Article
Geometric State Sum Models from Quasicrystals
Marcelo Amaral *
, Fang Fang
, Dugan Hammock
and Klee Irwin






Citation: Amaral, M.; Fang, F.;
Hammock, D.; Irwin, K. Geometric
State Sum Models from Quasicrystals.
Foundations 2021, 1, 155–168.
https://doi.org/10.3390/
foundations1020011
Academic Editor: Eugene Oks
Received: 9 September 2021
Accepted: 9 October 2021
Published: 13 October 2021
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4.0/).
Quantum Gravity Research, Los Angeles, CA 90290, USA; Fang@QuantumGravityResearch.org (F.F.);
Dugan@QuantumGravityResearch.org (D.H.); Klee@QuantumGravityResearch.org (K.I.)
* Correspondence: Marcelo@QuantumGravityResearch.org
Abstract: In light of the self-simulation hypothesis, a simple form of implementation of the principle
of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in
three dimensions. Emergence is discussed in the context of geometric state sum models.
Keywords: self-simulation hypothesis; principle of efficient language; quasicrystals; empires; game
of life; emergence; state sum models
1. Introduction
The self-simulation hypothesis (SSH) [1] posits that emergence is a core element in the
engine of reality, down to the underlying code—considering spacetime and particles as
secondary or emergent from this code. In 1938, Dirac [2] addressed the internal structure
of the electron and how it affects the spacetime structure itself. Feynman later evolved this
thinking to point out that a point of spacetime is like a computer, which lead Finkelstein
to propose that reality is a code in action in his 1969 spacetime code paper [3]. Wheeler
later wrote an interesting synthesis of this with ideas like the participatory universe as a
self-excited circuit, law without law and it from bit [4–6]. Second test edit. This information
theoretic line of thinking can lead to many ways of addressing the conundrum of quantum
gravity and unification physics problems. We focus here on the specific perspective that a
notion of pre-spacetime code or language in action leads to the physics and metaphysics
idea of reality as a self-simulation [7–9] and requires a new principle to drive the evolution—
the principle of efficient language (PEL) [1,7,9–11].
To address emergence of physical observables from an information theoretic frame-
work governed by the PEL, we need to set up a concrete, constrained and rigorous math-
ematical substrate. We will consider the Three-Dimensional Penrose tiling quasicrystal
(3DPT), also known as Amman or Ammann-Kramer-Neri tiling, projected from the Z6
lattice [12–14]—a generalization of the two-dimensional Penrose tiling [15]. On this point
set and associated tilings, we implement a state sum model [16,17], where the states are
given by objects inherent in the quasicrystalline geometry. We consider geometric realism
discussed in Section 2 as a new paradigm for state sum models following Einstein’s pro-
gram of geometrization of physics [18]. As we will see, quasicrystals are a natural substrate
for geometric realism, where self-referential geometric symbols [1,19] are given from first
principles for both kinematics and dynamics. Details on quasicrystals will be presented in
Section 3. Essentially, quasicrystalline structures [20–24] are structures that exhibit a new
kind of order—aperiodic order—which lies between disorder and periodicity. Statistical or
quantum mechanical models defined on lattices can be generalized in a straightforward
manner to quasicrystals [25]. The choice of Z6 root lattice and its associated 3DPT qua-
sicrystal provides a toy model for the conformal symmetry associated with the D6 root
system and the grand unified gauge theory associated with the exceptional Lie algebra
E8. Both D6 and E8 have similar quasicrystals associated with them [26–28]. The gauge
symmetry represented in the root system is transformed to a 3D quasicrystal network,
working as a toy model for quasicrystalline pre-spacetime code.
Foundations 2021, 1, 155–168. https://doi.org/10.3390/foundations1020011
https://www.mdpi.com/journal/foundations
Foundations 2021, 1
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Quasicrystals come with a natural non-local structure called an empire [29–32], which
can be considered as their defining property. The geometrical state sum model (GSS)
proposed will make use of the concept of empire overlaps (called hits in this paper) built
upon the rules driving the dynamics, which we will discuss in Section 4. One element that
implements the PEL is that an empire position can save resources in the GSS evolution.
A simple implementation of GSS is given in Section 4.1 with a new kind of cellular
automaton game of life, where different patterns emerge. Those patterns then act back
on the more general GSS model in the form of observables discussed in Section 4.2. In
Section 5, we conclude with some discussion about the notion of emergence within a GSS
model that implements the PEL.
2. Geometric Realism
The construction of general relativity (GR) marked a breakthrough for the so-called
program of geometrization of physics [18], which basically says that one should start with
geometry to understand the physical world—there should be a one-to-one correspondence
between physical quantities and geometric objects. Modern physics, starting with Einstein
himself, follows a path to apply that idea to generalize GR by going beyond Riemannian
manifolds (adding to the usual curvature variable, the torsion and nonmetricity variables).
