# Non-Local Game of Life in 2D Quasicrystals

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## Abstract

**:**

## 1. Introduction

## 2. Properties of Quasicrystals

- 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.

**The Cut-and-Project Method**

**The QC Window/Acceptance Domain**

**The Empire Window**

**The Possibility Space Window**

## 3. Rules of Life

**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.**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 ${E}_{0}$ to be the union of the empire of all existing dominant vertices, ${E}_{i}$ to be the empire of the ${i}_{th}$ neighboring vertex, and ${U}_{i}={E}_{0}\cap {E}_{i}$. If ${U}_{m}$ is the maximum of the list of ${U}_{i}$, then the ${m}_{th}$ neighbor will be the preferred dominant vertex in the next frame. If there is more than one maximum of the ${U}_{i}$ list, then a random choice will be made between the two maxima for the next step of the dominant vertex.

## 4. Walks of Life

#### 4.1. Solo Walk

**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 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.

#### 4.2. Two to Tango

## 5. The End Game—Discussions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QC | Quasicrystal |

VT | Vertex type |

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**Figure 1.**Cut-and-project windows in a rotated ${Z}_{2}$ lattice used for generating a 1D quasicrystal and its associated structures. (

**a**) The ${Z}_{2}$ lattice is rotated so that the quasicrystal space (E) coincides with the horizontal line and the perpendicular space (${E}^{\perp}$) lies along the vertical line. The blue vertical line is the QC window, which covers all points and square edges (unit cells of ${Z}_{2}$ 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.

**Figure 2.**Empire of a K vertex patch (red), its local empire (dark pink) and its non-local empire field (light pink).

**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.

**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.

**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.

**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.

**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.

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**MDPI and ACS Style**

Fang, F.; Paduroiu, S.; Hammock, D.; Irwin, K. Non-Local Game of Life in 2D Quasicrystals. *Crystals* **2018**, *8*, 416.
https://doi.org/10.3390/cryst8110416

**AMA Style**

Fang F, Paduroiu S, Hammock D, Irwin K. Non-Local Game of Life in 2D Quasicrystals. *Crystals*. 2018; 8(11):416.
https://doi.org/10.3390/cryst8110416

**Chicago/Turabian Style**

Fang, Fang, Sinziana Paduroiu, Dugan Hammock, and Klee Irwin. 2018. "Non-Local Game of Life in 2D Quasicrystals" *Crystals* 8, no. 11: 416.
https://doi.org/10.3390/cryst8110416