Tight Bounds on Periodic Cell Configurations in Life

This work was partially completed while Callahan was a postdoctoral assistant in the Institute for Theoretical Computer Science, ETH Z urich, Switzerland. Periodic configurations, or oscillators, occur in many cellular automata. In an oscillator, repeated applications of the automaton rules eventually restore the configuration to its initial state. This paper considers oscillators in Conway’s Life; analogous techniques should apply to other rules. Three explicit methods are presented to construct oscillators in Life while guaranteeing certain complexity bounds, leading to the existence of an infinite sequence Kn of oscillators of periods n = 58, 59, 60, . . . and uniformly bounded population, and an infinite sequence Dn of oscillators of periods n = 58, 59, 60, . . . and diameter bounded by b p log n, where b is a uniform constant.