A decomposition theorem for number-conserving multi-state cellular automata on triangular grids

Abstract

This paper concerns two-dimensional cellular automata on a triangular grid that preserve the sum of the states of all the cells. To study such cellular automata, we adapt the idea of the split-and-perturb decomposition of a number-conserving local rule, developed first for square grids, to the setting of triangular grids. As a result, we obtain a new mathematical tool that allows, for example, to enumerate all so-called k-ary (i.e., binary, ternary, quaternary, quinary, etc.) number-conserving cellular automata on a triangular grid, regardless of the value of k.