. They are observations that are equal to
symbols. Because those symbols are ontologically real due to actualization via
observation, they compose the next frame or state change in that region of the
QSN — a physically real region of space and time with particle patterns in it.
(4) Formally, the system is a spin network on a discretized moduli space, where the
operators are primitive measuring entities generating physically real information.
Fig. A.1. Golden division of tetrahedral edges with twisting.
Fig. A.2.
(a) An icosahedron divided into regular tetrahedron which are spaced from the inside sym-
metrically so there is space between the faces. (b) The same tetrahedron with a common vertix in the
center rotated with face kissing.
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(5) Its rules and syntactical degrees of freedom are derived by the geometric¯rst
principles of phason cutþ project dynamics related to the movement of a cut
window through the Elser and Sloan E8 to 4D quasicrystal.
A.4. Quantized space and \Time"
As explained above, space is quantized as 3-simplexes. And time is quantized like a
35 mm¯lm, as ordered sets of individual quasicrystalline frames of 3-simplexes
generated by ordered sets of trinary selection choices of the quantum viewers in
the QSN. Of course, this concept of a universal frame rate is anathema to key
assumptions in special relativity  the invariance of the speed of light and
the notion of smooth spacetime. The old relativistic notion is that, because
spacetime is smooth and structureless, nothing can have intrinsic time or motion but
only relative time and motion. The relativistic concept is well supported by experi-
ments, which show that, no matter how fast an observer chases a photon, it
always seems to elude him at the speed of light. Our solution to this is the electron
clock model.
A.5. Electron clock intrinsic time
We reject the assumption of structureless space. The Michelson–Morley experiment
of 1887 was not designed to test for a structure as described herein or any of the other
loop quantum gravity type theory, where spacetime has a discrete substructure. Prior
to 1887, the scienti¯c community presumed a speci¯c °uid type material called the
aether¯lled space.89 When experiments did not demonstrate this substance, a new
axiom was established  that there is no substructure to space. Of course, without
substructure, there can be no logical motion relative to space. An object would not
have intrinsic motion but only motion relative to another object °oating in the ocean
of the structure less vacuum. This key axiom undergirds relativity theory. The
second modern assumption is that fundamental particles, like the electron, have no
substructure and are instead dimensionless points. If this were true, such a particle
could not have an internal clock or any concept of rotation. All time or change that
would be ontologically real would be changed relative to another object changing 
another clock.
Louis de Broglie¯rst conceptualized the notion of the electron possessing an
internal clock.90 Later, David Hestenes made this idea more rigorous.91 In the
emergence theory framework, massive particles, like electrons are composites of
multiple Planck length 3-simplexes chosen as ordered sets in frames of the QSN.
There are two forms of dynamic pattern:
(1) Stepwise toroidal knot — This is a knot pattern much like a 3D trefoil knot that
has an asymmetric region that cycles around the geometry of the knot (as shown
in Fig. A.3). Multiple quasicrystal frames are required in order to complete a full
cycle around the knot — a tick of the internal electron clock.
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(2) Helical propagation: For simplicity, let us imagine it takes 10 frozen quasicrystal
states chosen by the quantum viewers to compose an animation of one knot cycle.
The entire knot can remain at one coordinate in the QSN or it can propagate
helically forward in a certain direction. However, if any of the 10 frames are used to
propagate the pattern forward, there will be fewer frames available to complete
time cycles of the internal toroidal knot-like clock. There must always be a rational
fraction of frames used for propagation and frames used for clock time. The two
patterns of \time" and propagation would be inversely proportional to one an-
other. And there would always be an absolute and intrinsic ratio of internal clock-
time to propagation with respect to the global frame rate of the QSN.
A photon in this model is a pattern of tetrahedra in the QSN that is only helical, not
toroidal. So the ratio of propagation to clock time in a given number of frames will
always be 100:0. That is, any non-massive particle (particles without internal knot
structure) will always propagate in an invariant manner with the same distance
covered over a given quantity of frames.
The traveler in a spacecraft moving at 99% of the speed of light will shift their
intrinsic clock cycles (as a ratio of total frames) to a very slow rate. This will include
all massive particles moving with it, including the measurement apparatus and the
operation of the brains of the scientists onboard the craft. The clock cycles or ex-
perience of change on the craft will be very slow and the photon will move at the
speed of light from the projector on the ship and will go to a mirror at some distance
before re°ecting back to the measurement apparatus to be compared to some
quantity of clock cycles. Very few clock cycles will have elapsed because time for
these travelers and their massive equipment will slow to a near halt. Accordingly, the
comparison of the distance traveled by the photon to the number of clock cycles will
indicate that the photon moved relative to the traveling craft at the same speed it
moved when the experiment was done while the vehicle was moving at 1% of the
photon's rate of propagation. However, the intrinsic or actual di®erence between the
Fig. A.3. A Hamiltonian path cycle through the 57 centers of 57 tetrahedron.
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speed of the vehicle moving at 99% of the speed of light and photon moving at 100%,
would in truth be 1% of the speed of light. Clearly, this viewpoint is far less enigmatic
and geometrically pleasing than the ordinary interpretation of these experiments via
the smooth spacetime ontology of special relativity.
A.6. Chirality
The conjecture that fundamental particles are dimensionless points without struc-
ture causes intuitive geometric confusion with other indications that particles deeply
relate to handedness or chirality. For example, a current of electrons has a well
understood geometric chirality feature. The right-handed rule of how a magnetic
¯eld is wrapped around the current in a chiral fashion tells us something deep about
handedness in nature. However, the notion of a right-handed or left-handed indi-
vidual particle is replaced by an abstract non-geometric sign value that is distinctly
non-geometric due to the conjecture of the dimensionless point particle identity of
the particle. For example, the point particle mathematical abstraction is one where
helicity is the sign of the projection of the spin vector onto the momentum vector,
where left is negative and right is positive. It is an outstanding mystery as to why the
weak interaction acts only on left-handed fermions such as the positron and not
right-handed ones like the electron.92
Quasiparticle patterns in the QSN have a fundamentally di®erent feature that
relates to chirality. In Fig. A.4, a left-handed group of 20 3-simplexes, where the
states of the tetrahedra by the quantum viewers on the simplexes are all \on-left" is
shown.
Ordinarily, a helix made of 3-simplexes, as shown on the left in Fig. A.5, will have
no periodicity because of the irrationality of the dihedral angle. However, in the
QSN, tetrahedra can only be related by the golden ratio-based angle:
1
2
cos1
1
4
 
 120


¼ ArcCos 
2
2
ffiffiffi
2
p
	


:
ðA:1Þ
Fig. A.4. A left-handed group of 20 3-simplexes, where the states of the tetrahedra by the quantum
viewers on the simplexes are all \on-left".
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The helix on the left is right-handed. So when the rotation of the golden ratio
angle is of opposite chirality, in this case, rotated by that value  right-handed 
the periodicity become 5-fold. And when it is rotated left, it becomes 3-periodic. The
deep reason for these two periodicities corresponds the E8 to 4D Elser–Sloan qua-
sicrystal, wherein the projection of the Gosset polytopes in the E8 crystal generates
600-cell made of 600 3-simplexes. Each simplex is part of a rings of 30 simplexes, as
shown in this diagram.
The periodicities of the tetrahedra ring in Fig. A.6 are a superposition of 3-fold
and 5-fold, where the orientation of 15 tetrahedra repeats 3-periodically and 15
repeat 5-periodically. The dihedral angle between any two adjacent tetrahedra is
1
2 cos
1 1
4
  120

 þ 60. In 4D, there is vectorial freedom for the 60 component of
the angle. When the relationships of 3-simplexes are represented in the QSN, we cast
out the 60 component because it is the portion related to the construction of a
simplex series, where each 60 of a new edge on an n-simplex to generate an nþ 1
simplex is 60 into an additional spatial dimension.
Realistic physics would not be able to be done if we projected the E8! 4D to 3D
or projected E8 directly to 3D. The key feature of the QSN is that, by making the
tetrahedra regular by taking a 3D slice of the 4D QC with regular tetrahedra and then
rotating copies of that slice by the same angle that relates adjacent tetrahedra in the 4D
QC but minus the 60 component, we introduce three crucial elements into the object:
(1) It generates an additional sign value necessary for Physics.
(2) It signi¯cantly increases the degrees of freedom in the code. In other words,
it transforms the code from a binary on/o® code to a trinary code of \on right",
Fig. A.5.
(a) The Boerdijk–Coxeter helix showing no Periodicity. (b) The Boerdijk–Coxeter helix
showing 5 Periodicity by same-handed golden twisting. (c) The Boerdijk–Coxeter helix showing 5
Periodicity by opposite-handed golden twisting.
