But I rather suspect that—much as in the analogy between birds and airplanes—it will in the end be possible to set up algorithms that achieve the same basic functions but work satisfactorily even on standard sequential-processing computers.

The following definition also handles the more general case of r neighbors: CAStep[TotalisticCARule[rule_List, r_Integer], a_List] := rule 〚 -1 - Sum[RotateLeft[a, i], {i, -r, r}] 〛 One can generate the representation of totalistic rules used by these functions from code numbers using ToTotalisticCARule[num_Integer, k_Integer, r_Integer] := TotalisticCARule[IntegerDigits[num, k, 1 + (k - 1)(2r + 1)], r]

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.

And already by the 1880s Georg Cantor and others had constructed completely discontinuous functions, in which the idea of treating numbers as continuous variables where only the size matters was called into question. But until almost the 1970s, and the emergence of fractal geometry and chaos theory, these functions were largely considered as mathematical curiosities, of no practical relevance.

The entropy as a function of time for systems of the type shown in case (b) from page 447 .

Thus for example, instead of getting results in terms of continuous functions, Cantor sets very often appear.

Standard examples of recursive sequences that do not come from linear recurrence relations include factorial f[1] = 1; f[n_] := n f[n - 1] and Ackermann functions (see below ).

For higher degrees it is known that such general formulas must involve other functions.

Then, for example, one can define CAStep[ElementaryCARule[rule_List], a_List] := rule 〚 8 - (RotateLeft[a] + 2 (a + 2 RotateRight[a])) 〛 CAStep[GeneralCARule[rule_, r_Integer:1], a_List] := Partition[a, 2r + 1, 1, r + 1] /. rule CAStep[FunctionCARule[f_, r_Integer:1], a_List] := Map[f, Partition[a, 2r + 1, 1, r + 1]] Note that the second two definitions have been generalized to allow rules that involve r neighbors on each side. … For efficiency in early versions of Mathematica, explicit rule lists in the second definition can be preprocessed using Dispatch[rules] , and functions in the third definition preprocessed using Compile[{{x, _Integer, 1}}, body] .

Various modifications can be made involving for example weighting with a risk function before maximizing.