crystals
Article
Non-Local Game of Life in 2D Quasicrystals
Fang Fang *
, Sinziana Paduroiu
, Dugan Hammock and Klee Irwin
Quantum Gravity Research, Los Angeles, CA 90290, USA; Sinziana@QuantumGravityResearch.org (S.P.);
Dugan@QuantumGravityResearch.org (D.H.); Klee@QuantumGravityResearch.org (K.I.)
* Correspondence: Fang@quantumgravityresearch.org; Tel.: +1-310-574-6934
Received: 25 September 2018; Accepted: 31 October 2018; Published: 6 November 2018






Abstract: On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game
of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any
local patch, the emperor, forces the existence of a large number of tiles at all distances, the empires.
Considering the emperor and its local patch as a quasiparticle, in this case a glider, its empire
represents its field and the interaction between quasiparticles can be modeled as the interaction
between their empires. Following a set of rules, we model the walk of life in different setups and we
present examples of self-interaction and two-particle interactions in several scenarios. This dynamic is
influenced by both higher dimensional representations and local choice of hinge variables. We discuss
our results in the broader context of particle physics and quantum field theory, as a first step in
building a geometrical model that bridges together higher dimensional representations, quasicrystals
and fundamental particles interactions.
Keywords: non-local; game of life; quasicrystals; empires; forced tiles; possibility space; cut-and-project;
particle self-interaction; particle interaction; higher dimensions
1. Introduction
First introduced by Conway [1], the ‘game of life’, a cellular automaton algorithm proposed
to simulate real life patterns with simple rules, quickly drew the attention of scientists in various
fields [2–5]. The ‘game of life’ variations explored so far have employed a periodic grid [6] or have
used only local rules [7,8]. For the first time, we simulate a game of life following non-local rules
on an aperiodic grid, the Penrose tiling [9], a two-dimensional quasicrystal (QC). The existence of
quasicrystals was first considered ruled out because of their forbidden symmetry, before Kleinert in
1981 [10] and Levine and Steinhardt in 1984 [11] predicted and theorized them. Shechtman discovered
them in 1982 and published his results two years later [12]. The special properties of quasicrystals,
their aperiodic order and their non-local nature dictated from higher dimensions [13], make them an
interesting research subject not just in the field of material science, but in many other fields, including
more recently quantum computing. They are thus an interesting candidate for modeling game of life
algorithms. The distribution and frequency of the vertex configurations or local patches are strictly
governed by the higher dimensional ‘mother lattice’ and the manner in which the quasicrystal is
generated from it. Each vertex configuration or local patch has an empire [14–17], a feature unique to
quasicrystals representing the totality of all the tiles whose existence and positions are forced by the
local patch. Therefore, the local patches propagate under the influence of the empire fields.
Respecting the inherent aperiodic order of quasicrystals and their non-local nature, we construct
a non-local game of life simulation on Penrose tiling, a well-known 2D quasicrystal. In this paper we
study one of the eight possible vertex configurations of the Penrose tiling, the K vertex type (VT). For
a certain vertex of this type, we consider the vertex patch formed by the tiles surrounding the vertex.
We treat the configuration formed by the vertex patch together with the tiles belonging to its local
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empire as a quasiparticle. We then simulate and track the evolution of this configuration, dynamically,
in several scenarios, finding patterns like rotations, quasi-translations and oscillations. These patterns
are dictated by the interaction of the quasiparticle with its own empire field, a self-interaction.
Furthermore, we explore the behavior of two such configurations as interactions between two
quasiparticles and their non-local fields. Our results are discussed in the context of higher dimensional
symmetries, which could prove to be crucial in describing by first principles the particles observed
in nature.
We review the important properties of quasicrystals and the methods to generate them in Section 2.
In Section 3 we describe the method used to simulate a non-local game of life on Penrose tiling.
The results for both a quasiparticle’s self-interaction and a two-particle interaction are shown in
Section 4. Finally, we discuss in a broader context the implications of our results and we present our
conclusions in Section 5.
