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Totalistic rules

By far the most symmetric restriction of a rule table is to consider only the number of neighbors of a cell, but not their spatial distribution. The result is called a totalistic rule, because of its dependence on the sum. For the Moore neighborhood, there are nine neighbors, therefore ten different sums. Sums of 0 and 9 can occur in only one way, but there are nine neighborhoods containing but one cell, and nine more with eight neighbors. In general, the binomial coefficients tell how many neighborhoods contain k cells.

A menu display for defining a totalistic rule will typically display the various sums along a row, the value which the cell will acquire in the new generation being shown in the matching position in the row below. The position of a movable cursor shows where new values may be inserted, thereby changing the rule.

Harold V. McIntosh