Rule 225 The width of the pattern after t steps varies between Sqrt[3/2] √ t (achieved when t = 3 × 2 2n + 1 ) and Sqrt[9/2] √ t (achieved when t = 2 2n + 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.

Comments on Mathematica functions CenterList works by first creating a list of n 0's, then replacing the middle 0 by a 1. (In Mathematica 4 and above PadLeft[{1}, n, 0, Floor[n/2]] can be used instead.) … The actual list of new cell values is then generated by using the fact that {i, j, k} 〚 {2, 1, 1, 3, 2} 〛 ⟶ {j, i, i, k, j} .

The necessary transformation is the so-called Lorentz transformation {t, x}  {t - v x/c 2 , x - v t}/Sqrt[1 - v 2 /c 2 ] And from this the time dilation factor 1/Sqrt[1 - v 2 /c 2 ] shown on page 524 follows, as well as the length contraction factor Sqrt[1 - v 2 /c 2 ] .

LFSR cryptanalysis Given a sequence obtained from a length n LFSR (see page 975 ) Nest[Mod[Append[#, Take[#, -n] . vec], 2] &, list, t] the vector of taps vec can be deduced from LinearSolve[Table[Take[seq, {i, i + n - 1}], {i, n}], Take[seq, {n + 1, 2n}], Modulus  2] (An iterative algorithm in n taking about n 2 rather than n 3 steps was given by Elwyn Berlekamp and James Massey in 1968.)

A simpler example is the rule Mod[RotateLeft[list] + RotateRight[list], 1] With a single nonzero initial cell with value 1/k the pattern produced is just Pascal's triangle modulo k .

Multiway systems based on numbers One can consider for example the rule n  {n + 1, 2 n} implemented by NestList[Union[Flatten[{# + 1, 2 #}]] &, {0}, t] In this case there are Fibonacci[t + 2] distinct numbers obtained at step t .

Thus for example unlike in 1D there is no guarantee in 2D that among repeating configurations of a particular period there is necessarily one that consists just of a repetitive array of fixed blocks. … Note that if one considers configurations in 2D that consist only of infinitely long stripes, then the problem reduces again to the 1D case.

In 4D the six regular polytopes have 4, 4, 6, 8, 4 and 12 edges meeting at each vertex, and in higher dimensions the simplex ( d + 1 vertices) and hypercube ( 2 d vertices) have d edges meeting at each vertex, while the co-cube ( 2d vertices) has 2(d - 1) .

The value of each cell can be any gray level between 0 and 1.