[No text on this page] Examples of multiway systems that generate different fractions of possible strings, and in effect range from being highly incomplete to highly inconsistent.

f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

Some models of crystal growth, however, call for long-range effects such as a temperature field which changes throughout the crystal in an effectively instantaneous way. It turns out, however, that many seemingly long-range effects can actually be captured quite easily in cellular automata.

Applying BitReverseOrder to this matrix yields a matrix which has an essentially nested form, and for size n = 2 s can be obtained from Nest[With[{c = BitReverseOrder[Range[0, Length[#] - 1]/ Length[#]]}, Flatten2D[MapIndexed[#1 {{1, 1}, {1, -1} (-1)^c 〚 Last[#2] 〛 } &, #, {2}]]] &, {{1}}, s] Using this structure, one obtains the so-called fast Fourier transform which operates in n Log[n] steps and is given by With[{n = Length[data]}, Fold[Flatten[Map[With[ {k = Length[#]/2}, {{1, 1}, {1, -1}} . {Take[#, k], Drop[ #, k](-1)^(Range[0, k - 1]/k)}] &, Partition[##]]] &, BitReverseOrder[data], 2^Range[Log[2, n]]]/ √ n ] (See also page 1080 .)

The areas of solid black thus correspond to ranges of parameters in the underlying rule for which the patterns obtained always reach a particular position. … And the presence of such structures implies that at least with some ranges of parameters, even very small changes in underlying rules can lead to large changes in certain aspects of the patterns that are produced.

The pictures on the facing page show typical examples of pigmentation patterns in animals, and demonstrate that even across a vast range of different types of animals just a few kinds of patterns occur over and over again. … A certain range of patterns emerges—almost all of which turn out to be quite similar to patterns that one sees on actual animals.

And indeed my guess is that the essential features of all sorts of intricate structures that are seen in living systems can actually be reproduced with remarkably simple rules—making it for example possible to use technology to repair or replace a whole new range of functions of biological tissues and organs. … But while this is enough to see a tremendous range of behavior, there is no guarantee that one will in fact run across whatever specific features one is looking for.

Forms of living systems This book has shown that even with underlying rules of some fixed type a vast range of different forms can often be produced. And this makes it reasonable to expect that with appropriate genetic programs the chemical building blocks of life on Earth should in principle allow a vast range of forms.

Starting from the set of all possible sequences, as given by AllNet[k_:2] := {Thread[(Range[k] - 1)  1]} this then yields for rule 126 the network {{0  1, 1  2}, {1  3, 1  4}, {1  1, 1  2}, {1  3, 0  4}} It is always possible to find a minimal network that represents a set of sequences. … = {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes. … To obtain such trimmed networks one can apply the function TrimNet[net_] := With[{m = Apply[Intersection, Map[FixedPoint[ Union[#, Flatten[Map[Last, net 〚 # 〛 , {2}]]]&, #]&, Map[List, Range[Length[net]]]]]}, net 〚 m 〛 /.

Very similar results seem to be obtained for constraints in a wide range of discrete systems.