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 . … After 2 steps, the polynomial is x 13 - 4 x 12 + 6 x 11 - 5 x 10 + 3 x 9 - 3 x 8 + 5 x 7 - 3 x 6 - x 5 + 4 x 4 - 2 x 3 + x 2 - x + 1 giving κ ≃ 1.732 . … The value of this for successive t never increases; for the first 3 steps in rule 126 it is for example approximately 1, 0.811, 0.793.

And in many cases these functions end up trying to prove theorems; so for example FullSimplify[(a + b)/2 ≥ Sqrt[a b], a > 0 && b > 0] must in effect prove a theorem to get the result True .

The pattern generated by rule 150R has fractal dimension Log[2, 3 + Sqrt[17]] - 1 or about 1.83. In rule 154R, each diagonal stripe is followed by at least one 0; otherwise, the positions of the stripes appear to be quite random, with a density around 0.44.

The facing page shows the first 4000 digits in the sequence, both in the usual case of base 10, and in base 2. … A pictorial representation of the first 20,000 digits of π in base 2. The curve drawn goes up every time a digit is 1, and down every time it is 0.

Defining v[u] = -Integrate[f[u], u] the field then has Lagrangian density (( ∂ t u) 2 - ( ∂ x u) 2 )/2 - v[u] and conserves the Hamiltonian (energy function) Integrate[(( ∂ t u) 2 + ( ∂ t u) 2 )/2 + v[u], {x, - ∞ , ∞ }] With the choice for f[u] made here (with a ≥ 0 ), v[u] is bounded from below, and as a result it follows that no singularities ever occur in u[t, x] .

Other examples [of substitution systems] (a) (Period-doubling sequence) After t steps, there are a total of 2 t elements, and the sequence is given by Nest[MapAt[1 - # &, Join[#, #], -1]&, {0}, t] . … The Thue–Morse sequence discussed on page 890 can be obtained from it by applying 1 - Mod[Flatten[Partition[FoldList[Plus, 0, list], 1, 2]], 2] (b) The n th element is simply Mod[n, 2] . … (d) The length of the sequence at step t satisfies a[t]  2a[t - 1] + a[t - 2] , so that a[t] = Round[(1 + √ 2 ) t - 1 /2] for t > 1 .

Mathematics of phyllotaxis A rotation by GoldenRatio  (1 + √ 5 )/2 turns is equivalent to a rotation by 2 - GoldenRatio  GoldenRatio -2 ≃ 0.38 turns, or 137.5°. Successive approximations to this number are given by Fibonacci[n - 2]/Fibonacci[n] , so that elements numbered Fibonacci[n] (i.e. 1, 2, 3, 5, 8, 13, ...) will be the ones that come closest to being a whole number of turns apart, and thus to being lined up on the stem.

And these amplitudes a i are assumed to be complex numbers with a continuous range of possible values, subject only to the conventional constraint of unit total probability Sum[Abs[a i ] 2 , {i, 2 n }]  1 . … It was found to be sufficient to do operations on just one and two spins at a time, and in fact it was shown that any 2 n × 2 n unitary matrix can be approximated arbitrarily closely by a suitable sequence of for example underlying 2-spin {x, y}  {x, Mod[x + y, 2]} operations (assuming values 0 and 1), together with 1-spin arbitrary phase change operations. … And with the setup described, even if a particular function is ultimately satisfiable the probability for a single output spin to be measured say as up can be as little as 2 -n —requiring on average 2 n trials to distinguish from 0 , just as in the classical probabilistic case.

Reversal-addition systems The operation that is performed here is n  n + FromDigits[Reverse[IntegerDigits[n, 2]], 2] After a few steps, the digit sequence obtained is typically reversal symmetric (a generalized palindrome) except for the interchange of 0 and 1, and for the presence of localized structures. … But with the initial condition n = 512 , no repetition occurs for at least a million steps, at which point n has 568418 base 2 digits. … If one always includes one new digit on the left at every step, even when it is 0, then a rather random pattern is produced.

Orthogonality is then the property that a 〚 i 〛 . a 〚 j 〛  0 for all i ≠ j . … Here a typical orthogonality property is Integrate[f[r, x] f[s, x], {x, 0, 1}]  KroneckerDelta[r, s] . As discovered by Joseph Fourier around 1810, this is satisfied for basis functions such as Sin[2 n π x]/ √ 2 .