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Cycles in space

Fundamental to finite automata is eventually periodic evolution, prolonged to the full number of states in the exceedingly rare event that their sequence is cyclic. The number of ``states'' of a cellular automaton is the number of configurations (not states per cell), providing an exponentially large bound relative to the automaton's length. In practice, many short cycles usually predominate over a few long ones, almost always reached through transients.

Longer automata admit longer cycles and longer transients too; the infinite limit may lack cycles. Cyclic boundary conditions locate behavior repeating over a finite range, leaving truly aperiodic configurations for a separate study.

Harold V. McIntosh