The encoded version of a purely nested sequence grows like Log[n] 2 .

There is even no need for the tiling to be repetitive; the picture below shows a cellular automaton on a nested Penrose tiling (see page 932 ).

As in other systems based on numbers, nested patterns are not common—though page 1160 shows how they can in principle be achieved with an equation whose solutions satisfy Mod[Binomial[x, y], 2]  1 .

(In Mathematica, the explicit form of such a tensor can be represented as a nested list for which TensorRank[list]  4 .) … Γ 〚 i 〛 , {i, d}, {j, d}, {k, d}] where the so-called Christoffel symbol Γ ij k is Γ = With[{gi = Inverse[g]}, Table[Sum[ gi 〚 l, k 〛 ( ∂ p 〚 j 〛 g 〚 i, l 〛 + ∂ p 〚 i 〛 g 〚 j, l 〛 - ∂ p 〚 l 〛 g 〚 j, l 〛 ), {l, d}], {i, d}, {j, d}, {k, d}]]/2 There are d 4 elements in the nested lists for Riemann , but symmetries and the so-called Bianchi identity reduce the number of independent components to 1/12 d 2 (d 2 - 1) —or 20 for d = 4 .

General recursive functions, however, also allow μ [f_] = NestWhile[# + 1 &, 0, Function[n, f[n, ##] ≠ 0]]& which can perform unbounded searches. … Note that functions of the form Nest[r[c[s, z], #] &, c[s, s], n] are given in terms of the original Ackermann function in the note above by f[n + 1, 2, # + 1] - 1 & .

With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] .

Random walks In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from NestList[(# + (-1)^Random[Integer])&, 0, t] or equivalently FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]] A generalization to d dimensions is then FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]] A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t .

Even on this one page there are perhaps a dozen other very similar nested structures. 12 th century (Italian). … The third picture—particularly the part magnified in the fourth picture—shows an approximate nested structure, presumably created as in the pictures below.

One marginally more complicated case effectively involving 13 neighbors is IsingEvolve[list_, t_Integer] := First[Nest[IsingStep, {list, Mask[list]}, t]] IsingStep[{a_, mask_}] := {MapThread[ If[#2  2 && #3  1, 1 - #1, #1]&, {a, ListConvolve[ {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, 2], mask}, 2], 1 - mask} where Mask[list_] := Array[Mod[#1 + #2, 2]&, Dimensions[list]] is set up so that alternating checkerboards of cells are updated on successive steps. … And what one sees at least roughly is that right around the phase transition there are patches of black and white of all sizes, forming an approximately nested random pattern.

The pattern corresponding to each point is the limit of Nest[Flatten[1 + {c #, Conjugate[c] #}]&, {1}, n] when n  ∞ .