Fig. A.6. 3D projection 30 tetrahedral ring from 600-cell.
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\on left" and \o®" in terms of the registration possibilities for a given tetrahe-
dron in a frame of the QSN.
(3) It improve the ratio of symbolism to meaning by reducing all of the tetrahedra to
the simplest possible 3D pixel of information, the 3-simplex. If the 3D QSN were
generated by projecting the E8 lattice or the 4D QC to 3D, it would generate
seven di®erent shapes of distorted 3-simplexes. It would change the ratio of
symbol simplicity rank to meaning in the code (see Part 3).
A.7. Conservation and the sum of squares law
Conservation is an inherent quality of irrational projection-based geometry. For
example, consider a tetrahedron with four lines running from the centroid to each
vertex. Assuming the edge length of the tetrahedron is one, we can project the four
inner lines to the plane with an in¯nite number of projection angles, such as in in
projection in Fig. A.7.
The sum of squares of each contracted length in the projection is always conserved
as 4 or the integer corresponding to the simplex-integer, in this case the 3-simplex
corresponding to the integer 4. The sum correlates in a mysterious way to the spatial
dimension of a projected polytope, as reported in two Quantum Gravity Research
papers, Julio Kovac's The Sum of Squares Law82 and Carlos Castro Perelman's et al.
The sum of the squares of areas, volumes and hypervolumes of regular polytopes from
Cli®ord polyvectors.83
Based on this same conservation principle, the \letters" or geometric symbol
types of a quasicrystal are conserved. For example, there are seven di®erent vertex
geometries in a Penrose tiling, as shown in Fig. A.8.
Each of their frequencies of occurrence are conserved as follows:
A = 1
B = ’
C = ’
D = ’2
E = ’3
F = ’4
G = ’5.
Fig. A.7. Projection of a tetrahedron.
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Similarly, the various legal particle con¯guration patterns made of relationships
between 3-simplexes chosen on the QSN have conserved quantities. We suggest that
the deep¯rst principles-based explanation for Noether's¯rst theorem, gauge sym-
metries and conservation laws in nature is hyperdimensional projective geometry,
where the full encoding and richness of hyperdimensional structure is transformed
into lower dimensional geometric symbolic code  quasicrystal language.
Quasicrystals have the fractal quality that any shape, such as the seven vertex
geometries in the Penrose tiling (see Fig. A.8), repeat according to a scaling algo-
rithm, typically the power series of the golden ratio, ’, ’2, ’3 ....
A.8. Alternative expression of geometric frustration
The term geometric frustration can be thought of as \trans-dimensional pressure"
resulting from a projection of an object to a lower dimension. For example, in 3D,
there is vectorial freedom or space for 12 unit length edges to be related by 90 in the
form of a cube. When projected along an irrational angle to 2D, the reduction of
vectorial freedom compresses or transforms the information into a 2D representation
that requires edges to contract and angles to change. The 2D projection or shape-
symbol is a map encoding (1) the information of the pre-projected object and (2) the
angular relationship of the projection space to the pre-projected object. The trans-
dimensional tension or pressure is immediately released or transformed into the
transformed lengths and angles of the projection.
An alternative form of transformation or transdimensional pressure expression is
rotation and translation. For example, consider the transformation of a 20-group of
tetrahedra sharing a common vertex in the 600-cell in 4D space. If we project it to 3D
along a certain angle, we can generate a group of 20 distorted tetrahedra with a
convex hull of a regular icosahedron and 12 inner edge lengths contracted by
ffiffiffiffiffiffiffiffiffi

ffiffiffi
5
p
q
2
¼ cos 18:
ðA:2Þ
Fig. A.8. The seven vertex con¯gurations of the Penrose tiling.
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We can understand the di®erence of dimensions as a curvature of one dimension into a
higher dimension. For example, a °at piece of paper can be curved into the 2nd dimension
such that it is a curved 2D object that requires three spatial dimensions to exist in.
So if we take our 4D 20-group, we can realize that it is bounded by a 3-sphere (4D
sphere), which is a curved 2-sphere (ordinary sphere). And we can slowly de-curve or
°atten the 3-sphere of space containing the 20-group until it is \°at", at which point
it is an ordinary 2-sphere. In this case, the 20 regular tetrahedra living in 4D that
have unit edge lengths would need to distort such that the 12 shared inner edges
contract to cos 18 (see Eq. A.2). This result is identical to the aforementioned
projection of the 4D object to 3D. An alternative method of encoding the projection
or uncurving action is to anchor the 20 tetrahedra around their common shared
vertex and rigidify them, such that they are not allowed to encode the geometric
frustration via edge contraction and angle change. This will force the tetrahedra to
express the information of their hyperdimensional relationships in lower dimensional
space by rotating along each of their 3-fold axes of symmetry that run through face
centers to opposing vertices (the shared center vertex).
Each of the 20 tetrahedra in the 4D space lives in a di®erent 3D space related to
the adjacent 3D space by ArcCosð½3’ 1=4Þ þ 60 ¼ ArcCosð1=4Þ. If we visualize
this as a gradual uncurving of the 4D space toward °at 3D space, we begin with zero
rotation of each tetrahedron.
As we initialize the uncurving, the faces will begin to rotate from one another such
that their 12 shared inner edges \blossom" into 60 unshared inner edges. As we do
this, we are gradually intersecting or converging the twenty separate 3D spaces into a
single 3D space. At the point where the 3-sphere bounding space is completely
°attened to an ordinary 3D sphere, the rotation value between the kissing inner faces
of the 20-group is ArcCosð½3’ 1=4Þ, which is the angular relationship between
kissing 3D spaces containing tetrahedra in the 4D space of the 600-cell, minus the 60
component that there is no room for in 3D (see`60 Construction' in Sec. 5.9).
Now, we have a curvature value of 0 and a rotational value of ArcCosð½3’ 1=4Þ
and have encoded the relationships of the 20 tetrahedra living in 4D into a geometric
symbol in 3D via rotation instead of edge contraction. We have converged 20 tetra-
hedra from 20 individual 3D spaces related to the other by ArcCosð½3’ 1=4Þ þ 60
into a single 3D space where they are related by the same fundamental irrational
Fig. A.9. The 20 group with the axis of rotation through the center of the face.
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component of their former relationships but without the 60 component that was used
to construct them by extruding successive spatial dimensions by a process of rotating
edge copies into the next spatial dimension by 60.
We can now reverse the process and slowly curve the °attened 4D object that is
now a 3D object back into a perfect 4D 20-group. As we do, the rotational value
decreases and the space curvature value increases. At the point in which the 2-sphere
and its rigid tetrahedra are curved into a perfect 3-sphere, the rotational value is 0
and the curvature value is 1=’. So, there is an inverse proportionality between the
curvature limit and the rotational limit as 0 rotation ! 1=’ spatial curvature and
1=’ spatial curvature ! 0 rotation.
In the QSN, every adjacent tetrahedral relationship is þ= ArcCosð½3’ 1=4Þ,
which is the 4D angular relationship between tetrahedra in the E8 to 4D quasicrystal
minus the 60 component not related to 3D.
This special non-arbitrary rotational value is powerful for modeling quantum
gravity and particle patterns for four reasons:
(1) It encodes the relationships of tetrahedra in a 4D space, which can be useful for
modeling 4D spacetime in three spatial dimensions.
(2) It encodes the relationships of tetrahedra in an E8 derived quasicrystal, which
can be useful to model gauge symmetry uni¯cation of gravity and the standard
model particles and forces.
(3) It introduces a binary sign value, chirality. The edge distortion method of
encoding geometric frustration does not generate the chirality value. This may be
useful for fundamental physics which uses three binary sign values (1) polarity,
(2) spin and (3) charge.
(4) The chirality sign value servers as an important degree of freedom in the qua-
sicrystalline code, as opposed to a more restrictive ordinary quasicrystalline. This
degree of syntactical freedom makes the geometric language more powerful.
This fundamental rotational value is the basis of action on the QSN. That is, the
Cli®ord rotor spin operations are this rotation, which will serve as the new }, the
reduced Planck constant or Dirac constant, in our emerging geometric¯rst principles
approach to fundamental physics.
There are three ways to visualize operations on the QSN:
(1) Graph theoretically: The QSN is a network of points and connections (edges).
It is simply an extended construction of the 20-twist discussed below. The 180
possible connections on the 60 points derived via any of the construction methods
discussed are part of the possibility space. So when graphed theoretically, we can
picture the 180 connections as a graph diagram in 3-space. And then we can do
graph operations to turn edges \on" or \o®" in order to make patterns.
