Transpose[m]  n IdentityMatrix[n] .

At step n , the complete array of cells is Table[If[FreeQ[Transpose[IntegerDigits[{i, j}, k, n]], form], 1, 0], {i, 0, k n - 1}, {j, 0, k n - 1}] where for the pattern on page 187 , k = 2 and form = {0, 1} .

Mod[Binomial[t, n], k] is given for prime k by With[{d = Ceiling[Log[k, Max[t, n] + 1]]}, Mod[Apply[Times, Apply[Binomial, Transpose[ {IntegerDigits[t, k, d] , IntegerDigits[n, k, d] }], {1}]], k]] The patterns obtained for any k are nested.

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 .

The top two (both with 120 comparisons) have a repetitive structure and correspond to standard sorting algorithms: transposition sort and insertion sort.

The following generates explicit lists of n -input Boolean functions requiring successively larger numbers of Nand operations: Map[FromDigits[#, 2] &, NestWhile[Append[#, Complement[Flatten[Table[Outer[1 - Times[##] &, # 〚 i 〛 , # 〚 -i 〛 , 1], {i, Length[#]}], 2], Flatten[#, 1]]] &, {1 - Transpose[IntegerDigits[Range[2 n ] - 1, 2, n]]}, Length[Flatten[#, 1]] < 2 2 n &], {2}] The results for 2-step cellular automaton evolution in the main text were found by a recursive procedure.

Much as for integers, finite lists of real numbers can be encoded as single real numbers—using for example roughly FromDigits[Flatten[Transpose[RealDigits[list]]]] —so that the number of such lists is 2 ℵ 0 .

Within say a surface whose points {x 1 , x 2 , … } are obtained by evaluating an expression e as a function of parameters p (so that for example e = {x, y, f[x, y]} , p = {x, y} for a Plot3D surface) the metric turns out to be given by (Transpose[#] . # &) [Outer[D, e, p]] In ordinary Euclidean space a defining feature of geometry is that the shortest path between two points is a straight line. … One can then compute the Ricci tensor (R ik = R ijk j ) using RicciTensor = Map[Tr, Transpose[Riemann, {1, 3, 2, 4}], {2}] and this has 1/2 d(d + 1) independent components in d > 2 dimensions.

Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants.

Universal cellular automaton The rules for the universal cellular automaton are {{_, 3, 7, 18, _}  12, {_, 5, 7 | 8, 0, _}  12, {_, 3, 10, 18, _}  16, {_, 5, 10 | 11, 0, _}  16, {_, 5, 8, 18, _}  7, {_, 5, 14, 0 | 18, _}  12, {_, _, 8, 5, _}  7, {_, _, 14, 5, _}  12, {_, 5, 11, 18, _}  10, {_, 5, 17, 0 | 18, _}  16, {_, _, x : (11 | 17), 5, _}  x - 1, {_, 0 | 9 | 18, x : (7 | 10 | 16), 3, _}  x + 1, {_, 0 | 9 | 18, 12, 3, _}  14, {_, _, 0 | 9 | 18, 7 | 10 | 12 | 16, x : (3 | 5)}  8 - x, {_, _, _, 8 | 11 | 14 | 17, x : (3 | 5)}  8 - x, {_, 13, 4, _, x : (0 | 18)}  x, {18, _, 4, _, _}  18, {_, _, 18, _, 4}  18, {0, _,4, _, _}  0, {_, _, 0, _, 4}  0, {4, _, 0 | 18, 1, _}  3, {4, _, _, _, _}  4, {_, _, 4, _, _}  9, {_, 4, 12, _, _}  7, {_, 4, 16, _, _}  10, {x : (0 | 18), _, 6, _, _}  x, {_, 2, 6, 15, x : (0 | 18)}  x, {_, 12 | 16, 6, 7, _}  0, {_, 12 | 16, 6, 10, _}  18, {_, 9, 10, 6, _}  16, {_, 9, 7, 6, _}  12, {9, 15, 6, 7, 9}  0, {9, 15, 6, 10, 9}  18, {9, _, 6, _, _}  9, {_, 6, 7, 9, 12 | 16}  12, {_, 6, 10, 9, 12 | 16}  16, {12 | 16, 6, 7, 9, _}  12, {12 | 16, 6, 10, 9, _}  16, {6, 13, _, _, _}  9, {6, _, _, _, _}  6, {_, _, 9, 13, 3}  9, {_, 9, 13, 3, _}  15, {_, _, _, 15, 3}  3, {_, 3, 15, 0 | 18, _}  13, {_, 13, 3, _, 0 | 18}  6, {x : (0 | 18), 15, 9, _, _}  x, {_, 6, 13, _, _}  15, {_, 4, 15, _, _}  13, {_, _, _, 15, 6}  6, {_, _, 2, 6, 15}  1, {_, _, 1, 6, _}  2, {_, 1, 6, _, _}  9, {_, 3, 2, _, _}  1, {3, 2, _, _, _}  3, {_, _, 3, 2, _}  3, {_, 1, 9, 1, 6}  6, {_, _, 9, 1, 6}  4, {_, 4, 2, _, _}  1, {_, _, _, _, x : (3 | 5)}  x, {_, _, 3 | 5, _, x : (0 | 18)}  x, {_, _, x : (1 | 2 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17), _, _}  x, {_, _, 18, 7 | 10, 18}  18, {_, _, 0, 7 | 10, 0}  0, {_, _, 0 | 18, _, _}  9, {_, _, x_, _, _}  x} where the numbers correspond to the icons shown in the main text according to The block in the initial conditions for the universal cellular automaton corresponding to a cell with color a is given by Flatten[{Transpose[{Join[{4, 18(1 - a), 6}, Table[9, {2 2 r + 1 - 3}]], 10 - 3 rtab}], Table[{9, 1}, {r}], 9, 13}] where r is the range of the rule to be emulated ( r = 1 for elementary rules) and rtab is the list of outcomes for that rule (starting with the outcome for {1, 1, (1) ...} ).