A Game of Life on Penrose tilings

TL;DR

This paper introduces cellular automata rules on Penrose tilings that are isomorphic to Conway's Game of Life, enabling local computation on nonperiodic quasicrystalline structures, with potential applications in unconventional computing.

Contribution

It defines a new class of cellular automata on Penrose tilings that are isomorphic to the Game of Life, extending cellular automata to quasiperiodic tilings.

Findings

Existence of glider-like structures in Penrose tilings.

Local rules enable computation on nonperiodic substrates.

Framework for computation in quasicrystals.

Abstract

We define rules for cellular automata played on quasiperiodic tilings of the plane arising from the multigrid method in such a way that these cellular automata are isomorphic to Conway's Game of Life. Although these tilings are nonperiodic, determining the next state of each tile is a local computation, requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. As an example, we show a version of a "glider" moving through a region of a Penrose tiling. This constitutes a potential theoretical framework for a method of executing computations in non-periodically structured substrates such as quasicrystals.