(2) Trinary code: We can turn entire tetrahedra \on" or \o®" in which case we can
think of a centroid of a tetrahedron as being selected and designated as either the
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right-rotated or left-rotated version or not on at all, for a total of three possible
choices.
(3) Cli®ord rotor/spin network: We can conceptualize the tetrahedra to rotate
smoothly in a classical sense, such that it is rotated from a left to a right position
via the ArcCosð½3’ 1=4Þ rotation value. We can further decompose these
rotations into individual edge rotations.
A.9. Simplex construction by 60
As mentioned in Sec. 3, the simplex series is constructed by starting with an edge, a
1-simplex, and rotating a copy on a vertex by 60 into the next spatial dimension to
form a 2-simplex or three equidistant points on the plane. A copy of one of those
edges is then rotated by 60 into the 3rd spatial dimension to form an equidistant
relationship of four points and a dihedral angle of ArcCos(1/3). The dihedral angle
series ranges from 60 in the 2-simplex to 90 in the in¯nite-simplex, spanning a total
of 30 and where each dihedral angle in the series between 30 and 90 is irrational as
the ArcCos of a successive fraction from the harmonic series 1/2, 1/3, 1=4 ....
We can think of the 60 component of each dihedral angle as being tied to the
action that extruded an additional spatial dimension necessary for the next point to
be added in such a manner that all points are equidistant. The remaining irrational
component of each dihedral angle is the more \meaningful" part, carrying the key
information of the given simplex-integer. For example, in the case of the 4D simplex,
the two parts of its ArcCosð1=4Þ dihedral angle are ArcCosð½3’ 1=4Þ 	 15:522
and the 60 component correlated to the extra-spatial rotation that extruded out the
next spatial dimension in the buildout process from 3D to 4D. The relationship
between kissing 3-simplexes in a 4D space is 60 þ 15:522. Accordingly, when one
uses the irrational component of this angle in a 3D construction of regular tetrahe-
dra, such as in our approach, it encodes the relational information between tetra-
hedra as they would have existed in, for example, the 4D Elser–Sloan quasicrystal
derived from E8.
And because 15:522 is inversely proportional to the 1=’ curvature value, as
explained above, it is most deeply a transformation of the information of a¯nite 4D
spaces (a 3-sphere of radius 1) into a¯nite 3D space  a 2-sphere of radius cos 18
(see Fig. 18).
This same construction approach can also be used to build out the E8 lattice,
which is a packing of 8-simplexes that leaves interstitial gaps in the shape of 8D
orthoplexes.
A.10. Specialness of 3D and 4D
In 2D there are an in¯nite number of regular polytopes, but they all have rational
angles and are trivial in some sense except for the ones based on the angles 60,
72 and 90 as the equilateral triangle, pentagon and square. These are the
polytopes corresponding to the¯ve Platonic solids, the only regular polytopes in 3D.
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For example, the equilateral triangle is the polytope in 2D corresponding to the
tetrahedron. Only the equilateral triangle and square can tile the plane, making them
the \crystal"-based 2D analogues of the platonic solids. The pentagon cannot tile the
plane and corresponds to the icosahedron and dodecahedron. Of the¯ve platonic
solids, three are based on the crystal group, the cube, tetrahedron and octahedron.
The remaining two, the icosahedron and octahedron, are the quasicrystal regular
solids. That is, they cannot tile space alone or in combination with other Platonic
solids. Virtually all quasicrystals discovered physically have the symmetry of the
dodecahedron and icosahedron called icosahedral symmetry. As we go to 4D, we
have four crystal symmetry polytopes, the 4D tetrahedron, 4D cube, 4D octahedron
and a crystal based polytope called the 24-cell. We also have two quasicrystalline
polytopes, the 4D icosahedron called the 600-cell, and the 4D dodecahedron called
the 120-cell.
With this, the quasicrystalline symmetry ends. It never appears again in any
dimension after 4D. In every higher dimension, the only regular polytopes are the
hyper-tetrahedron (n-simplex), hyper-cube and hyper-octahedron.
Some have wondered why 3D and 4D appear especially related to our physical
universe. If reality is based on quasicrystalline code, then this would perhaps be the
reason.
A.11. Principle of e±cient language
The principle of e±cient language is the guiding law or behavior of the universe in
the emergence theory framework. The old ontology of randomness and smooth
spacetime is replaced by a code-based ontology where symbolic information and
meaning become the new¯rst principles basis of our mathematical universe. As
discussed in Symbolic Power of Fibonacci Chain Networks in Sec. 4.6, meaning
comes in two fundamental categories: (1) ultra-low subjectivity physical meaning,
which is purely geometric and (2) ultra-high subjectivity or virtually transcendent
meaning, which includes things such as the meaning of irony and the myriad layers of
meaning imposed by an experimenter about, say, the notion of a particle being
measured as going through one slit or the other in a double-slit experiment. Inter-
estingly, it is impossible to imagine an instance of ultra-high subjective meaning
being disconnected from the underlying geometric code at the Planck scale. For
example, the experience of humor is always associated with countless changes in
particle position and alterations to the quantum, gravitational and electromagnetic
¯elds associated with that event. All forms of meaning are ultimately composed of
actions of the quantum viewing actions that animate the code. The inherent nonlocal
connectivity and distributed decision making actions of this neural-network like
formalism allow various emergent patterns of intelligent choice and actualization of
abstract meaning to be registered and considered within the degrees of freedom of the
code. Choices will be made in such a manner as to create maximal associations and
meanings where, in systems such as human beings, meaning is highly subjective.
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Consider for example, how a joke can be told and one individual will react with
massive levels of neural activity and associated meaning, while another person may
barely comprehend it. The¯rst person generates a much higher degree of correlated
and physically meaningful actions when considered at the Planck scale level of the
code operations. This feedback between the overall system (the universal emergent
neural-network) and the person generating a larger amount of meaning from the joke
plays a role in syntactically free choices of the code. We call these free choices the
hinge variable steps in the code. On average, physical laws and actions are preserved
because the physical meaning of the code (forces and physical laws) are the emergent
and non-¯rst principles manifestations of the underlying waveform language of the
quasicrystalline quasiparticle formulism.
A.12. Phason code
Phason quasiparticles have both a nonlocal wavelike quality and a local particle-like
propagation aspect called a supercell in crystallographic parlance. As mentioned
previously, there are three general ways matter can be organized: (1) Amorphous or
gaseous materials that have massive degrees of freedom and are therefore not nat-
urally codes. Geometric codes require a¯nite set of symbols, strict syntactical rules
and minimal degrees of freedom. (2) Crystalline materials have no degrees of freedom
unless there are local defects or phonon distortions. There are ultra¯ne scale vibra-
tions allowed, but not organized code-based larger scale oscillations. (3) Quasicrys-
tals are maximally restrictive without being ultimately restricted like in the case of a
crystal. For example, unlike a crystal, the assembly rules for a quasicrystal allow
construction choices within the rules that are not forced. A crystal allows only one
possible type of relationship between atoms. For example, all vertex types in a cubic
lattice are identical. In an amorphous or gaseous material, atoms can have a virtually
in¯nite number of relational objects or vertex types. In a quasicrystal, as with any
language, there is a rather small set of allowed combinations. For example, in the
Penrose tiling, which is found in nature, atoms form seven di®erent allowed vertex
geometries and the construction rules allow a very minimal level of freedom within
the construction syntax.
A.13. Empires and phason °ips
Because all quasicrystals are networks of 1D quasicrystals, understanding a phason
°ip and empires should start with how a quasicrystal is made via the cutþ project
method. An irrational projection of a cut or slice of any crystal to a lower dimension
produces a quasicrystal.
For example (see Fig. A.10(b)), one can select a rectangular cut window rotated
with an irrational angle to the 2D crystalline pointset. One projects the points
captured in the cut window to the 1D projection space to generate our 1D quasi-
crystal. In the second image, we translate the cut window, which projects a di®erent
set of points to the 1D space. When the cut windowmoves to a new coordinate, points
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instantly jump in or out of possible positions in the 1D space that we call the
possibility space. This instant change from one coordinate to the other in the pos-
sibility space is called a phason °ip.
When one point is captured in the cut window, there are an in¯nite number of
other points along the length of the cut window (if considering an ideal in¯nite point
space) that are also captured in the cut window at its new coordinate. This creates an
in¯nite number of phason °ips in the 1D possibility space. An arbitrarily large but
non-in¯nite quasicrystal can be built according to assembly rules instead of the cut +
project method. In this case, a user of the assembly language must choose a single
phason °ip, which is simply the designation of a point from the possibility space to be
\on" or \o®".
