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Nesting in numbers
Chapter 4 contains several examples of systems based on numbers that exhibit nested behavior.
For n = 2 , the patterns obtained are at most nested. … With a single cell seed, no pattern more complicated than nested can be obtained in such a system. … If the group is Abelian, so that f[i, j] f[j, i] , then only nested patterns are ever produced (see page 955 ).
Other integer functions
IntegerExponent[n, k] gives nested behavior as for decimation systems on page 909 , while MultiplicativeOrder[k, n] and EulerPhi[n] yield more complicated behavior, as shown on pages 257 and 1093 .
As discussed on page 955 , any rule based on addition modulo k must yield a nested pattern, and it therefore follows that any rule that is additive must give a nested pattern, as in the examples below.
There is still a nested structure but it is usually not visually as obvious as before.
The overall patterns produced by such cellular automata are essentially nested.
Cases like (c) and (d) show nested behavior reminiscent of a counter which generates digit sequences of successive integers.
Each rule yields a different sequence of elements, but all of them ultimately have simple nested forms.
Fractal dimensions
Certain features of nested patterns can be characterized by so-called fractal dimensions. … But even when this does not happen, the limiting behavior for small a is still (1/a) d for any nested pattern. … Fractal dimensions characterize some aspects of nested patterns, but patterns with the same dimension can often look very different.
[Patterns from] arbitrary digit operations
If the operation on digit sequences that determines whether a square will be black can be performed by a finite automaton (see page 957 ) then the pattern generated must always be either repetitive or nested. … But none of the patterns are purely nested.