This is in essence Riemann’s program as a unified view of geometry. Some variations of this
approach include elements from Klein’s program, which focus on symmetries and their
associated groups. However, there is another path that we will consider in light of recent
developments in theoretical physics, which brings the lesser known geometric program
view of Fedorov/Delone that can address both local and global geometry when considering
discrete systems [21]. For example, a problem that can be easily solved by this program is
which shapes tile space and how. The idea is that regular point systems are determined by
local settings. Global regularity results from the structure of local configurations.
Consider a clear route to the quantization of GR that is given by loop quantum gravity
(LQG) [33], where the classical 4-dimensional manifold is foliated in 3-dimensional (3D)
spacelike surfaces and the metric field gµν is decomposed in terms of connections and
tetrads, which, in the 3D foliation, reduces to 3D-connection and the triad field. Then the
connection and the triad are promoted to operators in a Hilbert space. This quantization
procedure leads to the result that the main kinematic objects are spin network states
spanning the Hilbert space. The dynamics can be achieved by the usual path integral
procedure and leads to spin foam transition amplitudes. These objects can be described
graph-theoretically with SU(2)-spin quantum numbers labeling the edges of the graphs
and another group or algebraic SU(2) data at the vertices. The spin foam path integral
of quantum gravity can be understood as a sum over spin network states. In summary,
the main object after the path integral quantization of GR is a sum over states of the
quantum geometry. In fact, this formulation is a cornerstone of modern physics. Making
use of the similarity of the path integral in quantum field theory and the partition function
in statistical mechanics [34], many concrete computations of physical observable with
different models, from Ising models to lattice gauge theory, condensed matter systems [35],
and quantum gravity, is done with a state sum over discrete lattices or graphs. So, the other
path to geometrization of physics is to address the geometry of state sum models at the
quantum or statistical mechanics regime.
Geometric realism will dictate for us that the labels, for example a spin state ±1, that
appear in those graphs, quasicrystals or lattices, must have a one-to-one correspondence
with the underlying discrete geometry, in our case, the quasicrystal one. That is, the labels
are directly related to the geometric building block of tilings. To be more concrete, let us
consider the state sum object
W4(sb) = N4∑
s
∏
e
Ae(s)∏
v
Av(s),
(1)
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In spin foam models or some lattice gauge models, W4(sb) is called the quantum
transition amplitude, or the partition function Z4 in some Ising like models where one
sum also over the boundary states sb. In Equation (1), N4 is a normalization constant
that depends on the discretization 4. There is weight or amplitude (Weight if W4 is
considered as a partition function or quantum amplitudes in a path integral picture.) Ae
for each edge of 4 and weight or amplitude Av for each vertex. The sum goes over
all the allowed configurations of states s. Usually Ae and Av are built from the group
and algebraic theoretic implementation of symmetries involved in the specific problem.
Geometric realism requires that Ae and Av are built from geometry of4.
Let us consider some examples of standard state sum models:
•
Ising models
The Ising model, constructed over a lattice4 can be described using only weights Ae
given by
Ae = e
βgs(e)g
−1
t(e)
(2)
with the sum over spin states s = ±1, gs(e) representing the spin at the starting vertex
and gt(e) representing the spin at the end vertex of every edge e. The β coupling constant
is proportional to the inverse of temperature. General Ising models can be achieved
by allowing the states, s, to take values in a large range of integers 0, 1, ..., n. Further
generalization can be considered by allowing the edge weights to locally vary as a general
function of the states gs(e)g
−1
t(e), Ae = f (gs(e)g
−1
t(e)).
•
Lattice gauge theory (LGT):
LGT gives a non-perturbative formulation of the path integral quantization of gauge
theories such as the standard model of particle physics. With LGT one gives up on Lorentz
symmetry and works with gauge symmetries at the vertices of the lattice 4. Gauge
invariant quantities are given by Wilson loops made up of edges around a two dimensional
face of4, g f = ∏e∈ f ge. For continuous groups of symmetry, the sum in Equation (1) is
converted on an integral that goes over the infinite of group elements (gauge) symmetry ge
and the products of amplitudes on edges and vertices are converted to a product over faces
A f = f (h f )
(3)
where the amplitude functions A f are class functions on the group of symmetry ( f (ghg−1) =
f (h)). The explicit form of the function f depends on the specific gauge symmetry model.
For example, for Yang-Mills theory, is given by
A f = e
β∑ f R(tr(U(h f )))
(4)
where U is a unitary finite dimensional matrix representation of the group, β here, is a
coupling constant. Another example is given by topological models where A f takes the
simpler form
A f = δ(h f )
(5)
where the delta function δ is taken with respect to the group measure.
•
Spin foam
In spin foam models, 4 is a triangulation of spacetime manifold or its dual. Spin foam
models also have amplitudes associated with faces (or edges), usually give by Equation (5),
but also amplitudes Av associated to vertices and constructed from more complicated
group invariant objects. In LQG, the group of symmetry is given by the little group (SU(2))
of spacetime symmetries (SL(2, C)). A generalization of spin foam for quantum gravity is to
include color charge symmetry (SU(3)), within the weights Ae and Av in Equation (1). This
can be done by defining the weights from invariants of SU(2)× SU(3) [17]. When W4(sb)
depends only on the boundary states, the spin foam is topological and the amplitudes are
constructed from topological invariants [16].
