Odd-Rule Cellular Automata on the Square Grid
Abstract
An "odd-rule" cellular automaton (CA) is defined by specifying a neighborhood for each cell, with the rule that a cell turns ON if it is in the neighborhood of an odd number of ON cells at the previous generation, and otherwise turns OFF. We classify all the odd-rule CAs defined by neighborhoods which are subsets of a 3 X 3 grid of square cells. There are 86 different CAs modulo trivial symmetries. When we consider only the different sequences giving the number of ON cells after n generations, the number drops to 48, two of which are the Moore and von Neumann CAs. This classification is carried out by using the "meta-algorithm" described in an earlier paper to derive the generating functions for the 86 sequences, and then removing duplicates. The fastest-growing of these CAs is neither the Fredkin nor von Neumann neighborhood, but instead is one defined by "Odd-rule" 365, which turns ON almost 75% of all possible cells.
Cite
@article{arxiv.1503.04249,
title = {Odd-Rule Cellular Automata on the Square Grid},
author = {Shalosh B. Ekhad and N. J. A. Sloane and Doron Zeilberger},
journal= {arXiv preprint arXiv:1503.04249},
year = {2015}
}
Comments
15 pages, 3 tables, 6 figures. Mar 20 2015: corrected a typo in Fig 1