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Of the 65,536 possible mobile automata with rules of the kind discussed so far it turns out that not a single one shows more complex behavior. So can such behavior then ever occur in mobile automata?

One can extend the set of rules one considers by allowing not only the color of the active cell itself but also the colors of its immediate neighbors to be updated at each step. And with this extension, there are a total of 4,294,967,296 possible rules.

If one samples these rules at random, one finds that more than 99% of them just yield simple repetitive behavior. But once in every few thousand rules, one sees behavior of the kind shown below—that is not purely repetitive, but instead has a kind of nested structure.

Captions on this page:

A mobile automaton with slightly more complicated rules that yields a nested pattern. Each column on the left shows 200 steps in the mobile automaton evolution. The compressed form of the pattern is based on a total of 8000 steps.

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