2. Properties of Quasicrystals
The forbidden symmetry of quasicrystals comes from the symmetry breaking from a higher
dimensional lattice, through projection. For example, five-fold symmetry is forbidden in 2D crystals,
but it exists in projection of the Z5 lattice. Penrose tiling can be obtained through projecting a layer of
the Z5 lattice along the five-fold axis, to two dimensions [18,19]. Although Penrose tiling does not have
the perfect periodic order that crystals possess, it has aperiodic order that is non-local and repetitive,
in other words, given any local patch of the tiling of any size, one could find an infinite number of
identical patches in a full (infinite) Penrose tiling. The following properties of quasicrystals make them
extremely interesting candidates for a game of life simulation:
•
Each local configuration type is distributed uniformly throughout the whole quasicrystal. This
matches the fact that particles tend to be distributed uniformly in a ground state or at very low
temperature when there is a lack of interactive forces.
•
The ratio of the frequencies of any two vertex types is a constant.
•
The empire of each local configuration is non-local. This property could prove important in
explaining the non-local interaction between particles.
For a better understanding of the aforementioned properties of quasicrystals, we will briefly
review a few important concepts.
The Cut-and-Project Method
There are several methods to generate a quasicrystal: the cut-and-project method [18–20],
the multigrid method [20] and the inflation/deflation or substitution rule method [20]. Here we will
focus only on the cut-and-project method not just because it is conceptually simple and computationally
efficient, but also because it provides better clarity on the concepts of empire and possibility space,
concepts crucial for our non-local game of life model.
The cut-and-project method for generating quasiperiodic tilings from a higher dimensional
lattice—the mother lattice—expanded to generating empires in quasicrystals is a well-developed
method [17]. We will review here some notions related to the non-local properties of quasicrystals,
namely the acceptance domain/QC window and the empire and possibility space windows using
the Z2 lattice and its 1D quasicrystal, the Fibonacci chain, as an example. Even though in this paper
we have performed the game of life on Penrose tiling projected from the Z5 lattice, for illustration
purposes we will show here the one-dimensional crystal projected from the two-dimensional lattice in
order to simplify the representation of the cut-and-project method and concepts. We review below
some key concepts used by the cut-and-project method and we give a graphic example in Figure 1 for
a simple case, the LS configuration, a vertex type in the Fibonacci chain.
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a
b
QC Window
Empire Window
Possibility Space Window
All Possible QC Windows
Quasicrystal Space E
LS
E
E⏊
E
E⏊
Voronoi Cell
LS
A Higher Dimensional Explanation
1D example
Figure 1. Cut-and-project windows in a rotated Z2 lattice used for generating a 1D quasicrystal and its
associated structures. (a) The Z2 lattice is rotated so that the quasicrystal space (E) coincides with the
horizontal line and the perpendicular space (E⊥) lies along the vertical line. The blue vertical line is the
QC window, which covers all points and square edges (unit cells of Z2 lattice) that lie between the two
dashed blue horizontal lines. All these points get projected to the horizontal space as the vertices of the
1D quasicrystal and all these edges get projected as the tiles of the quasicrystal. The red vertical line
defines the empire window for the LS vertex type, which covers all the edges between the two dashed
red horizontal lines. These edges, after being projected to the horizontal space, become the empire of
the LS vertex type. (b) The colored vertical lines give examples of possible QC windows that cover the
empire window. The union of all these QC windows gives the possibility space window, which covers
all the tiles that can possibly coexist with the LS vertex type.
The QC Window/Acceptance Domain
A one-dimensional quasicrystal can be obtained by projection from a two-dimensional lattice.
To avoid a dense set, the QC window or acceptance domain is defined in order to select a subset from
the two-dimensional lattice. Taking the Z2 lattice as an example, we denote the quasicrystal space as E
and the perpendicular space as E⊥. To select a subset that has uniform distribution, the acceptance
domain, W, is often defined as the projection of the Voronoi cell in E⊥. The blue vertical line shown in
Figure 1a is the QC window. This QC window captures all the Z2 points and edges between the two
dashed blue lines and they are projected to the quasicrystal’s space E as the points and tiles of the 1D
quasicrystal, respectively.
The Empire Window
The empire window is defined in relation to a vertex type that covers the mother lattice
representation of all the tiles forced by the given vertex type; it is the convex hull of the whole
vertex configuration in the mother lattice. Figure 1a shows the empire window of the labeled LS vertex
type, as the red vertical line. This empire window covers all the edges that when projected to E give
the forced tiles of the LS VT. These forced tiles represent the empire of the LS VT.
