In case (c), stripes appear at angles related to GoldenRatio .

Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

Examples of structures formed in various geometries by successively adding elements at a golden ratio angle 137.5°.

Exceptions are known to include so-called Pisot numbers such as GoldenRatio , √ 2 + 1 and Root[# 3 - # - 1 &, 1] (the numerically smallest of all Pisot numbers) for which Mod[h n , 1] becomes 0 or 1 for large n .

Note that in order to get an accurate approximation to a golden ratio angle there must be a fairly large number of cells.

For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

By the 1800s various mathematical features of phyllotaxis were known, and in 1837 Louis and Auguste Bravais identified the presence of a golden ratio angle.

Sequence (c) is the powers of two; (d) is the so-called Fibonacci sequence, related to powers of the golden ratio (1 + √ 5)/2 ≃ 1.618 .

From the result on page 890 , the number whose digits are obtained from {1  {1, 0}, 0  {1}} is given by Sum[2^(-Floor[n GoldenRatio]), {n, ∞ }] .

[Intractability in] systems of limited size In the system x  Mod[x + m, n] from page 255 the repetition period n/GCD[m, n] can be computed using Euclid's algorithm in at most about Log[GoldenRatio, n] steps.