The nested structure of the patterns produced is thus a direct consequence of the nesting seen in the patterns of these digit sequences, as shown on page 117 . Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences.

This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .

In cases (c) and (d), these fluctuations turn out to have a very regular nested form.

intricate nested pattern shown below arises from many different simple programs.

But what we see is that at the edges of these areas there are often intricate structures with an essentially nested form.

And sometimes nested or fractal behavior was seen.

We would not usually say, therefore, that either of the first two pictures at the top of the facing page seem random, since we can readily recognize highly regular repetitive and nested patterns in them.

And most methods manage to compress behavior that is repetitive, and at least to some extent behavior that is nested—exactly the two kinds of simple behavior that we have noted many times in this book.

For while ordinary human language has little trouble describing repetitive and even nested patterns, it seems to be able to do very little with more complex patterns—which is in a sense why this book, for example, depends so heavily on visual presentation.

And in the case of nested patterns even the specific structures seen are usually the same as for elementary rules.