Rule 22 With more complicated initial conditions the pattern is often no longer nested, as shown on page 263 .

In a nested sequence the number must always continue increasing roughly linearly, and must be greater than m for every m . (The differences of successive numbers themselves form a nested sequence.) … Up to limited m nested sequences can contain all k m possible blocks, and can do so with asymptotically equal frequencies.

Structures like spider webs, wasp nests, termite mounds, bird nests and beaver dams rely on behavior determined by animal brains.

So the result is that our ears are not sensitive to most of the elaborate structure that we see in the spectra of many nested sequences. … To get a spectrum with a more elaborate structure requires long-range correlations—as exist in nested sequences.

It turns out that the same is true for nested sequences. And in the picture below, sequences (b), (c) and (d) are all nested.

But the way this occurs is highly constrained, and in the end these systems can only produce patterns that are in essence purely nested—so that it is again not possible for universality to be achieved. … With simple initial conditions these rules always yield very regular nested patterns.

2D spectra The pictures below give the 2D Fourier transforms of the nested patterns shown on page 583 .

But with k = 3 possible elements, there are infinite nested sequences that can, such as the one produced by the substitution system {0  {0, 1, 2}, 1  {0, 2}, 2  {1}} , starting with {0} . … The constraint that no triple of identical blocks appear together turns out to be satisfied by the Thue–Morse nested sequence from page 83 —as already noted by Axel Thue in 1906. … But it also known that among the infinite sequences which do this, there are always nested ones (sometimes one has to iterate one substitution rule, then at the end apply once a different substitution rule).

With three possible states, only repetitive and nested patterns are ever ultimately produced, at least starting with all cells white.

Other [symbolic systems] rules If only a single variable appears in the rule, then typically only nested behavior can be generated—though in an example like ℯ [x_][_]  ℯ [x[ ℯ [ ℯ ][ ℯ ]][ ℯ ]] it can be quite complex. … Note that rules with no explicit e 's on the left-hand side always give trees with regular nested structures; x_[y_]  x[y][x[y]] (or x_  x[x] in Mathematica), for example, yields balanced binary trees.