Chapter 6: Starting from Randomness

Section 6: Special Initial Conditions

[Repetition in] 2D cellular automata

As expected from the discussion of constraints on page 942, the problem of finding repeating configurations is much more difficult in two dimensions than in one dimension. Thus for example unlike in 1D there is no guarantee in 2D that among repeating configurations of a particular period there is necessarily one that consists just of a repetitive array of fixed blocks. Indeed, as discussed on page 1139, it is in a sense possible that the only repeating configurations are arbitrarily complex. Note that if one considers configurations in 2D that consist only of infinitely long stripes, then the problem reduces again to the 1D case. (See also page 349.)

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