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.

Reversible mobile automata can for instance be constructed using Table[(IntegerDigits[i, 2, 3]  If[First[#]  0, {#, -1}, {Reverse[#], 1}]&)[IntegerDigits[perm 〚 i 〛 , 2, 3]], {i, 8}] where perm is an element of Permutations[Range[8]] .

And indeed, as we discuss in Chapter 12 , it seems likely that above a fairly low threshold the vast majority of underlying rules can in fact in some way or another support arbitrarily complex computations—potentially allowing something one might call intelligence in a vast range of very different universes.

But as soon as one allows dependence on slightly longer-range features of the network, much more complicated behavior immediately Evolution of network systems whose rules involve the addition of new nodes.

But now every day we see computations being done with a vast range of different kinds of data—from numbers to text to images to almost anything else.

In the mid-1960s David Raup used early computer graphics to generate pictures for various ranges of parameters, but perhaps because he considered only specific classes of molluscs there emerged from his work the belief that parameters of shells are greatly constrained—with explanations being proposed based on optimization of such features as strength, relative volume, and stability when falling through water. … And in fact, despite elaborate efforts of computer graphics it has proved rather difficult with parametrizations based on Frenet frames to produce shells that have a reasonable range of realistic shapes.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.

In such a rule, given a list of how many neighbors around a given cell (out of s possible) make the cell turn black the outer totalistic code for the rule can be obtained from Apply[Plus, 2^Join[2 list, 2 Range[s + 1] - 1]]

Of all k k 2r + 1 rules with k colors and range r it turns out that there are always exactly k 2r + 1 additive ones—each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k - 1 .

And to address this question, what I will do in this section is to consider a generalization of cellular automata in which each cell is not just black or white, but instead can have any of a continuous range of possible levels of gray.