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In fact, it turns out that in any cellular automaton it is inevitable that initial conditions which consist just of a fixed block of cells repeated forever will lead to simple repetitive behavior.

For what happens is that each block in effect independently acts like a system of limited size. The right-hand neighbor of the rightmost cell in any particular block is the leftmost cell in the next block, but since all the blocks are identical, this cell always has the same color as the leftmost cell in the block itself. And as a result, the block evolves just like one of the systems of limited size that we discussed on page 255. So this means that given a block that is n cells wide, the repetition period that is obtained must be at most 2^n steps.

But if one wants a short repetition period, then there is a question of whether there is a block of any size which can produce it. The pictures on the next page show the blocks that are needed to get repetition periods of up to ten steps in rule 30. It turns out that no block of any size gives a period of exactly two steps, but blocks can be found for all larger periods at least up to 15 steps.

But what about initial conditions that do not just consist of a single block repeated forever? It turns out that for rule 30, no other kind of initial conditions can ever yield repetitive behavior.

But for many rules—including a fair number of class 3 ones—the situation is different. And as one example the picture on the right below shows an initial condition for rule 126 that involves two different blocks but which nevertheless yields repetitive behavior.

Captions on this page:

Rule 126 with a typical random initial condition, and with an initial condition that consists of a random sequence of the blocks and . Rule 126 in general shows class 3 behavior, as on the left. But with the special initial condition on the right it acts like a simple class 2 rule. Note the patches of class 2 behavior even in the picture on the left.

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