The rule on page 82 can then be given simply as s[1, 0]  s[0, 1, 0] while the rule on page 85 becomes {s[0, 1, 0]  s[0, 0, 1], s[0]  s[0, 1, 0]} The Flat attribute of s makes these rules apply not only for example to the whole sequence s[1, 0, 1, 0] but also to any subsequence such as s[1, 0] . … And with this setup, t steps of evolution can be found with SSSEvolveList[rule_, init_s, t_Integer] := NestList[(# /. rule)&, init, t] Note that as an alternative to having s be Flat , one can explicitly set up rules based on patterns such as s[x___, 1, 0, y___]  s[x, 0, 1, 0, y] .

The nested structure of the patterns produced is thus a direct consequence of the nesting seen in the patterns of these digit sequences, as shown on page 117 . Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences.

In example (c), the elements are again correlated: the growth is by an average of ( √ 5 - 1)/2 ≃ 0.618 elements at each step, and the first elements on alternate steps form the same nested sequence as obtained from the substitution system {1  {1, 0}, 0  {1}} . … In example (e), the frequency of 1's is again about 3/4, but now {0, 0} occurs with frequency 0.05, {1, 1} occurs with frequency 0.55, while {0, 0, 0} and {0, 1, 0} cannot occur.

Most of the circles added at a given step are not the same size, however, making the overall geometry not straightforwardly nested. … In the limit of an infinite number of steps the filling fraction tends to 1, while the region left unfilled has a fractal dimension of about 1.3057. To achieve filling fraction 1 requires arbitrarily small circles, but there are many different arrangements of circles that will work, some not even close to nested.

The first one on the bottom (with 63 comparisons) has a nested structure and uses the method invented by Kenneth Batcher in 1964: Flatten[Reverse[Flatten[With[{m = Ceiling[Log[2, n]] - 1}, Table[With[{d = If[i  m, 2 t , 2 i + 1 - 2 t ]}, Map[ {0, d} + # &, Select[Range[n - d], BitAnd[# - 1, 2 t ]  If[i  m, 0, 2 t ] &]]], {t, 0, m}, {i, t, m}]], 1]], 1] The second one on the bottom also uses 63 comparisons, while the last one is the smallest known for n = 16 : it uses 60 comparisons and was invented by Milton Green in 1969.

The sequence on step t can be obtained from Nest[Join[#, 1 - #] &, {1}, t - 1] . … The color of the element at position n is given by 2 - (Floor[(n + 1) GoldenRatio] - Floor[n GoldenRatio]) (see page 904 ), while the position of the k th white element is given by the so-called Beatty sequence Floor[k GoldenRatio] . … The resulting curve has a nested form, with envelope n^Log[3, 2] .

And if one has nested lemmas one can end up with a proof that is in effect like a tree. … But while this is familiar from algebraic mathematics and from the operation of Mathematica it is not the model of proofs that has traditionally been used in mainstream mathematical logic.

. , or effectively Nest[ ω # &, ω , ω ] . ε 0 is the smallest solution to ω ε  ε . … And at least for a while the ordinary axioms of set theory can be used to study the sets that arise.

Then, for example, the rule for the mobile automaton shown on page 71 can be given as {{1, 1, 1}  {0, 1}, {1, 1, 0}  {0, 1}, {1, 0, 1}  {1, -1}, {1, 0, 0}  {0, -1}, {0, 1, 1}  {0, -1}, {0, 1, 0}  {0, 1}, {0, 0, 1}  {1, 1}, {0, 0, 0}  {1, -1}} where the left-hand side in each case gives the value of the active cell and its left and right neighbors, while the right-hand side consists of a pair containing the new value of the active cell and the displacement of its position. … With a rule given in this form, each step in the evolution of the mobile automaton corresponds to the function MAStep[rule_, {list_List, n_Integer}] /; (1 < n < Length[list]) := Apply[{ReplacePart[list, #1, n], n + #2}&, Replace[Take[list, {n - 1, n + 1}], rule]] The complete evolution for many steps can then be obtained with MAEvolveList[rule_, init_List, t_Integer] := NestList[MAStep[rule, #]&, init, t] (The program will run more efficiently if Dispatch is applied to the rule before giving it as input.)

And in a first approximation these rules can usually be thought of as specifying that every sentence must be constructed from various independent nested phrases, much as in a context-free grammar (see above ). … (One obvious general deviation from the context-free model is that in practice subordinate clauses can never be nested too deep if a sentence is expected to be understood.) … And following this, there were for a while many efforts to formulate precise models for human languages, and to relate these to properties of the brain.