The Possibility Space Window
For a given vertex type, from the empire window we get the forced tiles. The tiles that are not
forced, but are allowed to coexist with a given vertex configuration are covered by the possibility space
window. By definition, the possibility space window is the union of all possible QC windows that
cover the empire window. For the LS vertex type, this is shown as the black vertical line in Figure 1b.
Each tile covered by this window is just one possible way to tile that part of E, and therefore it overlaps
with other possibilities. These tiles form (shape) the possibility space of the LS vertex type.
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Thus, both the empire (forced tiles) and the possibility space (possible tiles) are non-locally
determined by the chosen vertex patch, or the chosen local patch, making them perfect tools in
modeling the non-local interactions between quasiparticles.
3. Rules of Life
To develop a non-local game of life movement in Penrose tiling, a 2D quasicrystal that can be
cut-and-projected from a Z5 lattice, we consider the movement of the smallest local patches, which
consist of all the tiles sharing one vertex. Each quasicrystal has a finite number of possible vertex types.
In Penrose tiling there is a total of eight VTs, each of them having its own forced tiles—its own empire
field—a disconnected large space (Figure 2) that can be used to interact with other vertices non-locally.
Figure 2. Empire of a K vertex patch (red), its local empire (dark pink) and its non-local empire field
(light pink).
A vertex patch has a local empire and a non-local empire, as one can see in Figure 2 for one K
vertex [21,22]. In Penrose tiling, there are two decagon shaped configurations, one with a five-fold
symmetry and one without, the Cartwheel decagon. This decagon configuration has the same local
and non-local empire as the K vertex patch shown in Figure 2. For this first simulation, we have chosen
the vertex type isomorphic to the decagon that does not have five-fold symmetry because it allows
a larger variety of movement patterns and it has an interesting configuration, being bordered by a
star configuration on one side and a sun on the other. The analysis of other vertex types, however,
is the subject of our ongoing work. In this study, we select a K vertex/vertex patch as the ‘living’ or
dominant vertex/patch. All tiles in the local empire live also inside the empire window, and therefore
any local patch that includes K and is included in the local empire would have the same empire field.
Hence, the local empire of the chosen K type vertex patch can be treated as part of its quasiparticle,
its ‘supporting’ tiles. The rest of the empire, the non-local part, will be considered as the field of the
quasiparticle and will be used to calculate its interaction with the environment. The gray tiles shown
in Figure 2 are the tiles that can possibly coexist with the K vertex configuration, forming thus the
possibility space.
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A vertex patch can only walk to its designated neighboring locations. Since a vertex tends to
maintain its identity during its movement, the neighbors of the vertex should be of the same VT. For a
living vertex patch we consider two types of movement: translation and rotation. When there is no
translational momentum, a vertex patch should rotate as though it has an internal clock. When there
is translational momentum, however, the movement is a sequence of rotations of different centers.
Due to the discreteness of quasicrystals and the lack of translational periodicity, a vertex patch cannot
perform a ‘straight’ walk, tile-by-tile.
While previous studies of cellular automata on nonperiodic grids have considered the classic
von Neumann and Moore neighborhoods [7,8], we consider the neighborhood to be dictated by the
higher dimensional representation, namely the position in the perpendicular space. From the vertex
patch’s nearest neighbors in the 2D tiling, we consider only those which are also in the same plane
in the perpendicular space. All Penrose tiling vertices are on four planes in the perpendicular space,
each plane distribution forming a pentagonal area [23]. Using this method, the neighbors of the K
vertex patch are the eight neighbors shown in Figure 3b, labeled 1–8. The neighbors around the
five-fold star vertex (red point) are labeled from 1 to 4 and the neighbors around the five-fold sun
vertex (blue point) are labeled from 5 to 8. Even though it appears the neighbors 2 and 3 are not the
closest ones in the 2D tiling, if we look at the nearest neighbors in the perpendicular space, neighbors
2 and 3 are actually closer to the vertex (Figure 3c). Indeed, all of the eight chosen neighbors are on the
same pentagonal area in the perpendicular space, forming two pentagons with the K vertex, as shown
in Figure 4. As a first step, the living vertex patch can walk only to one of the neighbors of the same
VT in Figure 3; the chosen neighbor will become a living vertex patch once that step is taken.
a
b
c
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Figure 3. Game of life variants in Penrose tiling. (a) The living vertex patch of the K type vertex is
colored red and the pink tiles surrounding it represent its local empire. These tiles act as supporting
tiles that always coexist with the chosen vertex patch. The gray tiles form the possibility space—tiles
that could potentially coexist with the vertex. (b) The eight vertex patches in green are the neighbors of
the dominant vertex patch that must live in the possibility space. (c) Overlaying together the living
vertex patch, its supporting tiles, its neighbors and its possibility space.
