Complex Systems

A New Candidate Rule for the Game of Three-Dimensional Life Download PDF

Carter Bays
Department of Computer Science
University of South Carolina, Columbia, SC 29208 USA


The two-dimensional cellular automaton "Life,'' originally described by Conway, has been expanded to three dimensions, with three different defining rules being noted (see [1, 2, 5]). Each rule can be written in the form , where is the minimum number of live neighbor cells that must touch a currently living cell in order to guarantee that remain alive the next generation; is the minimum number touching a currently dead cell in order that come to life next generation; and and are the corresponding upper limits. These rules are called the "environment'' and "fertility'' rules. According to this notation, Conway's Life would be written "Life 2333.''

Previously discovered three-dimensional life rules are Life 4555, Life 5766, and Life 5655. Each of these rules exhibits distinct characteristics. Life 5766 was shown to contain Conway's two-dimensional Life 2333; Life 4555 sports many period-four oscillators; and Life 5655, until now the most recently discovered rule, barely qualifies as a game of life, as its glider (translating oscillator) is quite rare in terms of its ability to condense out of "primordial soup'' (that is, random initial configurations).