Nested continuous functions Most standard continuous mathematical functions never show any kind of nested behavior. … Nested behavior is also found for example in EllipticTheta[3, 0, z] , which is given essentially by Sum[z n 2 , {n, ∞ }] .

In case (c), alternate steps in the leftmost column (which in all cyclic tag systems determines the overall behavior) have the same nested form as the third neighbor-independent substitution system shown on page 83 .

Nested structure of attractors Associating with each sequence of length n (and k possible colors for each element) a number Sum[a[i] k -i , {i, n}] , the set of sequences that occur in the limit n  ∞ forms a Cantor set. … In general, whenever the possible sequences correspond to paths through a finite network, it follows that the Cantor set obtained has a nested structure. … Note that if the possible sequences cannot be described by a network, then the Cantor set obtained will inevitably not have a strictly nested form.

Nested radicals Given a list of integers acting like digits one can consider representing numbers in the form Fold[Sqrt[#1 + #2]&, 0, Reverse[list]] . … (Note that Nest[Sqrt[# + 2] &, 0, n]  2 Cos[ π /2 n + 1 ] .) … For any number x the first n digits are given by Ceiling[NestList[(2 - Mod[-#, 1]) 2 &, x 2 , n - 1] - 2] Even rational numbers such as 3/2 do not yield simple digit sequences.

Note that the pattern in case (a) does eventually repeat, while the one in case (b) eventually shows a nested structure.

Penrose tilings The nested pattern shown below was studied by Roger Penrose in 1974 (see page 943 ). … In general, projections onto any regular lattice in any number of dimensions from hyperplanes with any quadratic irrational slopes will yield nested patterns that can be generated by subdividing some shape or another according to a substitution system. Despite some confusion in the literature, however, this procedure can reproduce only a tiny fraction of all possible nested patterns.

[Turing] machine 596440 For any list of initial colors init , it turns out that successive rows in the first t steps of the compressed evolution pattern turn out to be given by NestList[Join[{0}, Mod[1 + Rest[FoldList[Plus, 0, #]], 2], {{0}, {1, 1, 0}} 〚 Mod[Apply[Plus, #], 2] + 1] 〛 &, init, t] Inside the right-hand part of this pattern the cell values can then be obtained from an upside-down version of the rule 60 additive cellular automaton, and starting from a sequence of 1 's the picture below shows that a typical rule 60 nested pattern can be produced, at least in a limited region. The presence of glitches on the right-hand edge of the whole pattern means, however, that overall there is nothing as simple as nested behavior—making it conceivable that (possibly with analogies to tag systems) behavior complex enough to support universality can occur.

Nested patterns and numbers See page 931 .

Particularly dramatic are the concatenation systems discussed on page 913 , as well as successive rows in nested patterns such as Flatten[IntegerDigits[NestList[BitXor[#, 2 #] &, 1, 500], 2]] and sequences based on numbers such as Flatten[Table[If[GCD[i, j]  0, 1, 0], {i, 1000}, {j, i}]] (see page 613 ).

The nested structure on the diagonal essentially corresponds to a progression of base 2 digit sequences for positive and negative numbers.