Complex Systems

A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life Download PDF

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


In 1987, this author introduced three-dimensional (3D) versions of the two-dimensional (2D) cellular automaton (CA) made famous by John Conway [1]. At that time, some criteria were given to validate candidate 3D CA rules as "game of life" (GL) rules. Informally, we say that a CA rule is a GL rule if (a) when we count neighbors of a cell, we count all touching neighbors, and they are all treated the same; (b) the rule supports a "glider" (a translating oscillator); and (c) random patterns exhibit bounded growth. These informal criteria are spelled out more formally in [2], which also gives notation for writing rules, namely: E1, E2, . . /F1, F2, . . . Here, the Ei and Fi are listed in ascending order; the Ei specify the number of touching neighbors required to keep a living cell alive, and the Fi give the number required to bring a nonliving cell to life. Furthermore, when we write /F1, F2, . . we are implying that the E terms have not been specified.
Conway's rule is thus written 2,3/3 (and not 3,2/3). The two 3D GL rules introduced in 1987 were 4,5/5 and 5,6,7/6 [3]. There followed a series of notes in this journal wherein the discovery of two more GL rules was explored [4, 5]; the rules were 5,6/5 and 6,7,8/5. It should be noted that the rule 6,7,8/5 does not strictly adhere to the formal GL criteria given in [2]. Nevertheless it was well behaved enough to be considered, and this slight variation was mentioned in [5].