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State sum models are built from algebraic and group-theoretic elements. Even the
simple Ising model can be understood as a discrete Z2 model. The underlying group and
algebraic structures have a geometric correspondence. Geometric realism demands the
values on the GSS to come from the geometry, more concretely from tiling of4, which, for
the specific model presented in next sections, will be a 3D quasicrystal. The states, coupling
constants, potentials, weights and amplitudes themselves, will be determined by geometric
objects as lengths, volumes and volumetric intersections.
3. Kinematics: The 3D Quasicrystal, Empire and Hits
The construction of the quasicrystal of interest here, the 3DPT, will make use of the
canonical cut-and-project method [12,21]. To construct the 3DPT, 4, we consider the
canonical hypercubic lattice Z6 in Euclidean space R6. Let ε be an irrational 3-dimensional
subspace of R6 and ε⊥ be its orthogonal complement. Let P be the orthogonal projector
onto ε and P⊥ onto ε⊥. Now we fix a compact subset K of ε⊥, called the cut-window. The
canonical choice for the cut-window is the projection of the Voronoi cell of Z6 to ε⊥. The
Voronoi cell contains one lattice point, which lies at its center. Unlike the unit cells of lattices,
these cells are unique and their symmetry groups are the stabilizer groups of the lattice
points. The 3DPT quasicrystal 4 is constructed by projecting points λ ∈ Z6 to ε, P(λ),
such that P⊥(λ) lies inside K, the acceptance domain. Z6 lattice points are connected by
the unit length edges. If two points λ1 and λ2 are connected in Z6 and P⊥(λ1) and P⊥(λ2)
are accepted, then P(λ1) and P(λ2) are connected in 4. A vertex vi can have different
numbers of neighbors, here denoted vij, with j variating from 1 to the valence of vi (The
possible valences for any vertex at some 3DPT are: 4, 6, 8, 10, 12, 14, 16, 20.) The geometric
lengths of the connections are labeled as lij. A tiling T of 4 is a set of possible points
and connections given by this procedure. Different tiling configurations can be generated
by doing a shift on P⊥(λ) in ε⊥ before checking if P⊥(λ) lies inside K, so-called γ⊥. The
shift γ⊥ can be used to generate different tilings and it is a continuous parameter that can
be used to make the quasicrystal dynamic. A specific 3D tiling T of4 has two different
rhombohedral prototiles building blocks with 10 orientations each. Each vertex vi of one T
can be associated to different configurations of prototiles (up to 20 rhombohedral prototile
around one vertex vi). There are 24 possible different vertex types (VT), which appears
with different frequencies and valence in a tiling. Most of the 24 VT have valence 20, but
those with lower valence appear with more frequency in a tiling. See [12] for the explicit
form of VT and its frequencies. A small tiling and 3 of the 3DPT VTs are shown in Figure 1.
Figure 1. 3DPT tiling and some VTs.
The empire is an important property of quasicrystals that arises, in context, within the
empire problem [29–32]. A quasicrystal itself arises in context of the the ancient problem
of tiling space in an aperiodic way. To clarify the use of the 3DPT, we can think of 4
generated from K as a possibility point (tiling) space (PS) where points (or VT) can be
turned ON or OFF. Or we can think of4 as the selected points in Z6 but not projected yet.
We are interested in project subsets of those points, the empires, which we define below.
The empire problem asks what other vertices or VT of a quasicrystal tiling is forced to be
ON if a specific vertex or VT is chosen to be ON—it is projected or actualized. By defining
the empire problem and its solutions, one sets the kinematics of a given quasicrystal. For
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a specific vertex vi in4 there is associated a specific VT (from the 24 possible ones). We
can define a window E ⊂ K associated to any VT given at some vertex vi, and so, the
empire of a point P(λi) of4will be the sub-set of4 such that P⊥(λ) lies also inside E. The
cut-window K can be volumetric partitioned in sub-windows E, which can overlap. The
dynamic quasicrystal can be generated by projecting empires. Empires capture the non-
local aspect of quasicrystals [36] in the sense that when a vertex vi is ON, its empire—the
whole set of points defined by window Ei—is also ON. An additional question can be asked
now: If two vertices v1 and v2 and their empires are projected, what is the empire overlap
between them? The answer is that a measure of the overlap of points sets of different
empires is given geometrically by the overlap of empire windows E1 and E2 inside the
cut-window K. We call this overlap the hit H12. Consider the empire given by E1 to be
4E1 ⊂ 4 and for E2 to be4E2 ⊂ 4, then H12 = 4E1 ∩4E2 or in terms of vertex window
polytope intersection for any vi and vj we can compute a normalized measure of overlap by
Hij =
IVol(E1, E2)
IVol(E1)
(6)
where IVol is a function that gives the volume of the intersection of different polytopes,
computed here numerically, and if it has only one polytope as input, it returns the volume
of that polytope. Computing hits Hij between a VT at vi and different vj positions in one
tiling and then changing vi gives a hit map distribution for the specific tiling. One typical
example is shown in Figure 2. In what follows we will consider only nearest neighbors Hij.
