Candidates for the Game of Life in Three Dimensions

The game "Life" is defined in a strict sense and three candidates for t hree-d imensional versions are presented. One of these versions can be structured to contain an infinite number of parallel two-dimens ional universes, each of which allows for t he evolution of Conway life objects. Various oscillators are described, and a few interesting collisions between translating oscillators ("g lide rs") and other objects are mentioned. 1 . Introduction-Conway's Game of Life Most readers are probably familiar with John Conway 's two-dimensional cellular automaton known as the "Game of Life" 13,4J. The game is "played" by zero players on an arbitrarily large grid of square cells, where each cell is either "alive" or "dead". Essentially, the game works as follows. Start at generation one with some pattern of living cells (squares on the grid that are filled in). To obt ain the nex t generation , apply the following transition rules concur rent ly to each cell, C, on the gr id, whether filled in or not. Rule One: If C is living an d if it touc hes two or three living cells, it remains alive for th e next generation; otherwise , C dies [i.e ., erase the filled-in square for next generation) . Rule Two: If C is not living and if it touches .exact ly three living cells, C becomes alive [i.e., fill C in for next generation). Readers familiar with the game may recall t hat with ap propriate starting patterns , we can obtain a host of stable and oscillating shapes, which Conway and ot hers have given such whimsical names as "beehive", "blinker", "clock", "pulsar", etc . Severa l oscillators translate across the grid with successive generat ions; such oscillators are t rad it iona lly called gliders , a term which we shall use throughout this paper . 1.1 T he r ules of Life We can formalize the rules for Life as follows. Define environment E as the number of living neighbors required to prevent a cu rrently living cell ..A preliminary report on some of this work appeared in [1,2]. @ 1987 Complex Systems P ub licaticne, Inc .