Non-Local Game of Life in 2D Quasicrystals

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.

Empires: The Nonlocal Properties of Quasicrystals

“…This approach differs from previous studies that consider the nearest neighbors situated in the local 2D representation [27,28]. In Figure 4, one can see the K vertex type and 2D representation of the nearest neighbors in the perpendicular space that we have considered [17]. The distribution of the neighbors is interesting, as it surrounds an S vertex patch (sun) on one side and an S5 vertex patch (star) on the other side.…”

“…Several game-of-life [21] algorithms have been previously studied on Penrose tiling, but they have either considered a periodic grid [26] or they have considered only local rules [27,28]. Recently, for the first time, a game-of-life scenario has been simulated using nonlocal rules on a two-dimensional qusicrystal, the Penrose tiling, in [17]. In this simulation, for the [16], Figures 13 and 14.…”

“…For the case of two-particle interactions, one more arbitrary constraint is introduced, where the local patches of the two particles are not allowed to overlap. A detailed discussion of the algorithm and the simulation setup can be found in [17]. 2 2 Movies from the simulations can be watched on https://www.youtube.com/playlist?list=PL-kqKe jCypNT990P0h2CFhrRCpaH9e858.…”

“…More recently, the cut-and-project technique [14,15]-where the geometry of convex polytopes comes into play-has been implemented into a most efficient method of computing empires [16]. The cut-and-project method offers the most generality and has been used to calculate empires for the Penrose tiling and other quasicrystals that are projections of cubic lattices  n [16,17]. Quasicrystals that are projections of non-cubic lattices (e.g., the Elser-Sloane tiling as a projection of the E 8 lattice to  4 ) are more restrictive and some modifications to the cut-and-project method have been made in order to correctly compute the tilings and their empires [18].…”

“…The interest in the nonlocal nature of empires has led to further explorations into how multiple empires can interact within a given quasicrystal, where separate patches can impose geometric restrictions on each other no matter how far apart they may be located within the tiling. These interactions can be used to define rules (similar to cellular automata) to see what dynamics emerge from a game-of-life style evolution of the quasicrystal and for the first time such a simulation with nonlocal rules has been performed [17].…”