Implementation of digit sequences A whole number n can be converted to a sequence of digits in base k using IntegerDigits[n,k] or (see also page 1094 ) Reverse[Mod[NestWhileList[Floor[#/k] &, n, # ≥ k &], k]] and from a sequence of digits using FromDigits[list,k] or Fold[k #1 + #2 &, 0, list] For a number x between 0 and 1, the first m digits in its digit sequence in base k are given by RealDigits[x, k, m] or Floor[k NestList[Mod[k #, 1]&, x, m - 1]] and from these digits one can reconstruct an approximation to the number using FromDigits[{list, 0}, k] or Fold[#1/k + #2 &, 0, Reverse[list]]/k

The tetrahedron network from page 476 is for example given in this representation by {1  {2, 3, 4}, 2  {1, 3, 4}, 3  {1, 2, 4}, 4  {1, 2, 3}} The list of nodes reached by following up to n connections from node i are then given by NodeLists[g_, i_, n_] := NestList[Union[Flatten[# /. g]] &, {i}, n] The network distance corresponding to the length of the shortest path between two nodes is given by Distance[g_, {i_, j_}] := Length[NestWhileList[ Union[Flatten[# /. g]] &, {i}, !

Rule 94 [with simple initial conditions] With appropriate initial conditions this class 2 rule can yield both nested and random behavior, as shown below.

It exhibits a nested structure, and can be obtained as in the pictures below from the evolution of a 2D substitution system, or equivalently from a Kronecker product as in Nest[Flatten2D[Map[# {{1, 1}, {1, -1}} &, #, {2}]] &, {{1}}, s] with Flatten2D[a_] := Apply[Join, Apply[Join, Map[Transpose,a], {2}]] (c) is known as dyadic or Paley order. … Walsh functions can correspond to nested sequences. … However, the nested structure of m in natural order allows evaluation in only about n Log[n] steps using Nest[Flatten[Transpose[Partition[#, 2] . {{1, 1}, {1, -1}}]] &, data, Log[2, Length[data]]] This procedure is similar to the fast Fourier transform discussed below.

Simple geometries [in branching model] Page 357 shows how some of the nested patterns commonly seen in this book can be produced by the growth processes shown here.

And in fact, as the picture below demonstrates, these successive digit sequences form a pattern that shows an intricate nested structure. … The overall pattern has an intricate nested form.

All eight pictures on the facing page were generated from the two-dimensional substitution systems shown, and thus correspond to purely nested patterns. But except for the last picture on each row—which happen to be dominated by large areas of essentially uniform color—it is remarkably difficult for us to tell that the patterns are nested.

If this behavior always has an obvious repetitive or nested form then it will often be quite straightforward to analyze. … As mentioned in Chapter 10 , however, I do know of a few systems based on numbers for which a fairly complete analysis can be given even though the overall behavior is not repetitive or nested or otherwise obviously simple.

If one starts from more than a single non- 0 element, then it is still true that a nested pattern will be produced if f is both associative and commutative. And from the discussion on page 952 this means that any rule that shows generalized additivity must always yield a nested pattern. But if f is not commutative, then even if it is associative, non-nested patterns can be produced.

These operations can be thought of as finding elements in nested Pascal's triangle patterns produced by k -color additive cellular automata. Korec showed that finding elements in the nested pattern produced by the k = 3 cellular automaton with rule {{1, 1, 3}, {2, 2, 1}, {3, 3, 2}} 〚 #1, #2 〛 & (compare page 886 ) was also enough.