And with this kind of setup, it is ultimately inevitable that all the patterns produced must have a completely regular nested structure. … And the result of this has been that in traditional fractal geometry the idea of interaction between elements is not considered—so that all patterns that are produced have a purely nested form. … But the overall pattern obtained after a large number of steps still has a nested form.

Purely nested data can also be compressed nearly as much. … Examples of the pattern of repeats found in purely nested data. … Taking into account the length of the representation for each pointer, the compressed form of a nested sequence of length n will typically grow in length like Log[n] 2 .

So what about other nested patterns? It turns out that using an extension of the argument above it is always possible to take the rules An example of how the color of any square in a nested pattern can be found from its coordinates by a fairly simple mathematical procedure. … The nested pattern can be built up by a 2D substitution system with the rules shown.

Nested vibrations With an assembly of springs arranged in a nested pattern simple initial excitations can yield motion that shows nested behavior in time. … (Similar considerations apply to the motion of quantum mechanical electrons in nested potentials.)

And of the millions of these that I have tested, none have forced anything more complicated than the kind of nested behavior seen on the previous page . … The picture below then shows an example of such a system, in which by allowing only a specific set of 33 templates, a nested pattern is forced to occur. … This constraint, together with the requirement that the first template must appear at least somewhere, then turns out to force a nested pattern to occur.

Starting from random initial conditions this rule yields a collection of stripes which annihilate whenever they meet, leading to a sequence of progressively larger nested regions. … But even when the initial conditions are such that some stripes survive, nested regions are still formed by the stripes that do annihilate. … The picture The generation of a nested pattern by rule 184 starting from random initial conditions.

But what the picture suggests is that at some rough overall level these structures progressively tend to annihilate each other, and in doing so form an approximate nested pattern. … But the fact that they annihilate whenever they meet once again tends to lead to an overall nested pattern of behavior. … In both cases, the boundaries between domains appear to follow random walks, annihilating when they meet and thus forming a nested overall pattern.

With rule (a), however, it is fairly easy to see that a simple nested structure is produced, directly analogous to the one shown on page 509 . … And the behavior obtained never seems to repeat, nor do the networks produced exhibit any kind of obvious nested form. … It takes fewer steps for networks to be built up, but the results are qualitatively similar to those on the previous page : rule (a) yields a nested structure, rule (b) gives repetitive behavior, while rule (c) produces behavior that seems complicated and in some respects random.

In the first case, a regular nested pattern is obtained.

Forcing nested [2D] patterns It is straightforward to find constraints that allow nested patterns; the challenge is to find ones that force such patterns to occur. … One approach to finding constraints that can be satisfied only by nested patterns is nevertheless to start from specific nested patterns, look at what templates occur, and then see whether these templates are such that they do not allow any purely repetitive patterns. … It can nevertheless be shown that with a sufficiently large number of extra colors any nested pattern can be forced to occur.