CYCLIC CELLULAR AUTOMATA IN TWO DIMENSIONS Dedicated to T.E. Harris on his 70th birthday

Start by randomly populating each site of the two-dimensional integer lattice with any one of N types, labeled 0,1, , 1 The type at site can the type at neighboring site ( . y y eat x x (i.e., replace the type at with ) that mod We describe the dynamics x y provided y x 1 N. of , discrete-time deterministic systems which follow the rule: cyclic cellular automata (c.c.a. t (•) At any time , each type eats neighboring type that it can. t y every t These systems have remarkably complex dynamics. As becomes large they display a curious N metastability leading to large-scale locally-periodic structure. This article contains a preliminary account of our findings. For the most part, we rely on computations and computer graphics produced by the Cellular Automaton Machine. However we are able to give a simple proof that the infinite system is t asymptotically locally periodic for any Moreover, we identify a number of regularity properties N . of rule (•), mostly topological in nature, that offer some hope for a more detailed rigorous analysis. Research supported in part by NSF Grant DMS-87-0083