Other universal functions include rules 1, 45 and 202 ( If[a  1, b, c] ), but not 30, 60 or 110. For large n roughly 1/4 of all n -input functions are universal.

Mathematical interpretation of cellular automata In the context of pure mathematics, the state space of a 1D cellular automaton with an infinite number of cells can be viewed as a Cantor set. … The pictures above show representations of the mappings corresponding to various rules, obtained by plotting Sum[a[t + 1, i] 2 -i , {i, -n, n}] against Sum[a[t, i] 2 -i , {i, -n, n}] for all possible choices of the a[t, i] . … In the pictures below, this map has the form Mod[2x, 1] (compare page 153 ).

The vertical distance moved at the n th horizontal position is Floor[n h] - Floor[(n - 1) h] , and the sequence obtained from this (which contains only terms Floor[h] and Floor[h] + 1 ) provides a unique representation for h . … Given a sequence of length n , an approximation to h can be reconstructed using Max[MapIndexed[#1/First[#2] &, FoldList[Plus, First[list], Rest[list]]]] The fractional part of the result obtained is always an element of the Farey sequence Union[Flatten[Table[a/b, {b, n}, {a, 0, b}]]] (See also pages 892 , 932 and 1084 .)

Of all k k 2r + 1 rules with k colors and range r it turns out that there are always exactly k 2r + 1 additive ones—each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k - 1 .

Successive steps in the iterative procedure used on this page are given by Move[list_] := (If[Cost[#] < Cost[list], #, list] &)[ MapAt[1 - # &, list, Random[Integer, {1, Length[list]}]]] while those in the procedure on page 347 have ≤ in place of < . The third curve shown on page 346 is obtained from Table[Cost[IntegerDigits[i, 2, n]], {i, 0, 2 n - 1}] There is no single ordering that makes all states which can be reached by changing a single square be adjacent.

In the first case shown, the picture produced after t steps has 4 × 3 t - k - 1 regions with 3 × 2 k edges. … The number of nodes at distance up to r from a given node is at most 1 + Sum[c[i] + c[i - 1], {i, n}] where c[i_] := 2^DigitCount[i, 2] .

(The probability for s randomly chosen integers to be relatively prime is 1/Zeta[s] .) … In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

And in practice the n th digit can be found just by computing slightly over n terms of the sum, according to Round[FractionalPart[ Sum[FractionalPart[PowerMod[2, n - k, k]/k], {k, n}] + Sum[2 n - k /k, {k, n + 1, n + d}]]] where several values of d can be tried to check that the result does not change. … The same basic approach as for Log[2] can be used to obtain base 16 digits in π from the following formula for π : Sum[16 -k (4/(8k + 1) - 2/(8k + 4) - 1/(8k + 5)-1/(8k + 6)), {k, 0, ∞ }] A similar approach can also be used for many other constants that can be viewed as related to values of PolyLog .

The arrangement of triangles at step t can be obtained from a substitution system according to With[{ ϕ = GoldenRatio}, Nest[# /. a[p_, q_, r_]  With[{s = (p + ϕ q) (2 - ϕ )}, {a[r, s, q], b[r, s, p]}] /. b[p_, q_, r_]  With[{s = (p + ϕ r) (2 - ϕ )}, {a[p, q, s], b[ r, s, q]}] &, a[{1/2, Sin[2 π /5] ϕ }, {1, 0}, {0, 0}], t]] This pattern can be viewed as generalizations of the pattern generated by the 1D Fibonacci substitution system (c) on page 83 . As discussed on page 903 , this 1D sequence can be obtained by looking at how a line with GoldenRatio slope cuts through a 2D lattice of squares.

(The rule numbers here follow the scheme on page 927 with offsets {{-1, 0}, {0, -1}, {0, 1}, {1, 0}} ). … The pictures below show examples for a rule that allows growth except when there are exactly 1, 3 or 4 neighbors (totalistic constraint 242). … The pictures below show 1D generalized aggregation systems with various templates.