In Fig. A.10 note that, when the cut window changed location, some points in the
2D space (1) remained in the window, (2) some departed from the window and (3)
some entered the window. When a quasicrystal code user chooses a point to be \on"
(a)
(b)
Fig. A.10. A schematic diagram showing two ways of interpreting the cut-and-project method for gen-
erating a quasicrystal from a higher dimensional lattice. (a) shows that the points are selected for pro-
jection as long as there is non-trivial intersection between their Voronoi cell and the quasicrystal space.
The black points are the lattice points, and the hexagons are their Voronoi cells, E is the quasicrystal
space, E ? is the orthogonal space, and the solid blue and green segments are the projected tiles in the
quasicrystal space; (b) shows that the points are selected for projection as long as their projection on the
orthogonal space falls inside of W .
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from the possibility point space, it causes a certain group of other points in the
possibility space to also be turned on and other points to say on. These two sets of
points are called the empire of the selected \on" point.
A key concept is conservation. The number of points captured in the cut window is
conserved. As points enter the window, an equal number of points exit. A second key
concept is non-locality, the empire of forced points determined to be \on" or \o®" by
a single point selection of a code user is very large. A third key idea is discrete and
instant coordinate change. When the points are a model for particles, an ontology of
instant coordinate change in the \physical" projection space is recognized, much like
the notion of virtual particles in the Dirac sea, where particles are conserved such
that when one is annihilated, another instantly appears.
Quasicrystals in dimensions higher than 1D are more complex because they are
networks of 1D quasicrystals. So a phason °ip and empire of a single 1D quasicrystal
will have a massive empire that in°uences every other 1D quasicrystal in the net-
work. Figure A.11 shows an image from Laura E±nger-Dean's thesis, which shows
the empire of one of the vertex types of the Penrose tiling. We can see that the density
of the empire drops with distance from the vertex being designated as \on" at the
center. One can think of the possibility space as an aperiodic point space where any
point can be selected to be one of the allowed vertex types. In the Penrose tiling, there
are seven di®erent vertex geometries. As mentioned, once one vertex type is selected
for that vertex on the possibility space, it forces other vertices in the space to be
\on" or \o®" the empire. A key point for physics modeling, where forces drop with
Fig. A.11. The empire of a vertex con¯guration in Penrose tiling.
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distance, is that some empires drop in density with distance. We will connect this
with the idea of empire waves and the free lunch principle shortly.
Another key idea for physics using this formalism would be that the minimum
quantum of action notion of quantum mechanics would be replaced by the action of a
point being registered as \on" or \o®".
Phason quasiparticle behavior in any quasicrystal has two distinct sets of
construction rules:
(1) Quasicrystal Assembly Rules: These construction rules govern how a single
frozen state of selections on the possibility space can exist. The rules are de¯ned
by the angle, size and shape of the cut window in the higher-dimensional lattice.
(2) Ordering Rules for Two or More Quasicrystals: These rules govern the creation
of dynamical patterns generated by ordering two or more di®erent selection
states on the possibility space into a stepwise frame-based animation. The rules
are de¯ned by the way that a cut window can translate or rotate through the
higher-dimensional lattice and whether combinations of those actions is discrete
or continuous.
A.14. Empire waves
Just as the 2D Penrose tiling quasicrystal has empires (see Fig. A.11) that are
circular, with radial lines of higher density tiles evenly distributed from the empire
center point, a 3D quasicrystal has empires with radial lines of higher tile density
penetrating evenly distributed points on a sphere. As explained in Electron clock
Intrinsic Time (Sec. 5.5), a massive particle in our framework is composed of a vertex
type (a supercell of 20 3-simplexes) that dynamically animates over many coordinate
changes or frames to form (1) a toroidal knot cycle internally in the QSN and (2) a
propagation pattern through the QSN that changes coordinate along a stepwise
helical path. The interaction of these two forms of stepwise internal toroidal cycling
and forward propagating helical cycling generate a richly complex dynamical pattern
of empire waves  waves which extend to the end of the universal space of the QSN
but drop in density with distance. These waves are the geometric¯rst principles key
to modeling forces in this framework. However, it is helpful to explain that quantum
mechanics does not require the assumption of Bohr's conjecture known as comple-
mentarity  the core of the Copenhagen interpretation of quantum mechanics. This
is the view that a fundamental particle, such as an electron, is either a wave or
particle but never both. Neither experiment nor the mathematical machinery of
quantum mechanics compel this interpretation. The Broglie-Bohm theory states that
an electron, for example, is always a wave and particle at the same time and that the
wave aspects guides the particle coordinate, like a pilot wave. The cost of this ab-
solutely rigorous but less popular interpretation is the requirement of the assumption
of inherent nonlocality in nature. Empire waves are nonlocal according to the non-
enigmatic geometric¯rst principles of projective geometry.
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A.15. Free lunch principle  forces
With this general overview of empire waves established, it is now possible to
understand how forces can be modeled via geometric¯rst principles. Let us begin
with the analogy of the game Scrabble, where you gain points by making multiple
words diagonally, vertically or horizontally using letters from one or more other
words already on the game board. When you do this, you get \free lunch" by earning
points for each word your letter(s) played a position in. The Scrabble board is
analogous to the QSN possibility space. And the 26 Roman letters are like the¯nite
set of geometric relationships or vertex types in a quasicrystal  the geometric-
symbols of the language. The rules and freedom of English are like the rules and
freedom of the phason code in a quasicrystal language. In Scrabble, the commodity
that is to be conserved and used e±ciently is the number of turns each player gets.
Each turn needs to generate as much meaning as possible. In emergence theory, the
same principle applies. There are a certain number of quasicrystal frames or \turns"
that a system of, say, two particles can be expressed in over some portion of a
dynamical sequence. Let us consider that it takes 10 frames to model a cycle of
electron internal clock action or some total length of discrete transitions through the
space. The patterns of this object in the QSN always need some integer ratio of the
given number of total frames used for internal clock cycle steps versus helical
propagation steps.
The physical pattern is expressed as the trinary selections of 3-simplexes in the
QSN: on-right, on-left or o®. And just as the words \cat" and \rat" can share an \a"
for greater e±ciency and synergy, the system of two such patterns moving through
the QSN allow us to save steps. We can model free lunch in this geometric code
thanks to the empires. When the¯rst propagating electron is moving near the second
electron, the two begin to bene¯t from one another's empire waves. In the simplest
example, consider that it would ordinarily take two remotely separated electrons 10
frames each to express a certain amount of clock cycling and propagation. However,
the closer they are to one another, the more free lunch they will enjoy. The system
saves frames when a selected tetrahedron from one particle's empire is in the nec-
essary right or left \on" state to that matches the state necessary to¯ll a position in
the geometric pattern of a second electron, thereby saving a frame in the way that we
saved an \a" in our Scrabble game example.
The result is that the particles require fewer phason °ips or frames of trinary
selections on the universal QSN to express their clock cycles and their given number
of propagation steps along some direction. The physical meaning of this is that they
have advanced a further distance than they would have otherwise with 10+10, where
no free lunch is enjoyed. And because the density of free lunch opportunities
increases with approaching distance, the two particles will accelerate toward one
another as their separation decreases.
The empire wave around a massive particle in this framework is distinctly chiral
and behaves according to the right-hand rule, where the direction of propagation
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determines the direction of the chiral free-lunch empire wave system around it. A
\train" of these objects, such as electrons in the QSN, will fall in-line behind one
another and form a current because that positioning ensures the maximum amount
of free lunch. By all moving along the same helical path, a group empire wave system,
in the form of the chiral magnetic¯eld, emerges around them. However, most elec-
tron models are in either groups of free electrons or are in atomic systems that are
arranged with many di®erent orientations, such that the emergence of a chiral
magnetic¯eld does not occur. In other words, picture our model of the electron
approaching Earth. As it accelerates closer, the probability of¯nding free lunch
frame savings increases. Again, the empire wave¯eld of every massive fundamental
particle on Earth has no general similarity in their various orientations or directions
of propagation. And they are not strongly correlated. Accordingly, around Earth,
there is an enormous superposition of empire waves from every massive particle. One
can say that it is a noisy quantum¯eld of empire waves on the dynamical QSN.
There is a high degree of non-coherence, as compared to a current of electrons, where
there are coherent group patterns in the empire waves  like combed °owing hair as
opposed to tangled hair. Nonetheless, there will still be some opportunities for free
lunch around the tangled array of empire waves surrounding large groups of massive
particles for any approaching electron from outer space to enjoy as it nears Earth.
