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2D class 4 cellular automata No 5- or 9-neighbor totalistic rules nor 5-neighbor outer totalistic ones appear to yield class 4 behavior with a white background. But among 9-neighbor outer totalistic rules there are examples with codes 224 (Game of Life), 226, 4320 (sometimes called HighLife), 5344, 6248, 6752, 6754 and 8416, etc.
(Outer totalistic code 746.)
This rule is outer totalistic 9-neighbor code 224.
Totalistic rules depend only on the total number of black cells in a neighborhood; outer totalistic rules (as in the previous note) also depend on the color of the center cell. … 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]]
This is made possible by the fact that antlers, unlike horns, have a layer of soft tissue on the outside—which delivers the nutrients needed for growth to occur on the outer surface of the bone at their tips.
Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata. … And as the pictures below show, this is true even just for parts of the rules above ( s alone yields outer totalistic code 686 in 2D, and rule 90 in 1D). Ulam also in 1967 considered the pure 2D cellular automaton with outer totalistic code 12 (though he stated its rule in a complicated way).
Limiting shapes [in 2D cellular automata] When growth occurs at the maximum rate the outer boundaries of a cellular automaton pattern reflect the neighborhood involved in its underlying rule (in rough analogy to the Wulff construction for shapes of crystals).
The outer boundaries are somewhat smooth, though definitely not circular.
And after n steps the positions of all tips generated are given simply by Nest[Flatten[Outer[Times, 1 + #, b]] &, {0}, n]
The rules shown here are outer totalistic: (a) 4-neighbor code 468, (b) 4-neighbor code 686 and (c) 8-neighbor code 746.
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