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The point is that these initial conditions in effect contain only blocks for which rule 126 behaves like rule 90. And as a result, the overall patterns produced by rule 126 in this case are inevitably exactly like those produced by rule 90.

So what about other cellular automata that can yield similar patterns? In every example in this book where nested patterns like those from rule 90 are obtained it turns out that the underlying rules that are responsible can be set up to behave exactly like rule 90. Sometimes this will happen, say, for any initial condition that has black cells only in a limited region. But in other cases—like the example of rule 22 on page 263—rule 90 behavior is obtained only with rather specific initial conditions.

So what about rule 90 itself? Why does it yield nested patterns?

The basic reason can be thought of as being that just as other rules can emulate rule 90 when their initial conditions contain only certain blocks, so also rule 90 is able to emulate itself in this way.

The picture below shows how this works. The idea is to consider the initial conditions not as a sequence of individual cells, but rather as a sequence of blocks each containing two adjacent cells. And with an appropriate form for these blocks what one finds is that the configuration of blocks evolves exactly according to rule 90.

The fact that both individual cells and whole blocks of cells evolve according to the same rule then means that whatever pattern is

Captions on this page:

A demonstration of the fact that in rule 90 blocks of cells can behave just like individual cells. One consequence of this is that the patterns produced by rule 90 have a nested or self-similar form.

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