[Universality of] set theory Any integer n can be encoded as a set using for example Nest[Union[#, {#}] &, {}, n] .

The value at step t in the column immediately adjacent to the center is the nested sequence discussed on page 892 and given by Mod[IntegerExponent[t, 2], 2] .

If h is rational, the sequence is repetitive, while if h is a quadratic irrational, it is nested.

The pattern after n steps is then given by Nest[Flatten[f[#]] &, {0}, n] , where for the rule on page 189 f[z_] = 1/2 (1 -  ) {z + 1/2, z - 1/2} ( f[z_] = (1 -  ){z + 1, z} gives a transformed version).

Note that this solution in effect constructs a nested pattern of any width (it does this by optionally including or excluding one additional cell at each nesting level, using a mechanism related to the decimation systems of page 909 ).

Most of the circles added at a given step are not the same size, however, making the overall geometry not straightforwardly nested. … To achieve filling fraction 1 requires arbitrarily small circles, but there are many different arrangements of circles that will work, some not even close to nested. When actual granular materials are formed by crushing, there is probably some tendency to generate smaller pieces by following essentially substitution system rules, and the result may be a nested distribution of sizes that allows an Apollonian-like packing.

And just as in these systems, it is possible for intricate structures to be produced, but the structures always turn out to have a highly regular nested form.

Similarly, the nested structure of picture (b) implies that to reproduce this picture, all we need do is to specify the colors in a 3×3 block, and then say that as in a two-dimensional substitution system each black cell should repeatedly be replaced by this block.

So for example when presented with the 256 elementary cellular automaton patterns shown on page 55 mathematicians in my experience have two common responses: either to single out specific patterns that have a simple repetitive or perhaps nested form, or to generalize and look not at individual patterns, but rather at aggregate properties obtained say by evolving from all possible initial conditions.

And as a result, the pattern it produces must ultimately have a simple repetitive or nested form.