Figure 2. A typical hit map for the 3DPT quasicrystal. We consider a list with 1000 points of a 3DPT
tiling and compute vertex window polytope overlap between them, Equation (6).
With the main construction of the 3DPT presented, we can establish that the main
kinematics variables of interest are the hits Hij between vi and vj, the length lij of the
connection between vi and vj and the volume Vi of the VT polytope associated with vi.
We can turn now to the implementation of the dynamics aspects of the GSS W4(sb),
Equation (1), to be done in the next section.
4. Dynamics: Geometric State Sum Model and the PEL
To implement dynamics of the GSS, we consider the state sum in Equation (1). Follow-
ing the ideas from the previous sections, the weights Ae for an edge linking vertices vi and
vj will be given by
Ae = lijHij,
(7)
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where the states given at each edge sij = Hij are geometric quantities, specifically, volumet-
ric polytope intersections. The length lij plays the role of the coupling constant or inverse
of temperature. We can group these edge states by summing over nearest neighbors vij
Avi = ∑
j
lijHij.
(8)
We will add an additional term to each vertex vi, called the hit potential Yi, which
takes into account the PEL by implementing a look-ahead algorithm [37]. Consider a PS
tiling T with only the central vertex vc and its empire being ON, which means that the
vertex window polytope Evc is being used to select the possible points of the Z6 that can be
projected to the cut-window K. Now we will probe the quasicrystal possibility space4
with random walks of the vertex type at vc using rules based on Hij. So, we start on step 1,
with vc and its empire being ON. In step 2, one of the neighbors vcj will be ON according to
a non-deterministic rule R(Hij), which we will discuss in Section 4.2. Then we repeat these
steps until step N. This defines one animation A1. Next, we repeat this procedure, getting
a new animation A2, and so on until an animation AM, so that we end with M animations,
each with N steps. We call these animations possibility space random walks (PRW). The
hits potential Yi is defined at each vertex vi as the number of PRWs that use that position.
For consistency we will add a volume weight
Yi = ViYi,
(9)
where Vi is the volume of the VT polytope at vi. It encodes the coupling lij there. The
hit potential comes from the idea of minimizing the cost of projection resources to turn
ON points on4. The vertices that have more PRWs going over them have more potential
to save projection steps to generate animations for the look-ahead algorithm (or here:
Look-savings-ahead algorithm). As there are more walks going over those positions they
can be part of more possible emergent patterns. The hit potential can be defined from
weights of entire animations. We can count hits and associate an integer to one animation
Am by counting how many empire vertices the PRW of that animation encounters. These
vertices are already ON and don’t need to be turned ON on the PRW. The hits Hij and hit
potential Yi associated to a tiling Tk define one valid configuration of states given by
WTk = NTk ∏
i
AkviY
k
i ,
(10)
where the index k means that the geometric quantities are computed on the specific tiling
Tk. The defining object of interest for the GSS is then, in a partition function form,
W4 = N4∑
Tk
WTk ,
(11)
where the sum goes over the different allowed configurations, implemented by finding
new allowed tilings. One way to find new tilings, which involves changing only the states
Hij and not the couplings lij or Vi, is by using the shift γ⊥. To change the couplings lij or Vi
from quasicrystalline first principles, one can use inflation [27], which requires changing
the size of the windows. The state sum can also be defined as a transition function for state
Hab that fixes a subset of4, where Hab can be two disjointed subsets so that maintaining
both fixed, and summing over the remaining part, would give the transition function
between the states
W4(Hab) = N4 ∑
Tk(ij/ab)
WTk ,
(12)
where the notation (ij/ab) indicates that all tilings and their accompanying Hij are gener-
ated, but with Hab remaining fixed. This implementation is given by finding values of the
shift γ⊥ that generate new tilings sharing a fixed configuration.
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4.1. A New Kind of Game of Life in Quasicrystals
A concrete implementation of a dynamic GSS Equation (12) is given in the form of a
cellular automaton game of life (GoL). In this case we let the quasicrystal tiling space4
evolve according to local rules, which the same nature of R(Hij) we use to generate the hit
potential and which we discuss below.
Classical cellular automata are defined on regular lattices. The rules depend on the
state of each site and its neighbors. The neighborhood structure looks the same across the
lattice. For GoLs [38–42], which have outer totalistic cellular automaton rules, the next state
of a site depends only on its current state, and the total number of neighbor sites in certain
states. In 2-dimensional quasicrystalline GoLs [43–45] the neighborhoods are generalized
and not the same for each site, but the dynamics are implemented with similar rules to the
original GoL. GoL rules on Penrose tilings still have complex behavior.
