Cellular Automata in the Triangular Tessellation
Carter Bays
Department of Computer Science,
University of South Carolina, Columbia, SC 29208, USA
Abstract
For the discussion below the following definitions are helpful. A semitotalistic CA rule is a rule for a cellular automaton (CA) where (a) we tally the living neighbors to a cell without regard to their orientation with respect to that cell, and (b) the rule applied to a cell may depend upon its current status. A lifelike rule (LFR rule) is a semitotalistic CA rule where (1) cells have exactly two states (alive or dead); (2) the rule giving the state of a cell for the next generation depends exactly upon (a) its state this generation and (b) the total count of the number of live neighbor cells; and (3) when tallying neighbors of a cell, we consider exactly those neighboring cells that touch the cell in question. An LFR rule is written 
, where 
(the "environment'' rule) give the lower and upper bounds for the tally of live neighbors of a currently live cell C so that C remains alive, and 
(the "fertility'' rule) give the lower and upper bounds for the tally of live neighbors required for a currently dead cell to come to life. For an LFR rule to specify a game of Life we impose two further conditions: (A) there must exist at least one glider (translating oscillator) that is discoverable with probability one by starting with finite random initial configurations (sometimes called "random primordial soup''), and (B) the probability is zero that a finite random initial configuration leads to unbounded growth. Note that this second condition does not eliminate the possibility that some unusual highly organized configuration can be constructed where the growth is unbounded. Note also that we may be able to construct some extremely complex configuration that translates; however, if the possibility of discovering this with a random experiment is zero then condition (1) has not been met. We shall call LFR rules that satisfy (A) and (B) GL ("Game of Life'') rules; they will usually be written "Life '' (otherwise we simply write "rule ''). To date there has been only one GL rule discovered in two dimensions: that is of course the famous Conway game, Life 2333, which exists on a two-dimensional grid of square cells, where each cell has 8 touching neighbors.
