History of fractals The idea of using nested 2D shapes in art probably goes back to antiquity; some examples were shown on page 43 . In mathematics, nested shapes began to be used at the end of the 1800s, mainly as counterexamples to ideas about continuity that had grown out of work on calculus. … From the point of view of mathematics, however, nested shapes tended to be viewed as rare and pathological examples, of no general significance.

And this can be achieved in only two ways: either the pattern must be essentially uniform, or it must have a nested structure—just like we see in rule 90. … And in fact one can show that any rule that is additive will be able to emulate itself and will thus yield nested patterns.

And as one continues one sees various other repetitive and nested forms. … But it could also be that whatever one does rule 54 will always in the end just show purely repetitive or nested behavior—which cannot on its own support universality.

Nested lists One can think of structures that annihilate in pairs as being like parentheses or other delimiters that come in pairs, as in the picture below. A string of balanced parentheses is analogous to a nested Mathematica list such as {{{}, {{}}}, {}} . The Mathematica expression tree for this list then has a structure analogous to the nested pattern in the picture.

Sometimes the behavior that is generated in such systems has a simple repetitive or nested form.

In the bottom pictures, it is a nested pattern analogous to the two-dimensional one on page 171 .

Quasicrystals In some special materials it was discovered in 1984 that atoms are arranged not on a purely repetitive grid, but instead in a pattern with the nested type of structure discussed on page 932 . … And as we saw on page 932 it is indeed possible to get nested patterns by requiring that certain constraints be satisfied.

And with this representation, the evolution for t steps is given by SSEvolveList[rule_, init_List, t_Integer] := NestList[Flatten[# /. rule]&, init, t] where in the first example on page 82 , the initial condition is {1} . … In this case, the evolution can be obtained using SSEvolveList[rule_, init_String, t_Integer] := NestList[StringReplace[#, rule]&, init, t] For a neighbor-dependent substitution system such as the first one on page 85 the rule can be given as {{1, 1}  {0, 1}, {1, 0}  {1, 0}, {0, 1}  {0}, {0, 0}  {0, 1}} And with this representation, the evolution for t steps is given by SS2EvolveList[rule_, init_List, t_Integer] := NestList[Flatten[Partition[#, 2, 1] /. rule]&, init, t] where the initial condition for the first example on page 85 is {0, 1, 1, 0} .

Long halting times [in symbolic systems] Symbolic systems with rules of the form ℯ [x_][y_]  Nest[x, y, r] always evolve to fixed points—though with initial conditions of size n this can take of order Nest[r # &, 0, n] steps (see above ).

Multiway systems based on numbers One can consider for example the rule n  {n + 1, 2 n} implemented by NestList[Union[Flatten[{# + 1, 2 #}]] &, {0}, t] In this case there are Fibonacci[t + 2] distinct numbers obtained at step t . In general, rules based on simple arithmetic operations yield only simple nested structures.