On convergence of configurations

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Abstract

The transition rule F of a cellular automaton may sometimes be regarded as a “rule of growth” of a crystal from a “seed” ω. A study is made of the iterates ω, Fω, F2ω,…. For certain growth rules F it is proved that when ω is “sufficiently large” the sequence Fpω “converges” to a rational polytope W. The limiting shape W depends only on F.

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