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In general, if a period m is possible then so must all periods n for which p = {m, n} satisfies OrderedQ[Transpose[If[MemberQ[p/#, 1], Map[Reverse, {p/#, #}], {#, p/#}]] &[2^IntegerExponent[p, 2]]] Extensions of this to other types of systems seem difficult to find, but it is conceivable that when viewed as continuous mappings on a Cantor set (see page 869 ) at least some cellular automata might exhibit similar properties.
Implementation [of patterning model] Given a 2D array of values a and a list of weights w , each step in the evolution of the system corresponds to WeightedStep[w_List, a_] := Map[If[# > 0, 1, 0]&, Sum[w 〚 1 + i 〛 Apply[Plus, Map[RotateLeft[a, #]&, Layer[i]]], {i, 0, Length[w] - 1}], {2}] Layer[n_] := Layer[n] = Select[Flatten[Table[{i, j}, {i, -n, n}, {j, -n, n}],1], MemberQ[#, n| - n]&]
MemberQ[#, j] &]] - 1
MemberQ[c, #], Append[c, #], AStep[c]]& [f[c] + f[{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}]] f[a_]:=a 〚 Random[Integer, {1, Length[a]}] 〛 This implementation can easily be extended to any type of lattice and any number of dimensions.
If m is of the form 2 j , this implies a maximum period for any a of m/4 , achieved when MemberQ[{3, 5}, Mod[a, 8]] . … Maximal period is assured when in addition PrimeQ[2 n - 1] .) … This particular idea did not work well, but generalizations based on the recurrence f[n_] :=Mod[f[n - p] + f[n - q], 2 k ] have been studied extensively, for example with p = 24 , q = 55 .
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