The rule used is additive, and takes a cell to be black whenever an odd number of its neighbors were black on the step before (outer totalistic code 204).

With this setup, each step then corresponds to LifeStep[list_] := With[{p=Flatten[Array[List, {3, 3}, -1], 1]}, With[{u = Split[Sort[Flatten[Outer[Plus, list, p, 1], 1]]]}, Union[Cases[u, {x_, _, _}  x], Intersection[Cases[u, {x_, _, _, _}  x], list]]]] (A still more efficient implementation is based on finding runs of length 3 and 4 in Sort[u] .)

(Outer totalistic codes specify rules; the first rule makes a particular cell black when any of its five neighbors are black and has code 4094. … This tiling has two different shapes of tile, but here both are treated the same by the cellular automaton rule, which is given by an outer totalistic code number.

And very often the whole animal is covered by a fairly uniform outer skin.

Some bones in effect just expand by adding material to their outer surface.

Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants. … Given the original list s and the complete prime implicant list p the so-called Quine–McCluskey procedure can be used to find a minimal list of prime implicants, and thus a minimal DNF: QM[s_, p_] := First[Sort[Map[p 〚 # 〛 &, h[{}, Range[Length[s]], Outer[MatchQ, s, p, 1]]]]] h[i_, r_, t_] := Flatten[Map[h[Join[i, r 〚 # 〛 ], Drop[r, #], Delete[Drop[t, {}, #], Position[t 〚 All, # 〛 ], {True}]]] &, First[Sort[Position[#, True] &, t]]]], 1] h[i_, _, {}] := {i} The number of steps required in this procedure can increase exponentially with the length of p .

Properties [of difference patterns] In rule 126, the outer edges of the region of change always expand by exactly one cell per step.

Each collection of such functions can be obtained from lists of vectors representing 1D Walsh functions by using Outer[Outer[Times, ##] &, b, b, 1, 1] , or equivalently Map[Transpose, Map[# b &, b, {2}]] .

Note that the outer boundaries of patterns like those on page 330 formed by n random walks tend to become rougher when t is much larger than Log[n] . … As discussed in the note below, this can be viewed as a consequence of the fact that the probability distribution in a random walk depends only on Sum[Outer[Times, e 〚 s 〛 , e 〚 s 〛 ], {s, Length[e]}] and not on products of more of the e 〚 s 〛 .

Given a list of string specifications, a step in the evolution of the multiway system corresponds to Select[Union[Flatten[Outer[Plus, diff, list, 1], 1]], Abs[#]  # &]