But it will be exponentially less than the free lunch around the current of electrons.
Gravity would logically be orders of magnitude weaker than electromagnetic forces.
And it will be distinctly non-chiral, due to the fact that the average chirality is null,
with an approximately equal quantity of right and left-handed empire waves states
on the QSN around Earth (other than the Earth's magnetic¯eld).
A.16. A non-arbitrary length metric
The nearest neighbor lengths between points in the QSN are the Dirichlet integers 1
and 1. So if our framework is generally correct, it would more deeply explain why
black hole physics corresponds to the golden ratio and why quantum mechanics does
in the form of the ’5 entanglement probability discovered by Lucien Hardy.65
Accordingly, a new length system based on golden ratio values would simplify many
equations in physics. For example, the three most fundamental constants are the
speed of light, c, the gravitational constant, G, and Planck's constant, h. The only
number that uni¯es all three is a length called the Planck length, l, which happens to
be about 99.9% of the golden ratio in the metric system.
l ¼
ffiffiffiffiffiffiffi
}G
c3
r
:
ðA:3Þ
If spacetime had substructure built on our Planck length scale QSN, planetary
systems might evolve overtime to energetically favorable cyclical and length ratios
that approximate simple golden ratio fractions. And if we based our measuring
system on a physical valued tied to a planet, it would be less arbitrary than, for
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example, the yard, which was based on the distance from King Henry I of England's
nose to thumb distance.
Indeed, the metric system is less arbitrary because it is based on 14 the circum-
ference of Earth, where the distance from the Equator to the North pole is 10,002
kilometers, making the metric system unit value of 1, a full 99.98% of that distance
(disregarding where the decimal is). When the system was established, they could
not achieve the full accuracy of measuring this distance on Earth. So today, the
metric system unit is almost that distance. It is not well known, but the metric
system deeply relates to approximations of golden ratio values. The Earth and Moon
system is approximately a quarter of the age of the universe. So it has had a long time
to self-organize into optimal ratios that approximate the golden ratio. To an accu-
racy of 99.96%, the dimensionless ratios are
radius of Earth
radius of Earth
	


2
þ radius of Moonþ radius of Earth
radius of Earth
	


2
¼ ’2;
ðA:4Þ
or
radius of Moon
radius of Earth
	


¼
ffiffiffi
’
p  1:
ðA:5Þ
In other words, this is a double coincidence. It is not just that the sum of the Earth
and Moon diameters in the metric system are almost exactly the golden ratio
1.618..., but the breakdown of the two diameters that sum to that value is ’
ffiffiffi
’
p
for the Moon and
ffiffiffi
’
p
for Earth.
The master dimensionless ratio of fundamental physics is the¯ne structure con-
stant, a. Interestingly, it is also closely approximated with golden ratio expressions as
a ¼ ’2=2; ’2=360 ½to an accuracy of about 99:7%:
ðA:6Þ
A.17. A non-arbitrary \Time" metric as ordered quasicrystal frames
Much of the data we present in this paper includes time-based or planet and moon
cycle \coincidences" that seem to match far too closely to the golden ratio to be
explained away by anything other than the presumption of some unknown sub-
structure of spacetime in a new quantum gravity framework.
By combining both time and length-based values, the critical reader can perhaps
be interested in the following impressive number.
The gravitational constant, G, ties time and length based values together as
G ¼ c2=4:
ðA:7Þ
h ¼ 1:0000026 of the golden ratio as 0:6180382ð1059Þ cubic meters per second
[note 1=’ ¼ 0:61803 ... and ’ ¼ 1:61803 ... are the same ratio]. This deviation at the
millionth place after the decimal is remarkable.
Now, having put forth an argument why it is plausible that spacetime can have a
golden ratio-based substructure as a natural result of the projection of E8 to a lower
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dimensional quasiperiodic point space, we can speculate on the idea that the metric
system is deeply related to ’ and consider the idea of a¯rst principles analytical
expression of the constants c, G, h and a. But clearly, there is a problem. The¯rst
three constants are dimensional and tie into the speed of light. And the speed of light
is based on a length metric and a time metric. The length metric is being proposed
as nonarbitrary, according to this speculative argument related to our projective
approach to E8 uni¯cation physics.
However, the speed of light playing into these corresponding equations appears at
¯rst to be based on an arbitrary metric for time, the second. The QSN is based on the
numbers 2, 3 and the golden ratio because the 3-simplex building blocks are regular
or non-distorted. And the golden ratio is deeply related to 5 geometrically in the form
of the pentagon and to 5 algebraically as 12 of
ffiffiffi
5
p þ 1. From the analytical expressions
of the 3-simplex volume to its length values, such as height and centroid to vertex
distances, it is fundamentally built of the numbers 2 and 3 and their square roots. So
the QSN is deeply related to 2, 3 and 5. Incidentally, these are the symmetries that
de¯ne anything with icosahedral symmetry. And nearly every quasicrystal found in
nature (over 300) possesses icosahedral symmetry. It is interesting, then to note,
therefore, that the constant c (in the metric system) is 99.93% of the number 3,
disregarding where the decimal is placed. And the distance of the Earth to the Sun is
99.73% of 3/2.
The number of (presumably) ’-based meters traveled by a photon in vacuum in
one second is a close approximation of 3/2. Assuming hypothetically, that E8 qua-
sicrystalline physics is a good approach, why is this the case if the second is arbitrary?
The second is not arbitrary, of course. It is based on a cycle of the fundamental
Earth clock system, which itself is fundamentally based on ’, as argued above. It is
based on the clock cycles of the Earth rotating once on its axis, which is gravi-
tationally and electromagnetically tied to the Earth, Moon, and Sun system as a
whole. The number, of course, is 86,400 seconds in one of these non-arbitrary
physical cycles of the Earth clock. That is, 60 s  60min  24 h. Remarkably, the
modern precise average Earth day is 86,400.002 s. So the old number is unexpectedly
close to the accurate measurement. Again, we disregard where the decimal place is in
the context of thinking about the fundamental aspect of a number  its factori-
zation. Accordingly, 86,400 becomes 864 ¼ 25 33, a number deeply related to 2, 3
and 5.
Have we missed anything obvious? Yes, the Earth distorts along the equator. So if
we adjust for the meter to assume a non-rotating Earth with no distortion, we can see
if our number gets closer or further from the golden ratio being the Planck length.
Realizing that the pole-to-pole diameter of Earth is 12,713 km, and simplifying the
value by moving the decimal to 1.2713, we can calculate that a 14 of a circle inter-
secting a non-distorted sphere of this diameter is 0.9984766. This then is normalized
to 1. Note that this approach is not based on a metric. It is based on the ratio of the
Moon to Earth, where we get the dimensionless value. And here, we are not using the
ideal Phi values mentioned. We are using the actual values in their ratio. So this gives
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the dimensionless ratio value of 0.9984766 in the manner just described. We can then
normalize this to a standard unit of 1. Again the justi¯cation is the conjecture that
the substructure of space is based on a dimensionless ratio of one part being 1 and the
other part being 1=’. Now, what this means is that the Planck length now changes
slightly from the current value of 1.616199, which is based on the meter that is
measured from a distorted Equator to the normalized value based on the new di-
mensionless ratio-based length and based on the actual measurement of the Earth's
pole through pole diameter (not plugging the golden ratio approximation of that
diameter). We get a logically adjusted Planck length of 1.6183412... or 1.0002 of the
golden ratio.
A.18. Mass in the quasicrystalline spin network
We will now combine the following ideas in order to understand mass in the emer-
gence theory framework:
(1) Free lunch and empire waves
(2) Massive particle clock time to propagation inverse relationship
(3) Principle of e±cient language
Obviously, our vision of a geometric¯rst principles uni¯ed quantum gravity theory,
as explained thus far, reduces everything to length. Our formalism is Cli®ord rotor
operations on a spin network made of the two Dirichlet integer values 1 and 1=’.
Mass is the degree of resistance to a change in direction or acceleration of a massive
particle. If space and time are discretized, where space is divided into positions like on
a checker board and time is divided into turns of the players, where a piece can only
move to a connected square, an intuitive understanding of mass emerges.
In Fig. A.12, we see that putting a particle in motion along some direction
in spacetime as the square grid achieves an e±cient diagonal progression across
(a)
(b)
Fig. A.12. Two paths in a spacetime grid. (a) Shortest, most e±cient path between position 1 and 8.
(b) Seven e±cient paths of which one is illustrated.