The generalization to 3D that implements the GSS model is given by a 5-tuplet
G = (4, T, H, v,R) with elements as follows. 4 is a 3DPT; T includes the initial tiling
condition (where at least one VT is ON) and the set of steps to update 4—we usually
consider 1000 steps for 5000 3DPT point set; H is the set of states generalized to be a real
number between 0 and 1 according to Equation (6); v = vij are the neighbors of a vertex vi,
which vary between 4 and 20—we note that when a vertex is ON the whole VT associated
to it is considered to be ON, and also that sometimes we can consider the point set for
evolution to have only a specific VT instead of the full quasicrystal point set;R are the new
local adapted rules for this kind of GoL. The rules measure empire overlap: If there is too
much overlap with ON neighbor VTs then it turns or stays OFF (overpopulation condition);
if there is too little overlap it also will be OFF (under-population condition); but if there
is the right value of overlap with the mean value of all vij, then the current vertex will be
ON. Good values for the normalized Avi , Equation (8), are found to be between 0.7 and
0.9. We also consider the information entropy associated to hits to drive the evolution as in
Equation (14)—in this case the VT will be ON if its information entropy is close enough of
the mean of its neighbors, which are ON (one additional option with this kind of GoL is
that due to the non-local properties of empires we can allow the rules to be applied to the
whole quasicrystal and not only with the connected vertices of a VT, which we will leave
for future investigations).
These dynamics lead to different emergent patterns, most of which are oscillation
patterns as in Figure 3. There are also different kinds of propagation as in Figure 4. However,
varying the rules opens up possibilities of more complex dynamics, which are open for
future systematic investigations, but which may also encounter problems of optimization
of image processing.
Figure 3. One 3DPT GoL oscillation pattern. The pattern is a oscillator period 2 and the two frames
are shown.
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Figure 4. A recurrent pattern propagation for a 3DPT GoL made mainly from the VTs from Figure 3.
(1–8) show eight frames of the dynamics where the pattern highlighted moves from left to right.
4.2. GSS Observables and Emergence
In light of more general dynamics, we can interpret Equation (12) as one observable on
a GSS model. It is a fixed pattern over the space of geometric states. The specific emergent
pattern and its properties can be addressed with Equation (12). The hit potential Yi can,
in fact, be considered as derived from one observable VT that is ON and that is following
a family of PRWs. Let us consider the rules R(Hij) in this context. The rules are used to
guide the random walks to probe the possibility space and then define the hit potential.
What is moving in the quasicrystal, or, what are those VT, which are being turned ON or
OFF (being projected/actualized or not)? A GSS model aims to describe pre-spacetime
physics—the Planck scale quantum gravity regime. This regime is considered to have the
concept of holographic matter [46–49], which is proportional to information entropy
|Ivi − Ivj | = 4Iij = αm,
(13)
with α a constant of proportionality and m the mass crossing some horizon. In this Planck
scale regime picture, each connection is considered to be crossing a holographic horizon
and information is the stuff flowing between horizons and to be conserved. The local
information entropy is given by
Ii = −∑
j
PijlogPij
(14)
with Pij =
Hij
Ni . Different rules can be implemented under I. The above motivation leads
us to the notion of local conservation of I. We simply let the PRW, starting at the center of
a tiling, be guided by a probability distribution constructed by choosing each successive
position from a local subset whose values of I are within some selected range of the current
value of I, which sets the non deterministic look-saving-ahead algorithm. The resultant
hit potential Y(r) as a function of the distance from the center is given in Figure 5 for
1000 animations over 1000 steps.
The specific form presented in Figure 5 has a mean value that drops with the inverse
of distance away from the center but with a Gaussian contribution close to the center. In
general its form depends on the distribution of VTs on a specific tiling, the local rules and
random walk properties. We can also consider an additional second PRW starting at a
different position and allow the Y(r) to count overlap between the two walks—a synergistic
effect [50] on the whole emergent hit potential not accounted for on the underlying GoL.
See Figure 6.
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Figure 5. PRW hit potential.
Figure 6. PRW hit potential from two patterns evolution starting at different positions with one
at origin.
Note that we can use the PRWs to evolve4 determining H, v and evenR of the GoL
G. We can also allow local rules at site i to be determined from the full weights AviYi. This
generates a more sophisticated stratified recursive game
Gn+1 = G(Gn).
(15)
As a result of the GSS evolution, we consider a candidate for an order parameter,
which is analogous to magnetization in spin systems,
HT (l) =
1
NT ∑
i
1
Ni ∑
j
Hij(lij),
(16)
where l represents the length inflation scale that defines lij. As the cut-window K gets
smaller, the points in the projected space P(λ) get farther way from each other and in the
perpendicular space the points P⊥(λ) get closer. lij and Vi get bigger but the intersection
of vertex window polytopes in the perpendicular space also gets bigger, approaching 1.
As a result, over many inflations, we see that there are two dominant regimes. One is
“disordered”, where there is not much overlap of vertex window polytopes and there are
more local variations so that the information entropy rules depend on hits. The other
regime is “ordered”, where the overlaps approach 1 and therefore the information entropy
rules should depend only on the vertex valences and not on hits. See Figure 7.
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Figure 7. Average hits HT (l) evolution under inflations.
