With the state of a 2-color tag system encoded as an integer according to FromDigits[Reverse[list] + 1, 3] the following takes the rule for any such tag system (in the first form from page 894 ) and yields a primitive recursive function that emulates a single step in its evolution: TSToPR[{n_, rule_}] := Fold[Apply[c, Flatten[{#1, Array[p, # 2], c[r[z, c[r[p[1], s], c[r[z, p[2]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]], p[#2]]}]] & , c[c[r[p[1], s], p[1], c[r[p[1], r[z, c[s, c[s, s]]]], c[c[r[z, c[r[p[1], s], c[r[z, c[s, z]], c[r[p[1], r[z, c[r[p[1], s], c[r[z, p[2]], c[ r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]]], p[2], p[3]]], p[1]]], p[1], p[1]], p[1]], p[2]]], p[n + 1], MapIndexed[c[r[z, c[r[p[1], p[4]], p[2], p[3], p[4]]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[Length[#2] + 1]], # 1 〚 1 〛 , #1 〚 2 〛 ] & , Nest[Partition[#1, 2] & , Table[Nest[c[s, #] & z, FromDigits[Reverse[IntegerDigits[i, 2, n] /. rule] + 1, 3]], {i, 0, 2 n - 1}], n - 1], {0, n - 1}]], Range[n, 1, -1]] (For tag system (a) from page 94 this yields a primitive recursive function of size 325.)

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) ...} ).

In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

The pattern corresponding to each point is the limit of Nest[Flatten[1 + {c #, Conjugate[c] #}]&, {1}, n] when n  ∞ .

To next order the result is s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + (5 RicciScalar 2 - 3 RiemannNorm + 8 RicciNorm - 18 Laplacian[RicciScalar])r 4 /(360 (d + 2)(d + 4)) + …) where the new quantities involved are RicciNorm = Norm[RicciTensor, {g, g}] RiemannNorm = Norm[Riemann, {g, g, g, Inverse[g]}] Norm[t_, gl_] := Tr[Flatten[t Dual[t, gl]]] Dual[t_, gl_]:= Fold[Transpose[#1 .

As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83 , then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]] k ] is equal for i = 0 and i = 1 if s is a sequence of the form (b) on page 83 with length 2 m , m > k .

And rules that involve more than two cells can be obtained by having several instances of ⊕ —which can always be flattened.

(Starting with initial condition x the digit sequence at step n is essentially IntegerDigits[Mod[2 n Floor[2 53 x], 2 53 ], 2, 53] on the computer, and Flatten[IntegerDigits[IntegerDigits[ Mod[2 n Floor[10 12 x], 10 12 ], 10, 12], 2, 4]] on the calculator.