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# /. … (OddQ[Length[#]] &), {2}] (see page 358 ) • Flatten[Table[Map[{t, #} &, Fold[Flatten[{#1, #1 + #2}] &, 0, Flatten[2^(Position[ Reverse[IntegerDigits[t, 2]], 1] - 1)]]], {t, 2 n - 1}], 1] (see page 870 ) • Map[Map[FromDigits[#, 2] &, Transpose[Partition[#, 2]]] &, Position[Nest[{{#, #}, {#}} &, 1, n], 1] - 1] (see page 509 ) A formatting hack giving the same visual pattern is DisplayForm[Nest[SubsuperscriptBox[#, #, #] &, "1", n]]

Implementation [of patterning model] Given a 2D array of values a and a list of weights w , each step in the evolution of the system corresponds to WeightedStep[w_List, a_] := Map[If[# > 0, 1, 0]&, Sum[w 〚 1 + i 〛 Apply[Plus, Map[RotateLeft[a, #]&, Layer[i]]], {i, 0, Length[w] - 1}], {2}] Layer[n_] := Layer[n] = Select[Flatten[Table[{i, j}, {i, -n, n}, {j, -n, n}],1], MemberQ[#, n| - n]&]

Particularly dramatic are the concatenation systems discussed on page 913 , as well as successive rows in nested patterns such as Flatten[IntegerDigits[NestList[BitXor[#, 2 #] &, 1, 500], 2]] and sequences based on numbers such as Flatten[Table[If[GCD[i, j]  0, 1, 0], {i, 1000}, {j, i}]] (see page 613 ).

For all initial conditions this depth seems at first to increase linearly, then to decrease in a nested way according to FoldList[Plus, 0, Flatten[Table[ {1, 1, Table[-1, {IntegerExponent[i, 2] + 1}]}, {i, m}]]] This quantity alternates between value 1 at position 2 j and value j at position 2 j - j + 1 .

Groups and semigroups [and operator systems] With k possible values for each variable, the forms of operators allowed by axiom systems for group theory and semigroup theory correspond to multiplication tables for groups and semigroups with k elements.

Rule equivalences [for cellular automata] The table below gives basic equivalences between elementary cellular automaton rules.

The centers of an array of regular hexagons are given by Table[{i √ 3 , j}, {i, 1, m}, {j, Mod[i, 2], n, 2}] .

Implementation [of causal networks] Given a list of successive positions of the active cell, as from Map[Last, MAEvolveList[rule, init, t]] (see page 887 ), the network can be generated using MAToNet[list_] := Module[{u, j, k}, u[_] = ∞ ; Reverse[ Table[j = list 〚 i 〛 ; k = {u[j - 1], u[j], u[j + 1]}; u[j - 1] = u[j] = u[j + 1] = i; i  k, {i, Length[list], 1, -1}]]] where nodes not yet found by explicit evolution are indicated by ∞ .

The arrays of values generated can be thought of as being like truth tables.

= {} Given a set of sequences of values represented by a particular network, the set obtained after one step of cellular automaton evolution is given by NetCAStep[{k_, r_, rtab_}, net_] := Flatten[ Map[Table[# /. … = {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes. … Table[(a_  m 〚 i 〛 )  (a  i), {i, Length[m]}]]