Different emergent dynamics can be considered. For example, we can set a preferred
direction to follow the rules and also consider the central hit potential, which leads to
interesting curving path patterns as in Figure 8.
Figure 8. Evolution considering local information entropy conservation, initial preferred direction
and the central hit potential.
As a further result, we consider an oscillation pattern around the center of a tiling with
the large symmetric VT there. There are 12 of these same VTs around the center making the
geometry of an icosahedron. In each frame step only one of the 12 VTs is ON. This gives a
notion of an emergent quasiparticle (the whole icosahedron) with internal structure (the
12 symmetric VTs). See Figure 9.
Figure 9. Central icosahedron oscillating pattern.
With this object we can consider a 4-dimensional dynamic by considering the influence
of the empires of those 12 VTs at arbitrary 3D positions on4. The oscillation on the 12 VTs
plays the role of time. We consider a vertex P(λ1) around the icosahedron. The VT
associated to that position depends on the γ⊥ tiling Tk, VTk1 = VT(P(λ1)). By shifting the
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projections by specific γ⊥ values, we can change tiling k and VTk1 , preserving the oscillating
icosahedron pattern at the center. Figure 10 shows the cut-window K (the large window),
the empire vertex window for the icosahedron (the middle window), and the small window
limiting the available shifts.
Figure 10. Available points for shifts γ⊥ presented inside the small window.
So for each P(λ1) we compute Hij, where i refers to one of the 12 central VTs and j
to VTk1 . We average this computation over a large number of tilings Tk and over 12 cycles
of the central icosahedron, gettingH̄ij, and then we go to a new position. We define the
observable of interest to be
O(lij) = log(lijH̄ij),
(17)
which gives a different notion of emergent potential as shown in Figure 11. This one grows
with distance while the hit potential drops.
Figure 11. Potential due to composite quasiparticle oscillating at the center of the tiling.
As a last result, we consider how information entropy grows with the number of steps
for patterns made of different VTs. We define the hit section (hs) to be a certain number of
the same VTs within some distance r from the center of some tiling. We compute Hij, where
i refers to a certain VT at the center of this tiling and j refers to the same VT at different
position p on that tiling, p < r, with j ∈ hs,
Ii(r) = −
hs(r)
∑
j
PijlogPij.
(18)
The result is that Ii(r) distinguishes the patterns made of different VTs, which should
be proportional to the frequency of appearance of the respective VTs, see Figure 12.
Foundations 2021, 1
166
Figure 12. Information entropy order parameter for different VTn, where the integer n number the
3DPT VTs and we show only 5 of the 24. hs grows different with distance for the different VTs.
5. Discussions and Outlook
In this paper, we discussed state sum models under geometric realism. The SSH
paradigm is looking to understand the emergence of spacetime and matter from a pre-
spacetime code that takes into account stratification and recursion. GSS is discussed as a
framework in this direction. State sum models implement the principles of locality and
superposition. GSS adds geometric realism at the state level and the PEL for dynamics.
With the GSS formulation, we made concrete the idea that emergence happens due to
structure and not necessarily randomness. In the current implementation, the structure is
given by a 3D quasicrystal with aperiodic order projected from a 6-dimensional lattice. By
having dynamics where different points can be projected or actualized at different steps
with their empires, quasiparticle patterns emerge in a simple GoL simulation as presented
in Section 4.1. The patterns that emerge are built from known quasicrystal structures, the
VTs. A stratified recursive feedback loop can then be established between, on the one
hand, the quasicrystal level of projections, non-local empire overlaps and local information
entropy rules and, on the other hand, the emergent level of quasiparticle patterns made
of VTs. This is expressed as a potential derived from many evolutions of PRWs going
over the underlying quasicrystal points, generating an emergent pattern. Overlap of PRW
positions or PRWs and empire vertices is understood as potential optimization; it gives an
opportunity to economize resources, which in this case are projections from 6 dimensions
to 3 dimensions. This implements the PEL by allowing the simulation to express more
emergent patterns over less projection step resources. The PEL is implemented on the
structure of PRWs, which are considered here as virtual walks to probe the possibility
space of walks. It aims to create “cognitive” structure, as in artificial neural networks, so
that the simulation itself can decide the best sequence of movements for the emergent
pattern. This kind of structure under the emergence paradigm within complex systems
is what is observed in general: The emergent structure, with its new emergent properties,
always occurs under the dynamics of a large underlying number of building blocks that
have structure governing their dynamic interactions. This is seen from atoms to DNA to
neurons to stars and galaxies.
In this work, we investigated some general code theoretic properties of general emer-
gent patterns on the GSS. For future investigations, we aim to derive the physical emergent
laws governing the dynamics of the emergent patterns, making more concrete the emer-
gence of spacetime and matter. An important hint in this direction is that the cut-and-project
method used to derive the quasicrystal makes clear the connection with the higher di-
mensional lattices. The main quasicrystals of interest are derived from the so-called root
Foundations 2021, 1
167
lattices of Lie algebras. Lie algebras and groups are the language of symmetry and so a
core element in fundamental physics, as in quantum mechanics. The dynamics derived
in this paper can be thought of as a dynamics that tails back to the root system of Lie
algebras [17,27], suggesting a generalization from Z6 to D6 and E8.