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the board, using the shortest path between position 1 and 8. In the left grid
(Fig. A.12(a)), there is only one shortest path. In the right grid (Fig. A.12(b)), there
are seven paths of which one is illustrated.
As explained in Sec. 5.3, mass is connected to curvature. In our framework of
discretized spacetime, curvature is derived from the angle 15.522 as explained in
Fang's et al. paper`Encoding Geometric Frustration'.104
A.19. Generation of the quasicrystalline spin network
A.19.1. A fang methods
In Method 1, Fang Fang initially constructed the QSN by modifying the icosagrid
with Fibonacci chain spacing to make it a quasicrystal. In doing this, the alternative
Method 2 was discovered. Packings of tetrahedra in the form of the FCC lattice
are Fibonacci chain spaced. Then¯ve copies are rotated from one another by the
15.522 angle.
A.19.1.1. Method 1. Fibonacci icosagrid
This approach is inspired by the pentagridmethod of constructing the Penrose tiling.
A 3D analogue is the icosagrid construction method for icosahedrally symmetric
quasicrystals. 10 sets of equidistant planes parallel to the faces of an icosahedron are
established with periodically repeating parallel planes in each set that, together, form
the icosahedrally symmetric icosagrid. The intersecting planes segment the 3-space
into an in¯nite number of 3D cell sizes. The icosagrid is not a quasicrystal due to the
arbitrary closeness of its edge intersections and the resulting in¯nite number of
prototile sizes. We converted it into an icosahedral quasicrystal by changing the
equal spacing between parallel planes to have a long and short spacing, L and S with
L/S ¼ golden ratio and the order of the spacing follow the Fibonacci sequence.
Therefore, we call this kind of spacing the Fibonacci spacing. This Quasicrystal
turned out to be a 3D network of Fibonacci chains and we would like to name it QSN.
A.19.1.2. Method 2. Golden composition of the¯bonacci tetragrid
Similar to the icosagrid, a tetragrid is made of four sets of equidistant planes that are
parallel to the faces of a tetrahedron. Applying the Fibonacci spacing to this
structure will also give us a quasicrystal with tetrahedral symmetry (Fibonacci tet-
ragrid)  again we focus mostly on the regular tetrahedral cells. In order to obtain
icosahedral symmetry, we need to implant the 5-fold symmetry. We applied a Golden
Composition process to this Fibonacci tetragrid and achieved the same QSN structure.
The Golden Composition is described as follows:
(1) Start from a point in a Fibonacci tetragrid and identify the eight tetrahedral
cells sharing this point with 4 in one orientation and the remaining 4 in another
orientation.
(2) Pick the four tetrahedral cells of the same orientation and duplicate another four
copies.
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Put two copies together so that they share their center point, the adjacent tetra-
hedral faces are parallel, touching each other and with a relative rotation angle of
ArcCosð½3’ 1=4Þ, the golden rotation. Repeat the process three more times to add
the other three copies to this structure. A twisted 20-tetrahedra cluster, 20G, is
formed in the end. Now expand the Fibonacci tetragrid associated with each of the
4-tetrahedron sets by turning on the tetrahedra of the same orientation as the four,
an icosagrid of one chirality is achieved. Similarly, if the tetrahedral cell of the other
orientation are turned on, an icosagrid of the opposite chirality will be achieved.
In either case, there is a 20G at the center of the structure.
A.19.2. Cli®ord rotor induction method
As explained in the paper`Emergence of an Aperiodic Dirchlet Space from the
Tetrahedral Units of an Icosahedral Internal Space'102 an inductive framework has
been established to link higher-dimensional geometry from the basic units of an
icosahedron using spinors of geometric algebra and a sequence of transformations
of Cartan sub-algebra. Spinors are linear combination of a scalar and bivector
components de¯ned (see Eq. A.8).:
s ¼ 	D þ 
De12 þ De23 þ De31:
ðA:8Þ
The subscript D denotes that the spinorial coe±cients live in a Dirichlet coordinate
system, i.e., 
D :¼ 
1 þ 
2, where  is the golden ratio. This approach presents, for
the¯rst time to our knowledge, a direct inductive and Dirichlet quantized link
between a three-dimensional quasicrystal to higher-dimensional Lie algebras and
lattices that are potential candidates of uni¯cation models in physics. Such an
inductive model bears the imprints of an emergence principle where all complex
higher-dimensional physics can be thought to emerge from a three-dimensional
quasicrystalline base.
A.19.3. Dirichlet integer induction method
The need for quasicrystalline coordinates brought us naturally to consider a class of
number which are more rich than the rational integers (useful for crystals), but more
constrained than the real numbers, the quadratic integers. From this class, the¯ve-
fold symmetry of our quasicrystal guides our choice to the ring living in the quadratic
¯eld associated to 5, which is sometimes noted as Z½’, the quadratic ring of
\Dirichlet integers" referencing to their use in Dirichlet's thesis and following works,
or in short D.
Then we use a digital space D3, to host triplets of Dirichlets integers, and a digital
spacetime D4, to host quadruplets of Dirichlets integers. D4 could have a quaternion
structure, and written H D. Where D3 the space part, is the imaginary part.
Furthermore, the structure can be complexi¯ed to biquaternion, and also put in
bijection with octonion and sedenion.
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Each point can also be seen as a 4 by 4 matrix of integers, a digital tetrad,M4ðZÞ,
or as HH Z.
The 16 numbers are integers.
1
i
j k 
aw bw cw dw
ax bx
cx dx
ay
by
cy
dy
az
bz
cz
dz

1
’
I
I’
:
There is one line per dimension, and the¯rst dimension, indexed by w is hidden in the
space construction. a and b are combined to make the real part of the Dirichlet complex.
The generators satis¯es
ijk ¼ i2 ¼ j2 ¼ k2 ¼ I2 ¼ ’ ’2 ¼ 1.
In a¯rst
approach, the imaginary part will be set to 0 (so all c and d are null). A point in the
realized space will just show three coordinates:
ax þ ’bx;
ay þ ’by;
az þ ’bz;
...where ’ ¼ 1þ
ffiffi
5
p
2
, and is equivalent to the non-golden part made of the a, and the
golden part made of the b.
In a Euclidian spacetime, aw and bw can correspond to time, while it is cw and dw
in a Lorentzian spacetime, and all four are used in a Kaluza–Klein model.
A set of eight integers, (the a and b, or the c and dÞ, can encode a position in an E8
lattice (with a doubling convention).
Let us focus on how these numbers emerge. Our model is from the¯rst principle
built from regular tetrahedral in an Euclidian 3D space, because the simplex is the
simplest geometric symbol, and the space is observed as tridimensional.
We ask the question: which sets of vertices in D3 can hold regular tetrahedra of
the same size having one vertex in the center (0,0,0)? The equation is the equation
of the sphere, written in D3, which holds two equations by separating the golden and
the non-golden parts. From this, the result is
. Amaximal possibility space bigger than the QSN but smaller than Dirichlet space.
. Vertex¯gure: New polyhedron with 108 vertices and 86 faces.
. Tetrahedron centers¯gure: New polyhedron with 32 vertices which is not the
icosidodecahedron.
. 72 possible tetrahedra around a vertex (see Fig. A.17).
Having built a¯rst-principle version of Dirichlet space where 20G emerge naturally
(but as 4 copies), I have the intuition that E8 physics can also emerge naturally as
encoded by the possible tetrahedron con¯gurations. We will focus on rule emergence.
Some come from Physics, like the hadronic rule saying that the combined color of
three quarks in a neutron or photon is neutral, also known as the SU(3) symmetry,
quantum chromodynamics; some from information theory and mathematics; some
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from matching with quasicrystal study when importing CQC simulation with dy-
namic window and observing phason occurrence, to deduce phason rules.
(The Dirichlet Integer Induction Method was introduced in an internal commu-
nication from Raymond Aschheim to Klee Irwin via email on May 19, 2016).
A.19.4. Projection and graph diagram method
To illustrate the correspondence of the 20G twist to the E8 lattice, I developed the
following method:
(1) Project E8 to 4D to generate the Elser–Sloan quasicrystal. It is made entirely of
600-cells. Alternatively, we may project one of the 240 vertex root vector poly-
topes of E8 to 4D to generate two 600-cells scaled by the golden ratio.
(2) Select 20 tetrahedra sharing a common vertex in a 600-cell and project the cluster
to 3D such that the outer 12 points form the vertices of a regular icosahedron.
(3) Induce its dual, the dodecahedron, which has 30 points.
(4) Use the 30 points to create a graph diagram by connecting points separated by a
distance of ’ (Sqrt2) times the dodecahedral edge length. This creates a 3D
graph diagram equal to two superimposed tetrahedron 5-compounds, one right
and one left-handed.
