In a rotating but perfectly spherical black hole (Kerr metric), the situation is nevertheless different, and in this case the whole infinite future of the outside universe can indeed in principle be seen in the finite time between crossing the outer and inner event horizons.

The transitions between these states have probabilities given by m[Map[Length, list]] where m[s_] := With[{q = FoldList[Plus, 0, s]}, ReplacePart[ RotateRight[IdentityMatrix[Last[q]], {0, 1}], 1/Length[s], Flatten[Outer[List, Rest[q], Drop[q, -1] + 1], 1]]] The average spectrum of sequences generated according to these probabilities can be obtained by computing the correlation function for elements a distance r apart ξ [list_, r_] := With[{w = (# - Apply[Plus, #]/Length[#] &)[ Flatten[list]]}, w .

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.

 [i, j, k] || If[EvenQ[n + i - j],  [i, j], False] ||  [i + 1, j, k], {i, 0, t - 2}, {j, Max[1, n - i], n + i}, {k, 0, ktot - 1}], Table[Map[Function[ z, Outer[!

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!

. # &) [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.

Of the 262,144 9-neighbor outer totalistic rules the only ones that conserve e[s] are identity and complement.