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

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

#### Abstract

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: *E*_{1}, *E*_{2}, . . /*F*_{1}, *F*_{2}, . . . Here, the *E*_{i} and *F*_{i} are listed in ascending order; the *E*_{i} specify the number of touching neighbors required to keep a living cell alive, and the *F*_{i} give the number required to bring a nonliving cell to life. Furthermore, when we write /*F*_{1}, *F*_{2}, . . 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].