In Fig. A.13, the right chirality 5-compound is shown. The Cartesian coordinates
of the 30 vertices are the cyclic permutations of:
ð1;1;1Þ
ð0;1=’;’Þ
ð1=’;’; 0Þ
ð’; 0;1=’Þ:
(5) Select either the right or left-handed tetrahedron 5-compound. And from it, we
select one tetrahedron and translate a copy of it away from the center of the
cluster along one of its 3-fold axes of symmetry by a distance of Sqrt(3/8) times
its edge length — the distance necessary to translate one of its vertices to be
coincident with the center of the tetrahedron 5-compound. We do the same
Fig. A.13. Compound of 5 tetrahedron with right chirality.
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copy-and-translate action along the other three of the selected tetrahedron's 3-
fold axes of symmetry. This generates four new tetrahedra that share a common
vertex with the centroid of the cluster. Their 12 outer vertices form the points of
a cuboctahedron (as shown in Fig. A.14).
We repeat this 4-step process with the remaining four tetrahedra from the
initial tetrahedron 5-compound (see Fig. A.15). We then remove the original¯ve
tetrahedra as well as the dodecahedron.
Thus far, this induction process generated 20 tetrahedra sharing a common
vertex at their group center. It is the 20G twist with 60 outer vertices equal to a
cuboctahedron 5-compound. It has Cartesian coordinates that are the cyclic
permutations of
ð2; 0;2Þ;
ð’;’1;ð2’ 1ÞÞ;
ð1;’2;’2Þ:
Fig. A.14. One tetrahedron is translated along each of three edges sharing a vertex to give four tetrahedras.
Fig. A.15.
(a) A small tetragrid local cluster with eight tetrahedral cells, four "up" and four "down".
(b)–(f) The golden composition process: (b) 1 tetragrid. (c) 2 tetragrids, shown in red and orange.
(d) 3 tetragrids, shown in red, orange and green. (e) 4 tetragrids, shown in red, orange, green and blue.
(f) 4 tetragrids, shown in red, orange, green, blue and purple.
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(6) Finally, in Fig. A.16, we repeat the entire process with the starting tetrahedron
5-compound of the opposite chirality. The right-handed 20G twist [f6a] and left-
handed one [f6c] are superimposed in the QSN to form the basic building block
set of possibilities from 60+1 points and 180 possible 3-simplex edges or
connections [f6b].
A.20. Hyperdimensional information encoded in 3D
One of the principles of the emergence theory approach is simplicity. We question the
physical realism of hyperdimensional spaces implied by models such as general rel-
ativity and string theory.
However, it is clear that the gauge symmetry uni¯cation of all particles and
forces of the standard model, which is everything except gravity, are described by
the six-dimensional root vector polytope of the E6 lattice. Full gauge symmetry
uni¯cation with gravity seems to be possible with the eight-dimensional E8 lattice,
which embeds E6. This uni¯cation can be achieved with or without geometry by
selecting either the pure algebraic Lie algebras or the geometric analogues of
hyperdimensional crystals and geometric algebras. Even without considering the
gauge theory implications of hyper geometry, general relativity alone relies on a 4D
geometric structure.
Fig. A.16.
(f6a) The right twisted 20G. (f6b) The superposition of the left-twisted and right-twisted 20G.
(f6c) The left twisted 20G.
Fig. A.17. Seventy two possible tetrahedra around a vertex. Four combinations of non-intersecting
tetrahedron subsets of the 72. (a) 2 left twisted tetrahedra (see Fig. A.16 f6c). (b) 2 right twisted tetra-
hedra (see Fig. A.16 f6a). (c) 2 right twisted tetrahedra rotated 90 degrees compared to (b) (see Fig. A.16
f6a). (d) 2 left twisted tetrahedra rotated 90 degrees compared to (a) (see Fig. A.16 f6c). From left to right,
the 2 tetrahedron facing left, right, right, left chirality. Here you see we can mix chiralities.
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Clearly, fundamental physics implies a deep tie to hyperdimensional math.
But we have never measured any geometric dimensions beyond 3D. So the
Occam's razor approach is to see if we can derive all the hyperdimensional infor-
mation from a purely 3D framework without having to adopt the ontological
realism of, for example, curled up spatial dimensions or the 4D spacetime of general
relativity.
Because we can measure reality to be 3D and because it is simpler than hyper-
spaces, we seek to model the implied math of hyperdimensional geometry while
restricting ourselves to Euclidean 3-space.
The quintessential example of a lower-dimensional object encoding the informa-
tion of a higher-dimensional object is irrational angle-based projective geometry-
quasicrystallography.
The vertex Cartesian coordinates are the cyclic permutations of
ð1;1;3Þ
ð’1;ð’2Þ;2’Þ
ð’;ð2’1Þ;’2Þ
ð’2;ð’2Þ;2Þ
ðð2’ 1Þ;1;ð2’ 1ÞÞ:
A.21. Emergence of a self-actualized code operator
Frank Wilczek challenged physicists to develop a conscious measurement operator
that comports with the formalism of quantum mechanics.50 This is daunting for
social reasons. Discussions of consciousness in academic circles of physicists is gen-
erally scorned, with few exceptions. And many physics journals reject such notions
under the unwritten premise that philosophy and physics should not be combined.
However, blunt logical deduction, free of social fears, points to the idea that
consciousness is a fundamental element, as though it is the substrate of reality.
J.B.S. Haldane93 said:
We do not¯nd obvious evidence of life or mind in so-called inert
matter...; but if the scienti¯c point of view is correct, we shall ultimately
¯nd them, at least in rudimentary form, all through the universe.
Erwin Schr€odinger94 said:
For consciousness is absolutely fundamental.
Andrei Linde,95 co-pioneer of in°ationary big bang theory, said:
Will it not turn out, with the further development of science, that the study
of the universe and the study of consciousness will be inseparably linked,
and that ultimate progress in the one will be impossible without progress in
the other?
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David Bohm96 said:
The laws of physics leave a place for mind in the description of every
molecule... In other words, mind is already inherent in every electron,
and the processes of human consciousness di®er only in degree and not in
kind.
Freeman Dyson96 said:
That which we experience as mind... will in a natural way ultimately
reach the level of the wave-function and of the 'dance' of the particles.
There is no unbridgeable gap or barrier between any of these levels... It is
implied that, in some sense, a rudimentary consciousness is present even
at the level of particle physics.
Werner Heisenberg97 said:
Was [is] it utterly absurd to seek behind the ordering structures of this
world a consciousness whose \intentions" were these very structures?
The growing credibility of the digital physics argument still leaves one with the sense
of audacious improbability. These scientists claim that the universe is a simulation in
the quantum computer of an advanced being or society. Although they could be
correct, this has a similar level of outlandishness as the idea that a creator God from
outside the universe is the source of everything. Of course, this is a popular religious
view. But the idea that something from outside the universe created the universe
implies a new de¯nition of the term universe. That term is supposed to mean
everything. The idea of a self-actualized universe may be more sensible.
The mounting evidence that the universe is made of information and is being
computed includes the aforementioned mathematical proof of the Maldacena con-
jecture and the discovery of error correction codes. There are many other pieces of
evidence that add to the argument. But for those new to the thought process, here is
a simple way to deduce that something like a computer or mind is needed: Every-
thing we know about physics, including classic physics, indicates that reality or
energy is information. And information cannot exist without something to actualize
it. It is abstract and relates deeply to a mind-like entity, whether that be a biological
neural network or an arti¯cial intelligence.
However, there is a more plausible explanation than the digital physics computer
simulation hypothesis. In his submission to the FQXi Essay contest, mathematical
physicist, Raymond Aschheim,7 a scientist at Quantum Gravity Research, said:
Can reality emerge from abstraction, from only information? Can this
information be self-emergent? Can a structure be both the software and
the hardware? Can it be ultimately simple, just equivalent to a set? Can
symmetry spontaneously appear from pure mathematical consideration,
from the most symmetric concept, a Platonic \sixth element"? Would this
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symmetry be just structuring all the particles we know? Can all this be
represented? Can standard physics be computed from this model? Eight
questions: eight times yes.
The notion of a self-emergent computational but non-deterministic neural-network
universe is more plausible than the idea of a simulation creator from outside the
universe. In fact, the emergence of free will and consciousness need not be a specu-
lation. It is proven to exist, at least in humans. So it is one of the interesting
behaviors of the universe locally in the region of our physical bodies. In my paper, A
New Approach to the Hard Problem of Consciousness: A Quasicrystalline Language
of \Primitive Units of Consciousness" in Quantized Spacetime,98 I discuss in detail
the plausibility of a self-emergent mind-like universe. The¯rst question is to consider
whether or not physics imposes a limit on self-organizing evolution of consciousness.
