Higher-dimensional generalizations [of substitution systems] The state of a d -dimensional substitution system can be represented by a nested list of depth d . The evolution of the system for t steps can be obtained from SSEvolve[rule_, init_, t_, d_Integer] := Nest[FlattenArray[# /. rule, d] &, init, t] FlattenArray[list_, d_] := Fold[Function[{a, n}, Map[MapThread[Join, #, n] &, a, -{d + 2}]], list, Reverse[Range[d] - 1]] The analog in 3D of the 2D rule on page 187 is {1  Array[If[LessEqual[##], 0, 1] &, {2, 2, 2}], 0  Array[0 &, {2, 2, 2}]} Note that in d dimensions, each black cell must be replaced by at least d + 1 black cells at each step in order to obtain an object that is not restricted to a dimension d - 1 hyperplane.

Examples of cellular automata that produce nested or fractal patterns.

Implementation [of symbolic systems] The evolution for t steps of the first symbolic system shown can be implemented simply by NestList[#/. ℯ [x_][y_]  x[x[y]]&, init, t]

Nesting in rule 45 As illustrated on page 701 , starting from a single black cell on a background of repeated blocks, rule 45 yields a slanted version of the nested rule 90 pattern.

This is similar to picture (c) on page 131 , and is a digit-by-digit version of FoldList[Plus, 0, Table[Apply[Plus, 2 Rest[IntegerDigits[i, 2]] - 1], {i, n}]] Note that although the picture above has a nested structure, the original concatenation sequences are not nested, and so cannot be generated by substitution systems. The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .

Implementation [of 3/2 system] The evolution for t steps of the system at the top of the page can be computed simply by NestList[If[EvenQ[#], 3#/2, 3(# + 1)/2] &, 1, t]

Implementation [of cyclic tag systems] With the rules for the cyclic tag system on page 95 given as {{1, 1}, {1, 0}} , the evolution can be obtained from CTEvolveList[rules_, init_, t_] := Map[Last, NestList[CTStep, {rules, init}, t]] CTStep[{{r_, s___}, {0, a___}}] := {{s, r}, {a}} CTStep[{{r_, s___}, {1, a___}}] := {{s, r}, Join[{a}, r]} CTStep[{u_, {}}] := {u, {}} The leading elements on many more than t successive steps can be obtained directly from CTList[rules_, init_, t_] := Flatten[Map[Last, NestList[CTListStep, {rules, init}, t]]] CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]],Flatten[rules 〚 Mod[Flatten[Position[list, 1]], Length[rules], 1] 〛 ]}

But the point is that when the initial number of black and white cells is exactly equal—corresponding to a phase transition point—a typical configuration of rule 184 will contain domains with a nested distribution of sizes. … A typical result is that correlations between colors of different cells fall off like a power of distance—with the specific power depending only on general features of the nested patterns formed, and not on most details of the system.

But in one case a nested pattern is produced. … Note the appearance of a slanted version of the nested pattern from rule 90.

Rule 218 [with simple initial conditions] If pairs of adjacent black cells appear anywhere in its initial conditions this class 2 rule gives uniform black, but if none do it gives a rule 90 nested pattern.