Chapter 5: Two Dimensions and Beyond

Section 2: Cellular Automata

Numbers of possible [2D cellular automaton] rules

The table below gives the total number of 2D rules of various types with two possible colors for each cell. Given an initial pattern with a certain symmetry, a rule will maintain that symmetry if the rule is such that every neighborhood equivalent under the symmetry yields the same color of cell. Rules are considered rotationally symmetric in the table below if they preserve any possible rotational symmetry consistent with the underlying arrangement of cells. Totalistic rules depend only on the total number of black cells in a neighborhood; outer totalistic rules (as in the previous note) also depend on the color of the center cell. Growth totalistic rules make any cell that becomes black remain black forever.

In such a rule, given a list of how many neighbors around a given cell (out of s possible) make the cell turn black the outer totalistic code for the rule can be obtained from

Apply[Plus, 2^Join[2 list, 2 Range[s + 1] - 1]]

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]