In particular, if a state of a cellular automaton has a certain spatial period, given by the minimum positive m for which RotateLeft[list, m]  list , then this state can never evolve to one with a larger spatial period.

Starting with a list of the initial conditions for s steps, the configurations for the next s steps are given by Append[Rest[list], Map[Mod[Apply[Plus, Flatten[c #]], 2]&, Transpose[ Table[RotateLeft[list, {0, i}], {i, -r, r}], {3, 2, 1}]]] where r = (Length[First[c]] - 1)/2 .

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# /.

Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

A single step in evolution of a general cellular automaton with state a and rule number num is then given by Map[IntegerDigits[num, k, k^Length[os]] 〚 -1 - # 〛 &, Apply[Plus, MapIndexed[k^(Length[os] - First[#2]) RotateLeft[a, #1] &, os]], {-1}] or equivalently by Map[IntegerDigits[num, k, k^Length[os]] 〚 -# - 1 〛 &, ListCorrelate[Fold[ReplacePart[k #1, 1, #2 + r + 1] &, Array[0 &, Table[2r + 1, {d}]], os], a, r + 1], {d}]

Random walks In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from NestList[(# + (-1)^Random[Integer])&, 0, t] or equivalently FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]] A generalization to d dimensions is then FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]] A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t .

The basic examples are then rules of the form RotateLeft[a] ⊕ RotateRight[a] —analogs of rule 90, but with other addition operations (compare page 886 ).

The undecidability of PCP can be seen to follow from the undecidability of the halting problem through the fact that the question of whether a tag system of the kind on page 93 with initial sequence s ever reaches a halting state (where none of its rules apply) is equivalent to the question of whether there is a way to satisfy the PCP constraint TSToPCP[{n_, rule_}, s_] := Map[Flatten[IntegerDigits[#, 2, 2]] &, Module[{f}, f[u_] := Flatten[Map[{1, #} &, 3u]]; Join[Map[{f[Last[#]], RotateLeft[f[First[#]]]} &, rule], {{f[s], {1}}}, Flatten[ Table[{{1, 2}, Append[RotateLeft[f[IntegerDigits[j, 2, i]]], 2]}, {i, 0, n - 1}, {j, 0, 2 i - 1}], 1]]], {2}] Any PCP constraint can also immediately be related to the evolution of a multiway tag system of the kind discussed in the note below. … When a sequence of blocks leads to upper and lower strings that disagree, the rectangle is left white.

Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] .

But in fact even the vacuum Einstein equations for complete universes (with no points left out) have solutions that show curvature. … And a variety of inhomogeneous solutions with no singularities are also known—an example being the 1962 Ozsváth–Schücking rotating vacuum.