On convergence of configurations

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 F^pω “converges” to a rational polytope W. The limiting shape W depends only on F.