Cellular Automata Operations on Penrose Tiling

A cellular automaton (CA) is a collection of colored cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states (i.e. colors) of neighboring cells. The rules are then applied iteratively for as many time steps as desired (see [Wolfram Mathworld]). The general case is a regular periodic grid, and my interest is in how the system behaves on a non-periodic, irregular one. When analyzing a CA, it is convenient for a grid to fill the space that it occupies. Penrose Tiling is an irregular, non-periodic, space-filling tiling. This tiling is closely related to the study of quasi-crystals, for which the 2011 Nobel Prize in Chemistry was awarded. The goal of this research has been to discover interesting differences between the evolution of CA on an aperiodic grid as compared to evolution on a periodic one. Applications of this research are broad in scope and include Material Science and Biotechnology