Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Relation to 2D cellular automata

The kind of constraints discussed are exactly those that must be satisfied by configurations that remain unchanged in the evolution of a 2D cellular automaton. The argument for this is similar to the one on pages 941 and 954 for 1D cellular automata. The point is that of the 32 5-cell neighborhoods involved in the 2D cellular automaton rule, only some subset will have the property that the center cell remains unchanged after applying the rule. And any configuration which does not change must involve only these subsets. Using the results of this section it then follows that in the evolution of all 2D cellular automata of the type discussed on page 170 there exist purely repetitive configurations that remain unchanged.

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