Notes

Chapter 6: Starting from Randomness

Section 6: Special Initial Conditions


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, #] &, #]], 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+Sqrt[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

From Stephen Wolfram: A New Kind of Science [citation]