In a rectangular region, the position is given by Mod[a t, {w, h}] and every point will be visited if the parameters have irrational ratios.

MapIndexed[ #1  First[#2] &, Union[Map[# 〚 1, 1 〛 &, #]]] &[ With[{b = Ceiling[Log[2, k]] - 1}, Flatten[Table[ {Table[{Table[{{m, i, n, d}, c}  {{m, Mod[i, 2 n - 1 ], n - 1, d}, Quotient[i, 2 n - 1 ], 1}, {n, 2, b}, {i, 0, 2 n - 1}], Table[{ {m, i, 1, d}, c}  {{m, -1, 1, d}, i, d}, {i, 0, 1}], Table[ {{m, -1, n, d}, c}  {{m, -1, n + 1, d}, c, d}, {n, b - 1}], {{m, -1, b, d}, c}  {{0, 0, m}, c, d}}, {d, -1, 1, 2}], Table[{{i, n, m}, c}  {{ i + 2 n c, n + 1, m}, c, -1}, {n, 0, b - 1}, {i, 0, 2 n - 1}], With[{r = 2 b }, Table[ If[i + r c ≥ k, {}, Cases[rule, ({m, i + r c}  {x_, y_, z_})  {{i, b, m}, c}  {{x, Mod[y, r], b, z}, Quotient[y, r], 1})]], {i, 0, r - 1}]]}, {m, s}, {c, 0, 1}]]]] Some of these states are usually unnecessary, and in the main text such states have been pruned.

The distance gone by the ball at a given time is x = v t - a t 2 /2 , and its orientation is Mod[x, 2 π r] .

(b) (Thue–Morse sequence) The color s[n] of the element at position n is given by 1 - Mod[DigitCount[n - 1, 2, 1], 2] . … The first 2 m elements in the sequence can be obtained from (see page 1081 ) (CoefficientList[Product[1 - z 2 s , {s, 0, m - 1}], z] + 1)/2 The first n elements can also be obtained from (see page 1092 ) Mod[CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/ (2(1 + x)), {x, 0, n - 1}], x], 2] The sequence occurs many times in this book; it can for example be derived from a column of values in the rule 150 cellular automaton pattern discussed on page 885 . … (d) (Cantor set) The color of the element at position n is given by If[FreeQ[IntegerDigits[n - 1, 3], 1], 1, 0] , which turns out to be equivalent to If[OddQ[n], Sign[Mod[Binomial[n - 1, (n - 1)/2], 3]], 0, 1] There are 3 t elements after t steps, of which 2 t are black.

Even in the last case shown, the arrangement of stripes eventually becomes completely regular, with the n th new stripe being produced at step n 2 + 21n/2 - {6, 5, -4, 3} 〚 Mod[n, 4] + 1 〛 /2 .

Implementation [of cyclic tag systems] With the rules for the cyclic tag system on page 95 given as {{1, 1}, {1, 0}} , the evolution can be obtained from CTEvolveList[rules_, init_, t_] := Map[Last, NestList[CTStep, {rules, init}, t]] CTStep[{{r_, s___}, {0, a___}}] := {{s, r}, {a}} CTStep[{{r_, s___}, {1, a___}}] := {{s, r}, Join[{a}, r]} CTStep[{u_, {}}] := {u, {}} The leading elements on many more than t successive steps can be obtained directly from CTList[rules_, init_, t_] := Flatten[Map[Last, NestList[CTListStep, {rules, init}, t]]] CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]],Flatten[rules 〚 Mod[Flatten[Position[list, 1]], Length[rules], 1] 〛 ]}

Pattern (a) on page 213 can be specified as {{2, -1, 2, 3}, {{0, 0, 0, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}}} Given this, a complete nx by ny array filled with this pattern can be constructed from c[{d1_, d2_, d3_, d4_}, {x_, y_}] := With[{d = d1 d2 + d1 d4 + d3 d4}, Mod[{{d2 x + d4 x + d3 y, d4 x - d1 y}}/d, 1]] Fill[{dlist_, data_}, {nx_, ny_}] := Array[c[dlist, {##}] &, {nx, ny}] /.

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 .

(An example is NestList[Mod[2 #, 1]&, N[ π /4, 40], 200] ; Map[Precision, list] gives the number of significant digits of each element in the list.) … As an example, consider the iterated map x  Mod[2x, 1] discussed in the main text. … (Starting with initial condition x the digit sequence at step n is essentially IntegerDigits[Mod[2 n Floor[2 53 x], 2 53 ], 2, 53] on the computer, and Flatten[IntegerDigits[IntegerDigits[ Mod[2 n Floor[10 12 x], 10 12 ], 10, 12], 2, 4]] on the calculator.

(h) Mod[Quotient[s, 2 n ], 2] extracts the digit associated with 2 n in the base 2 digit sequence of s .