For the dynamic advancement of the quasiparticle in our game of life model, we employ the
following general rules:
•
Intrinsic rotation: A living vertex patch never stays in a fixed location for two consecutive
outputs—frames. Depending on its intrinsic properties, it will perform either a clockwise
(to neighbors 1, 2, 3 or 4) or a counterclockwise rotation (to neighbors 5, 6, 7, 8), rotation that we
refer to as an intrinsic clock.
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•
Least change principle: In our game of life algorithm, the chosen general rule by which the
quasiparticle moves is based on the least change principle, which states that the preferred path is
the path where the number of tiles in the empire field that are required to change as a result of
that move, is minimal. According to the least change principle, gliders move forward following
a maximum trits-saving path, where trits, or bits in the 2D case, refer to the number of tiles
changed during the motion. That is, each timestep the particle will take the path that causes the
least ‘amount’ of change in the empire field, while it cannot stay in the same position for two
consecutive timesteps. In the higher dimensional approach, in which the shift of the cut window
is guided by the empire window and the possibility space window in the higher dimensional
mother lattice, the ‘trits’ correspond to the number of shifts of the cut window. For example,
let us define E0 to be the union of the empire of all existing dominant vertices, Ei to be the empire
of the ith neighboring vertex, and Ui = E0 ∩ Ei. If Um is the maximum of the list of Ui, then the
mth neighbor will be the preferred dominant vertex in the next frame. If there is more than one
maximum of the Ui list, then a random choice will be made between the two maxima for the next
step of the dominant vertex.
a
b
c
Figure 4. Representation of the neighbors in the perpendicular space. (a) The face-on view of the
perpendicular space, where purple represents the K vertex patch, the green represents the nearest
neighbors, the cyan point is the K vertex and the orange points are the eight neighbors’ vertices. (b) An
edge-on view showing the tiles belonging to the K vertex patch and the tiles belonging to the nearest
neighbors. (c) The same edge-on view, including this time, in yellow, the area corresponding to the
neighbors that appear closer in the 2D projection, but are in fact further away in the perpendicular space.
With these two rules, our game of life ensures first a continuous motion of our quasiparticles
as well as a syntactical freedom present in the code, for the case in which the particle is guided by a
random choice, resulting in a non-deterministic path for the quasiparticle.
For the self-interaction case, however, besides modeling the ‘least change’ walk, we have also
modeled several scenarios in which we force the particle’s first step to be in the direction of one of the
neighbors or a sequence of two of the neighbors, in order to explore different moving patterns
and trajectories. In this case, choosing a vertex without five-fold symmetry allows for various
moving patterns.
4. Walks of Life
In this Section we present the results, first for one quasiparticle, corresponding to the K VT, in several
possible dynamic scenarios of self-interactions, followed by the case of two quasiparticles with their
interacting empire fields. As previously mentioned, the quasiparticle moves under the influence of its
non-local empire and possibility space, following the least change principle, if not instructed otherwise.
For the two-particle interactions, there is one extra factor that influences the movement: the initial
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configuration, the distance between the particles and the initial positions-orientation with respect to
each other (see Supplementary Materials section).
4.1. Solo Walk
First we applied the algorithm to a single living vertex patch, which can only interact with its
own empire field, such that when computing the Um, we use only the empire of that vertex patch as
the U0. The vertex patch is shown in Figure 3 and several scenarios are analyzed, depending on the
choices made between the neighbors. The most ‘efficient’ walk, with respect to the principle of least
change, implies two equal choices between the 5th and the 8th neighbors and it is shown in Figure 5.
The pattern of the walk will then be determined by the choices made between these two neighbors.
