Fractal dimensions [of additive cellular automata]

The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952) from a single initial 1 can be found using

g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}]

The fractal dimension of this pattern is then given by the large m limit of

Log[k,g[w, k,k^{m + 1}]/g[w, k, k^{m}]]

When k is prime it turns out that this can be computed as

d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[{s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k^{s} - 1]]]]]]]

For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58. For rule 150 d[{1, 1, 1}] = Log[2, 1 + √5] ≃ 1.69. (See page 58.) For the other rules on page 952:

d[{1, 1, 0, 1, 0}] = Log[2, Root[4 + 2 # - 2 #^{2} - 3 #^{3} + #^{4} &, 2]] ≃ 1.72

d[{1, 1, 0, 1, 1}] = Log[2, Root[-4 + 4 # + #^{2} - 4 #^{3} + #^{4} &, 2]] ≃ 1.80

Other cases include (see page 870):

d[{1, 0, 1}, k] = 1 + Log[k, (k + 1)/2]

d[{1, 1, 1}, 3] = Log[3, 6] ≃ 1.63

d[{1, 1, 1}, 5] = Log[5, 19] ≃ 1.83

d[{1, 1, 1}, 7] = Log[7, Root[-27136 + 23280 # - 7288 #^{2} + 1008 #^{3} - 59 #^{4} + #^{5} & , 1]] ≃ 1.85