Cellular automaton combinators With k and s[k] representing respectively cell values 0 and 1 , a combinator f for which f[a -1 ][a 0 ][a 1 ] gives the new value of a single cell in an elementary cellular automaton with rule number m can be constructed as Apply[p[p[p[#1][#2]][p[#3][#4]]][p[p[#5][#6]][p[#7][ #8]]] /. {0  k, 1  s[k]} &, IntegerDigits[m, 2, 8]] //. crules where p = ToC[z[y][x], {x, y, z}] //. crules The resulting combinator has size 61, but for specific rules somewhat smaller combinators can be found—an example for rule 90 is s[k[k]][s[s][k[s[s[s[k][k]][k[s[k]]]][k[k]]]]] with size 16.

Doing a synthesis of a chemical is much like constructing a network by applying a specified sequence of transformations.

Gödel originally based his theorem on Peano arithmetic (as discussed in the context of Principia Mathematica ), but expected that it would in fact apply to any reasonable formal system for mathematics—and in later years considered that this had been established by thinking about Turing machines.

The ordinary axioms of arithmetic do not apply, but there are still fairly straightforward rules for manipulating such expressions.

The Riemann Hypothesis is also equivalent to the statement that a bound of order √ n Log[n] 2 exists on Abs[Log[Apply[LCM, Range[n]]] - n] .

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# /.

The number of sequences s n of length n that can actually occur is given by Apply[Plus, Flatten[MatrixPower[m, n]]] where the adjacency matrix m is given by MapAt[(1 + #) &, Table[0, {Length[net]}, {Length[net]}], Flatten[MapIndexed[{First[#2], Last[#1]} &, net, {2}], 1]] For rule 32, for example, s n turns out to be Fibonacci[n + 3] , so that for large n it is approximately GoldenRatio n .

Learning the new kind of science There will, I hope, be many who want to learn about what is in this book, whether out of general intellectual interest, to apply it in some way or another, or to participate in its further development.

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row.

And in general, the probabilities for all 8 possible combinations of 3 cells are given by probs = Apply[Times, Table[IntegerDigits[8 - i, 2, 3], {i, 8}] /. {1  p, 0  1 - p}, {1}] In terms of these probabilities the density at the next step in the evolution of cellular automaton with rule number m is then given by Simplify[probs .