Other moving patterns are described below:
Figure 5. Single particle game of life based on the non-local empire field and the least change principle
with alternating hinge variable choices. The order of the frames is from left to right, upper panels to
lower panels. In the chosen Penrose tiling patch, the particle moves from the lower right corner of the
patch to the upper left corner. The dynamical evolution of the quasiparticle can be seen in the movie at
https://youtu.be/_LvJudfJq7s.
• Rotations: Several initial conditions, determined by the choice of the first steps between the
neighbors, give a rotation pattern. Choosing the neighbors 1–2 results in the quasiparticle having
a clockwise rotation around a configuration of tiles (situated between 2 and 3), as can be seen
in the movie at https://youtu.be/HjvJEs_kUxU. Choosing the neighbors 3–4 gives the same
rotation, but counterclockwise. Choosing the 3–8 neighbors, the quasiparticle rotates around a
circular configuration of tiles, tangent to one of the particle’s tiles. The minimum area covered by
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rotation is achieved when only the neighbor 8 is chosen, while the maximum area covered by
rotation is achieved for 3–6.
• Quasi-translation: When choosing the sequence 4–8, the quasiparticle has the closest path to
straight gliding, a quasi-translation movement (https://youtu.be/fJT40Ec3ejE, e.g.).
• Oscillations: When the steps are chosen to alternate between 5 and 8 or an equal sequence of 5s
and 8s, an oscillation pattern results, as shown in the movie at https://youtu.be/tq3pkKI4-3Y.
• Random Walks: We have also performed simulations where the steps are chosen between
the neighbors by a random number generator for each step. A special case of random walk,
where between two neighbors, 5 and 8, for example, the choice was randomly made each step,
was also performed.
• Combined Motion: Most of the other choices of neighbors result in a combination of translation
and rotation, like in the example displayed in the movie at https://youtu.be/_LvJudfJq7s.
It is important to point out that as the living vertex patch walks to its neighbor in the possibility
space, both the empire and the entire possibility space will change and update their positions from
frame to frame. Therefore, the tiling background space does not remain fixed, but is in dynamic mode,
showing that any self-interaction of the quasiparticles instantly changes the environment at infinite
distance, step-by-step.
4.2. Two to Tango
When two or more living vertices are present, each dominant vertex patch can walk to one of its
eight neighbors, following the least change principle. In this case, the E0 is the union of the empires
of all dominant vertices. As a separate rule, we also restrict the movement of the quasiparticles by
not allowing their local empires to overlap. For the case of two vertex patches, depending on the
initial conditions, meaning the position and orientation of the particles with respect to each other,
the two quasiparticles will eventually evolve into a stable oscillation or rotation pattern, as shown in
Figure 6. For one of the stable patterns (Figure 6a), the dynamical evolution is shown in the movie at
https://youtu.be/L1KPckojco4. For a different initial configuration, where the local empires briefly
overlap in the first timesteps, the particles still get locked in a quasi-stable motion (Figure 6c), as can
be seen in the movie at https://youtu.be/5FCJWayzDfY from which several detailed steps are shown
in Figure 7.
1
2
4
3
1
4
2
3
1
2
1
2
Teeter-totter
Expansion-contraction
Cycling-chasing
(a)
(b)
(c)
Figure 6. Stabilized patterns for different interactions between two quasiparticles. The (a) panel
shows the quasiparticles get locked in a teeter-totter type of oscillation; as the first particle (red) moves
from position 1 to 2 as in the figure, the second particles (blue) moves into the positions 1 and 2;
the dynamical evolution can be watched in the movie at https://youtu.be/L1KPckojco4. The (b) panel
shows a movement in which the particles come closer in one frame and get separated in the next,
pattern that keeps repeating. The (c) panel shows the particles chasing each other in circles.
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Figure 7. Illustration of the stabilized pattern for one type of interaction between two quasiparticles;
the pattern is a cycling-chasing one like depicted in Figure 6c. The entire evolution can be seen in the
movie at https://youtu.be/5FCJWayzDfY.
5. The End Game—Discussions and Outlook
In this paper we present the results of the first non-local game of life in 2D quasicrystals. Using
the Penrose tiling, we model and track the dynamical behavior of a vertex configuration belonging to
the K VT based on its non-local empire field, following a newly developed algorithm.