One different consideration to be addressed for GSS models is the computational
efficiency on current classical computers. The values on usual state sum models come
from algebraic or group-theoretic objects. There are more computational costs to obtain the
values from the geometry.
Author Contributions: Conceptualization, M.A., F.F. and K.I.; methodology, M.A.; software, M.A.
and D.H.; validation, F.F.; formal analysis, M.A.; investigation, M.A., F.F. and D.H.; writing—original
draft preparation, M.A.; writing—review and editing, M.A.; visualization, F.F.; supervision, M.A. and
K.I.; project administration, K.I.; funding acquisition, K.I. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments: We acknowledge the many discussions had with David Chester, Raymond As-
chheim and Richard Clawson and we thank them for their generous feedback in editing discussions.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
SSH
Self-simulation hypothesis
PEL
Principle of Efficient Language
3DPT
3-Dimensional Penrose Tiling quasicrystal
PEL
Geometrical State Sum (GSS)
GR
General Relativity
LQG
Loop Quantum Gravity
3D
3-dimensional
LGT
Lattice Gauge Theory
VT
Vertex Type
PS
Possibility Space
PRW
Possibility Random Walk
GoL
Game of Life
References
1.
Irwin, K.; Amaral, M.; Chester, D. The Self-Simulation Hypothesis Interpretation of Quantum Mechanics. Entropy 2020, 22, 247.
[CrossRef]
2.
Dirac, P.A.M. Classical theory of radiating electrons. Proc. Roy. Soc. Lond. A 1938, 167, 148–169.
3.
Finkelstein, D. Space-time code. Phys. Rev. 1969, 184, 1261–1279. [CrossRef]
4.
Wheeler, J.A. Beyond the Black Hole. In Some Strangeness in the Proportion: A Centennial Symposium to Celebrate the Achievements of
Albert Einstein; Woolf, H., Ed.; Addison-Wesley: Reading, MA, USA, 1980.
5.
Wheeler, J.A. Hermann Weyl and the Unity of Knowledge. Am. Sci. 1986, 74, 366–375.
6.
Wheeler, J.A. Information, physics, quantum: The search for links. In Complexity, Entropy, and the Physics of Information;
Addison-Wesley: Boston, MA, USA, 1990.
7.
Langan, C.M. The Cognitive-Theoretic Model of the Universe: A New Kind of Reality Theory. Prog. Complex. Inf. Des. 2002, 1,
2–3.
8.
Aschheim, R. Hacking Reality Code. FQXI Essay Contest 2011, Category: Is Reality Digital or Analog? Essay Contest (2010–2011),
Number 929. Available online: https://fqxi.org/community/forum/category/31417 (accessed on 8 October 2021).
9.
Irwin, K. A New Approach to the Hard Problem of Consciousness: A Quasicrystalline Language of Primitive Units of
Consciousness in Quantized Spacetime. J. Conscious Explor. Res. 2014, 5, 483–497. [CrossRef]
10.
Irwin, K. The Code-Theoretic Axiom: The Third Ontology. Rep. Adv. Phys. Sci. 2019, 3, 1950002. [CrossRef]
11.
Irwin, K.; Amaral, M.M.; Aschleim, R.; Fang, F. Quantum walk on spin network and the golden ratio as the fundamental constant
of nature. In Proceedings of the Fourth International Conference on the Nature and Ontology of Spacetime, Varna, Bulgaria, 30
May–2 June 2016; pp. 117–160.
12. Hammock, D.; Fang, F.; Irwin, K. Quasicrystal Tilings in Three Dimensions and Their Empires. Crystals 2018, 8, 370. [CrossRef]
Foundations 2021, 1
168
13. Katz, A. Theory of Matching Rules for the 3-Dimensional Penrose Tilings. Commun. Math. Phys. 1988, 118, 263–288. [CrossRef]
14.
Takakura, H.; Gomez, C.; Yamamoto, A.; De Boissieu, M.; Tsai, A.P. Atomic structure of the binary icosahedral Yb-Cd quasicrystal.
Nat. Mater 2007, 6, 58–63. [CrossRef]
15.
Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 1974, 10, 266–271.
16.
Barrett, J.W. State sum models for quantum gravity. arXiv 2000, arXiv:gr-qc/0010050.
17. Amaral, M.; Aschheim, R.; Irwin, K. Quantum Gravity at the Fifth Root of Unity. arXiv 2019, arXiv:1903.10851.
18. Wanas, M.I. The accelerating expansion of the universe and torsion energy. Int. J.Mod. Phys. A 2007, 31, 5709–5716. [CrossRef]
19.
Irwin, K. Toward the Unification of Physics and Number Theory. Rep. Adv. Phys. Sci. 2019, 3, 1950003. [CrossRef]
20.
Baake, M.; Grimm, U. Aperiodic Order; Cambridge University Press: Cambridge, UK, 2013.
21.
