The program in the multiregister machine can be converted to a program for the 2-register machine according to RMToRM2[prog_] := Module[{segs, adrs}, segs = MapIndexed[seg, prog]; adrs = FoldList[Plus, 1, Map[Length, segs]]; MapIndexed[#1 /.

The Thue–Morse sequence discussed on page 890 can be obtained from it by applying 1 - Mod[Flatten[Partition[FoldList[Plus, 0, list], 1, 2]], 2] (b) The n th element is simply Mod[n, 2] .

Among such computations are Plus , Times , Divide , Det and LinearSolve for integers, as well as determining outcomes in additive cellular automata (see page 609 ).

TMToRM[rules_] := Module[{segs, adrs}, segs = Map[TMCompile, rules] ; adrs = Thread[Map[First, rules]  Drop[FoldList[Plus, 1, Map[Length, segs]], -1]]; MapIndexed[(# /.

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 .

Sierpiński pattern Other ways to generate step n of the pattern shown here in various orientations include: • Mod[Array[Binomial, {2, 2} n , 0], 2] (see pages 611 and 870 ) • 1 - Sign[Array[BitAnd, {2, 2} n , 0]] (see pages 608 and 871 ) • NestList[Mod[RotateLeft[#] + #, 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • NestList[Mod[ListConvolve[{1, 1}, #, -1], 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • IntegerDigits[NestList[BitXor[2#, #] &, 1, 2 n - 1], 2, 2 n ] (see page 906 ) • NestList[Mod[Rest[FoldList[Plus, 0, #]], 2] &, Table[1, {2 n }], 2 n - 1] (see page 1034 ) • Table[PadRight[ Mod[CoefficientList[(1 + x) t - 1 , x], 2], 2 n - 1], {t, 2 n }] (see pages 870 and 951 ) • Reverse[Mod[CoefficientList[Series[1/(1 - (1 + x)y), {x, 0, 2 n - 1}, {y, 0, 2 n - 1}], {x, y}], 2]] (see page 1091 ) • Nest[Apply[Join, MapThread[ Join, {{#, #}, {0 #, #}}, 2]] &, {{1}}, n] (compare page 1073 ) The positions of black squares can be found from: • Nest[Flatten[2# /.

The number of sequences s n of length n that can actually occur is given by Apply[Plus, Flatten[MatrixPower[m, n]]] where the adjacency matrix m is given by MapAt[(1 + #) &, Table[0, {Length[net]}, {Length[net]}], Flatten[MapIndexed[{First[#2], Last[#1]} &, net, {2}], 1]] For rule 32, for example, s n turns out to be Fibonacci[n + 3] , so that for large n it is approximately GoldenRatio n .

Any real number x can be represented as a set of integers using for example Rest[FoldList[Plus, 1, ContinuedFraction[x]]] but except when x is rational this list is not finite.

With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] .

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!