Implementation [of 2D cellular automata] An n × n array of white squares with a single black square in the middle can be generated by PadLeft[{{1}}, {n, n}, 0, Floor[{n, n}/2]] For the 5-neighbor rules introduced on page 170 each step can be implemented by CAStep[rule_, a_] := Map[rule 〚 10 - # 〛 &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}] where rule is obtained from the code number by IntegerDigits[code, 2, 10] . For the 9-neighbor rules introduced on page 177 CAStep[rule_, a_] := Map[rule 〚 18 - # 〛 &, ListConvolve[{{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}, a, 2], {2}] where rule is given by IntegerDigits[code, 2, 18] . In d dimensions with k colors, 5-neighbor rules generalize to (2d + 1) -neighbor rules, with CAStep[{rule_, d_}, a_] := Map[rule 〚 -1 - # 〛 &, a + k AxesTotal[a, d], {d}] AxesTotal[a_, d_] := Apply[Plus, Map[RotateLeft[a, #] + RotateRight[a, #]&, IdentityMatrix[d]]] with rule given by IntegerDigits[code, k, k(2d(k - 1) + 1)] . 9-neighbor rules generalize to 3 d -neighbor rules, with CAStep[{rule_, d_}, a_] := Map[rule 〚 -1 - # 〛 &, a + k FullTotal[a, d], {d}] FullTotal[a_, d_] := Array[RotateLeft[a, {##}] &, Table[3, {d}], -1, Plus] - a with rule given by IntegerDigits[code, k, k((3 d - 1)(k - 1) + 1)] .

[No text on this page] Patterns generated by a sequence of two-dimensional cellular automaton rules. … In each case the base 2 digit sequence for the code number specifies the rule as follows.

[No text on this page] A two-dimensional cellular automaton first shown on page 178 with the rule that if out of the eight neighbors (including diagonals) around a given cell, there are exactly three black cells, then the cell itself becomes black on the next step. … This simple rule produces randomness through the mechanism of intrinsic randomness generation, and this randomness in turn leads to a pattern of growth that takes on an increasingly smooth more-or-less circular form.

Both rules 254 and 90 have fairly simple overall behavior, and yield comparatively small Boolean expressions. Rule 30 has much more complicated behavior and yields Boolean expressions whose size grows rapidly from one step to the next.

For rules (a) and (b) the patterns produced always exhibit features that remain aligned with directions in the underlying grid. But with rule (c) essentially the same rounded pattern is obtained regardless of orientation. The rules shown here are outer totalistic: (a) 4-neighbor code 468, (b) 4-neighbor code 686 and (c) 8-neighbor code 746.

The specific pictures at the bottom of the facing page are for elementary cellular automata with two possible colors for each cell and nearest-neighbor rules. But the same basic idea can be used for cellular automata with rules of any kind. … Examples of Turing machines that emulate cellular automata with rules 90 and 30.

Searching [for doubling rules] No symmetric k = 3 , r = 1 rule yields doubling. General rules can show subtle bugs; rule 1340716537107 for example first fails at n = 24 . … Several different rules that work can behave identically, since up to 6 of the 27 cases in each rule are not sampled with the initial conditions used.

The maximum length DNF for elementary rules after 1 step is 4, and this is achieved by rules 105, 107, 109, 121, 150, 151, 158, 182, 214 and 233. These rules have behavior of quite varying complexity. … In practice, after 2 steps, the maximum length is 9, achieved by rules 107, 121 and 182; after 3 steps, it is 33 achieved by rule 182; after 4 steps, 78 achieved by rule 129; after 5 steps 256 achieved by rules 105 and 150.

One can update the gray level of each cell by using rules that are in a sense a cross between the totalistic cellular automaton rules that we discussed at the beginning of the last chapter and the iterated maps that we just discussed in the previous section . … The picture on the facing page shows what happens with a slightly more complicated rule in which the average gray level is multiplied by 3/2 , and then only the fractional part is kept if the result of this is greater than 1. … The rule shown here takes the new gray level of each cell to be the average of its own gray level and those of its immediate neighbors.

And to see an example of this consider setting up rules like those on page 509 that preserve the planarity of networks. … But the way the rules are set up, all the networks they produce must still be planar. … These are 3-color totalistic rules with codes 294 and 1893.