In other words, are humans the limit or can intelligence tend toward in¯nity? From
what we know about classic and quantum physics, there is no limit. It can tend
toward in¯nite awareness and intelligence. The next question is, \What percentage
of the energy in the universe can self-organize into conscious systems and networks of
conscious systems?" Of course, the answer is the same as the¯rst question. Physics
imposes no upper limit. So the answer is that, in principle, 100% of the energy of the
universe can self-organize into a conscious network of conscious sub-systems. The
¯nal consideration in the deduction relates to the axiom:
Given enough time, whatever can happen will happen.
By this axiom, somewhere ahead of us in spacetime, 100% of the universe has self-
organized into a conscious system. It certainly need not be anthropomorphized. We
can leave the detail of what this entity would be like out of the deduction. For
example, there is no reason to presume that it cannot exist trans-temporally and
have an extremely di®erent quality than what we conceptualize as consciousness.
The next step of deduction is to question whether or not trans-temporal feedback
loops are disallowed by current physics paradigms.
Stephen Hawking of Cambridge and Thomas Hertog of the European Laboratory
for Particle Physics at CERN say that the future loops back to create the past.99 The
delayed choice quantum eraser experiment also indicates that the future loops back
to create the past. And Daryl Bem of Cornel has published several experimental
results demonstrating retro-causality.57 In 2014, Brierley et al.100 demonstrated
quantum entanglement of particles across time. In fact, there is an old wives tale that
general relativity prohibits trans-temporal feedback loops. This is not true. General
relativity simply states that communication between events cannot occur via photon
mediation. In fact, general relativity predicts wormholes through time and space.
The inherent non-locality of quantum reality does not require signals for things to be
connected; any more than rotating a penny while looking at the heads side requires
time to transmit the torque to the other side of the penny. It is a simultaneous or
null-speed correlation. The truth is that until we have a predictive¯rst principles
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theory that uni¯es general relativity and quantum mechanics, one cannot aggres-
sively invoke interpretations of either of these two place-holder theories to say with
con¯dence what can and cannot occur. Both will turn out to be °awed or incomplete
in certain ways when a full theory of everything is discovered.
So for now, let us develop the most conservative argument as follows:
(1) Like the exponential explosion of any doubling algorithm, high-level forms of
consciousness and networked consciousnesses will envelope the universe. There
are no hypotheses that can reasonably challenge this idea. We have hard evi-
dence that consciousness emerges because our minds are sharing the words of this
sentence. The idea of consciousness exponentially spreading throughout the
universe is plausible due to the extraordinary behavior of doubling algorithms.
For example, if we doubled a penny as fast as we can hit the \x2" button on an
iPhone calculator in 30 s, we would have more pennies than all the atoms in the
entire universe. The reason we do not see doubling algorithms in nature go more
than a few iterations because resources halt the doubling algorithm very early.
(2) The question is whether or not a species with high consciousness and evolving
consciousness can leave their biosphere and continue doubling and staying non-
locally networked. Humans made it to another cosmological body in 1969, when
we landed on the Moon. It is only a matter of time before technology and our built-
in compulsion to explore takes us out into the universe, where resource limitation
halting will not occur until all the energy of the universe is exhausted. Again, the
challenge is not to argue why this will occur. That is established by the axiom
\Given enough time, whatever can happen will happen". The onus of logic falls on
those who guess humans will destroy themselves or that society will collapse or and
that all other potential species in the universe will have the same fate.
(3) Now, if the universe is expanding faster than the speed of light, then exponen-
tially expanding consciousness can never sequester all energy into a universal
scale conscious neural network of quantum entangled conscious sub-systems. As
mentioned, general relativity allows wormholes, and quantum mechanics is in-
herently non-local. So until a predictive theory of everything is discovered, it is
not clear whether or not non-local information exchange or teleportation can
occur, where a consciousness can relocate trans-temporally or trans-spatially in
instant-time (perhaps without atomic form) to in°uence matter and energy in
distant regions of the universe. However, it is worthwhile to play the what-if
game to see where the idea leads. If a new¯rst-principles quantum gravity theory
inspired a technology that allowed consciousness to project into spacetime
coordinates non-locally, where would we go¯rst?What if you were given 100 free
airline vouchers to °y anywhere in the world? Would you explore ballistically by
¯rst traveling 100 miles from your home, then 200 miles and so on until you
explored the far reaches of the world? Or would you make a favorites list and
bounce around arbitrarily depending on whether Beijing, Sydney or Rio made it
near the top of your wish list? If humans or any other intelligent life in the
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universe discovers non-local information exchange that a consciousness can ex-
ploit, we will bounce around the universe and plant consciousness in various
parts of the cosmos. At¯rst, the transplanted consciousness outposts will have
the sparse pattern of a sponge — or neural network — throughout the whole
cosmos. So the expansion rate of the universe would not be a problem for the
deduction that, given enough time, high consciousness will eventually envelop all
energy in the universe. The sparse sponge-like pattern of outposts of networked
consciousness will¯ll-in as they approach maximum density at 100% of the
energy of the universe.
(4) What would this high consciousness be like? It is hard to say. But it would not be
very much like us. We are related to snails and horses and dinosaurs, but we are
not very much like them. However, the one thing we would share in common is
that we would understand the¯rst-principles theory of everything that would be
a prerequisite for the exploitation of non-local mental and physical technology.
When a¯rst-principles theory of everything is discovered, it will not be replaced
by something else. To say otherwise indicates a misunderstanding of what the
term¯rst-principles means in this context. The Pythagorean Theorem is based
on¯rst principles. It will not be replaced. We are not talking about a model of
how the universe works. We are speaking of discovering the simulation code of
geometric symbolism itself and interacting with it.
So, we have told an audacious story, even though it may be logically inevitable.
However, it should be noted that the big bang theory is audacious and probably true
at the same time. The emergence of this very conversation, dear reader, and the
human consciousness that it exists within is audacious. And so too is the notion of
the universe being a simulation from a creator outside the universe. So if auda-
ciousness is evil, then we are seeking the lesser of all evils. The deduction herein is in
fact conservative. And yet it is audacious at the same time. It is not just plausible. It
is inevitable.
The punchline of the deduction is this: Because this is an inevitable outcome, the
simplest answer on how an information theoretic universe can exist and what its
substrate would be if it self-actualized is the entire system  reality  is a mind-
like mathematical (geometric) neural network. Just as our now limited consciousness
can hold within it the notion of a square, we can allow a self-organizing game or
language of squares to emerge in our mind. A far greater neural network could hold
within it the relatively simple geometry of E8 and the 4D and 3D quasicrystals we
have discussed. Primitive quanta or measuring entities (quantum viewers) at the
Planck scale substructure of the imagined possibility space, which are essentially
vantage points of the universal emergent consciousness, would actualize geometric
symbols by observations (projective transformations) within the quasicrystalline
possibility space. Each observation of a Planck scale quantum viewer generates a
projective transformation equal to a rotation of the 3-simplex it is associated with.
These primitive geometric binary choice states on the possibility space are part of a
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code that forms a neural network based on 3D simplex-integers in an E8 derived
quasicrystal. By geometric¯rst principles, the code has a free-variable called the
phason °ip. And the universal consciousness operating the details of the code obeys
the principle of e±cient language, taking instructions from conscious sub-systems
like us, who are engines of emergent meaning.
The universe would not exist if it weren't for intermediary emergent entities like
us. It would also not exist if it weren't for the maximally simple golden ratio-based
quasicrystalline E8 code that self-organizes quarks and electrons into 81 stable atoms
and into countless compounds and planets and people and societies and overly-wordy
sentences and on up through to the collective consciousness of the universe. And that
primitive starting code and the simplex-integers and the quantum viewer operators
needed to animate the whole thing would not exist without the collective emergent
consciousness. Retrocausality allows the whole idea to be logically consistent, where
the future creates the past and the past creates the future. The simple creates the
complex and the complex creates the simple  a cosmic scale evolving feedback loop
of co-creation. This framework is both explanatory and conservative. And it requires
no magical moments that are unexplainable, like the moment of the big bang or a
creator-God. It uses¯rst-principles logic where A co-creates B, which co-creates C,
which co-creates A. Non-linear causality is mathematically and logically rigorous.
The entire framework is based on two fundamental and inarguable behaviors of
nature: (1) emergent complexity and (2) feedback loops.
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