Senechal, M.J. Quasicrystals and Geometry; Cambrige University Press: Cambrige, UK, 1995.
22.
Levine, D.; Steinhardt, P.J.; Quasicrystals, I. Definition and structure. Phys. Rev. B 1986, 34, 596. [CrossRef]
23. Gardner, M. Extraordinary nonperiodic tiling that enriches the theory of tiles. Sci. Am. 1977, 236, 110. [CrossRef]
24.
de Bruijn, N.G. Algebraic theory of Penrose’s non-periodic tilings. Nederl Akad Wetensch Proc. 1981, 84, 1–7. [CrossRef]
25. Grimm, U. Aperiodicity and Disorder-Do They Play a Role? In Computational Statistical Physics: From Billiards to Monte Carlo;
Hoffmann, K.H., Schreiber, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2002; pp. 191–210.
26.
Elser, V.; Sloane, N.J.A. A highly symmetric four-dimensional quasicrystal. J. Phys. A 1987, 20, 6161–6168. [CrossRef]
27. Chen, L.; Moody, R.V.; Patera, J. Non-crystallographic root systems. In Quasicrystals and Discrete Geometry; Fields Institute
Monographs; American Mathematical Society: Providence, RI, USA, 1998; Volume 10.
28.
Fang, F.; Irwin, K. An Icosahedral Quasicrystal and E8 derived quasicrystals. arXiv 2016, arXiv:1511.07786.
29. Conway, J.H. Triangle tessellations of the plane. Amer. Math. Mon. 1965, 72, 915.
30. Grunbaum, B.; Shephard, G.C. Tilings and Patterns; W. H. Freeman and Company: New York, NY, USA, 1987.
31.
Effinger-Dean, L. The Empire Problem in Penrose Tilings. Bachelor’s Thesis, Williams College, Williamstown, MA, USA, 2006.
32.
Fang, F.; Hammock, D.; Irwin, K. Methods for Calculating Empires in Quasicrystals. Crystals 2017, 7, 304. [CrossRef]
33. Rovelli, C.; Vidotto, F. Covariant Loop Quantum Gravity, 1st ed.; Cambridge University Press: Cambridge, UK, 2014.
34. Ortiz, L.; Amaral, M.; Irwin, K. Aspects of aperiodicity and randomness in theoretical physics. arXiv 2020, arXiv:2003.07282.
35.
Bahr, B.; Dittrich, B.; Ryan, J.P. Spin foam models with finite groups. J. Grav. 2013, 2013, 549824. [CrossRef]
36.
Fang, F.; Paduroiu, S.; Hammock, D.; Irwin, K. Empires: The Nonlocal Properties of Quasicrystals. In Electron Crystallography,
Devinder Singh and Simona Condurache-Bota; IntechOpen: London, UK, 2019.
37. Alexander, S.; Cunningham, W.J.; Lanier, J.; Smolin, L.; Stanojevic, S.; Toomey, M.W.; Wecker, D. The Autodidactic Universe. arXiv
2021, arXiv:2104.03902.
38. Gardner, M. Mathematical games—The fantastic combinations of John Conway’s new solitaire game of ‘life’. Sci. Am. 1970, 223,
120–123. [CrossRef]
39. Klein, J. Breve: A 3d environment for the simulation of decentralized systems and artificial life. In Proceedings of the 8th
International Conference on Artificial Life, Dortmund, Germany, 14–17 September 2003.
40.
Bak, P.; Chen, K.; Creutz, M. Self-organized criticality in the Game of Life. Nature 1989, 342, 780. [CrossRef]
41.
Bays, C. Candidates for the game of life in three dimensions. Complex Syst. 1987, 1, 373–400.
42. Couclelis, H. Cellular worlds: A framework for modeling micro—macro dynamics. Environ. Plan. A 1985, 17, 585–596. [CrossRef]
43.
Bailey, D.A.; Lindsey, K.A. Game of Life on Penrose Tilings. arXiv 2017, arXiv:1708.09301.
44. Owens, N.; Stepney, S. Investigations of Game of Life Cellular Automata Rules on Penrose Tilings: Lifetime, Ash, and Oscillator
Statistics. J. Cell. Autom. 2010, 5, 207–225.
45.
Fang, F.; Paduroiu, S.; Hammock, D.; Irwin, K. Non-Local Game of Life in 2D Quasicrystals. Crystals 2018, 8, 416. [CrossRef]
46.
Bekenstein, J.D. Black holes and entropy. Phys. Rev. D 1973, 7, 2333–2346. [CrossRef]
47. Verlinde, E. On the origin of gravity and the laws of Newton. J. High Energ. Phys. 2011, 2011, 29. [CrossRef]
48. García-Islas, J.M. Entropic Motion in Loop Quantum Gravity. Can. J. Phys. 2016, 94, 569–573. [CrossRef]
49. Amaral, M.M.; Aschheim, R.; Irwin, K. Quantum walk on a spin network. arXiv 2016, arXiv:1602.07653.
50. Goertzel, B. Toward a Formal Model of Cognitive Synergy. arXiv 2018, arXiv:1703.04361.