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But in both these cases it turns out to be not too long before these fluctuations essentially repeat. The picture below shows an example where a larger amount of apparent randomness is seen. Yet even in this case one finds that there ends up again being essential repetition—although now only every 1071 steps.

Captions on this page:

A multiway system with behavior that shows some signs of apparent randomness. The rule for this system involves three possible replacements. Note that the first replacement only removes elements and does not insert new ones. In the pictures sequences containing zero elements therefore sometimes appear. At least with the initial condition used here, despite considerable early apparent randomness, the differences in number of elements do repeat (shifted by 1) every 1071 steps.

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