Considering a quasiparticle composed by the emperor (vertex patch) and its local empire, for a
K vertex type, which is projected from a higher dimensional Z5 lattice and which does not present
five-fold symmetry, we analyze the quasiparticle’s self-interaction with its non-local empire field in
various scenarios. We choose the neighbors of the quasiparticle to be the ones that are the closest to the
vertex patch in 2D and also on the same plane in the perpendicular space. These are the K type vertex
patches where the living vertex patch can move, the ones that can become alive once the movement
is made. Depending on which one of these neighbors is chosen for the first steps movements, we
find that a particle can rotate, oscillate or move in a quasi-translation pattern in the possibility space.
The most ‘efficient’ walk, subjected to the least change principle rule, engages both rotation and
forward movement. The choice between two equally efficient steps is made by a random hinge
variable, resulting in an unpredictable path for the particle. Since the non-local empire changes with
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the change in the particle’s position each step, so does the possibility space, creating a mutual influence
between the particle and its field on one hand and the possibility space, the environment, on the other.
This influence propagates instantly and at all distances. Indeed, after one step, the possibility space
changes because of the change in the empire field. As a consequence, for the next step, the movement
is restricted depending on the new spatial configuration, creating a feedback loop of influence.
Furthermore, we explore the interactions between two quasiparticles of the same vertex type,
considering a scenario in which the overlapping of their local empires is forbidden. We find that
depending again on the choice of the initial conditions, the particles eventually end up bound in a
‘dance’, either in oscillation-like patterns or coupled rotations.
While this is purely a game of life algorithm applied to one vertex type on a 2D quasicrystal,
these explorations can prove of extreme importance to the field of particles physics, having as the end
game the first principles explanation of the observed particle physics rich phenomenology. Indeed,
gauge theory and group theory have been used extensively in describing algebraically the standard
model of particle physics. Several attempts of connecting the geometry with the algebra have also been
made [24–27]. Symmetries corresponding to higher dimensional lattices, like Z5, have been studied in
the context of grand unified theories [28,29]. From our preliminary results, we can draw some general
parallels with results from the particle physics and quantum field theories, but these connections
remain open to further exploration and constitute the subject of future planned research.
Firstly, in the quantum field theory, a particle and its field are considered ‘interchangeable’, in the
sense in which the interaction of particles with the spacetime itself creates a field. In a similar manner,
for a discretized space configuration, a vertex configuration projected from a higher lattice has a large
empire, which acts as a field. Any change in the position of a quasiparticle exerts an influence at
all distances, not only on the particle’s own field, but also on the aperiodic grid itself. Because of
the forbidden symmetries of quasicrystals, only certain configurations can be found in projection,
constraining perhaps the symmetries allowed in nature.
Our next goal is performing similar studies for several vertex types, analyzing also the interactions
between quasiparticles with different vertex configurations in both 2D and 3D quasicrystals, while also
varying the rules assumed. For example, when allowing the local empires to overlap, depending on
the initial conditions, two quasiparticles can collide and the interacting vertex patches are expected to
merge into configurations corresponding to other vertex types. Furthermore, the overlaps of empires
correspond to local deflations in the tiling space or local curvatures in the QC window. As a next step,
we intend to model many particle interactions, using this uneven distribution as a source of interaction
forces between particles and to simulate the game of life based on the least action principle.
This study bridges several fields of physics, from high dimensional representations to quasicrystal
math, phason dynamics and particle phenomenology, and opens a new avenue of research with
multiple ramifications.
Supplementary Materials: Several movies of the game of life in different scenarios can be watched on this youtube
playlist https://www.youtube.com/playlist?list=PL-kqKejCypNT990P0h2CFhrRCpaH9e858. The Mathematica
notebooks used in this paper can be provided on demand.
Author Contributions: F.F. developed the algorithm of the game of life and performed and analyzed several
simulations; S.P. performed simulations for different game of life self-interaction scenarios; D.H. provided the
code for calculating the possibility space; K.I. proposed this research initially. F.F. and S.P. co-wrote the paper.
Funding: This research received no external funding.
Acknowledgments: The authors thank Todd Rowland for his help in optimizing the code and for useful
discussions. We also thank Paul Steinhardt for useful comments and suggestions.
Conflicts of Interest: The authors declare no conflict of interest.
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Abbreviations
The following abbreviations are used in this manuscript:
QC Quasicrystal
VT
Vertex type
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