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For large n the ratio f[n]/f[n - 1] approaches GoldenRatio or (1 + √ 5 )/2 ≃ 1.618 .
… The number GoldenRatio appears to have been used in art and architecture since antiquity. 1/GoldenRatio is the default AspectRatio for Mathematica graphics. 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 ).

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°. … As mentioned on page 891 , having GoldenRatio turns between elements makes them in a sense as evenly distributed as possible, given that they are added sequentially.

Projections of [phyllotaxis] patterns
The literature of phyllotaxis is full of baroque descriptions of the features of projections of patterns with golden ratio angles. In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.

Lengths of [number] representations
(a) n , (b) Floor[Log[2, n] + 1] , (c) Tr[FixedPointList[Max[0, Ceiling[Log[2, #]]] &, n + 2]] - n - 3 , (d) 2 Ceiling[Log[3, 2n + 1]] , (e) Floor[Log[GoldenRatio, √ 5 (n + 1/2)]] . Large n approximations: (a) n , (b) Log[2, n] , (c) Log[2, n] + Log[2, Log[2,n ]] + … , (d) 2 Log[3, n] , (e) Log[GoldenRatio, n] .

The arrangement of triangles at step t can be obtained from a substitution system according to
With[{ ϕ = GoldenRatio}, Nest[# /. a[p_, q_, r_] With[{s = (p + ϕ q) (2 - ϕ )}, {a[r, s, q], b[r, s, p]}] /. b[p_, q_, r_] With[{s = (p + ϕ r) (2 - ϕ )}, {a[p, q, s], b[ r, s, q]}] &, a[{1/2, Sin[2 π /5] ϕ }, {1, 0}, {0, 0}], t]]
This pattern can be viewed as generalizations of the pattern generated by the 1D Fibonacci substitution system (c) on page 83 . As discussed on page 903 , this 1D sequence can be obtained by looking at how a line with GoldenRatio slope cuts through a 2D lattice of squares. Penrose tilings can be obtained by looking at how a 2D plane with slopes based on GoldenRatio cuts through a lattice of hypercubes in 5D.

The second example involves two distinct shapes: a square and a GoldenRatio aspect ratio rectangle.

Among the relations satisfied by Lucas numbers are:
• Lucas[n_] := Fibonacci[n - 1] + Fibonacci[n + 1]
• GoldenRatio n (Lucas[n] + Fibonacci[n] √ 5 )/2

The color of the element at position n is given by 2 - (Floor[(n + 1) GoldenRatio] - Floor[n GoldenRatio]) (see page 904 ), while the position of the k th white element is given by the so-called Beatty sequence Floor[k GoldenRatio] . The ratio of the number of white elements to black at step t is Fibonacci[t - 1]/Fibonacci[t - 2] , which approaches GoldenRatio for large t .

Generalized Fibonacci sequences
Any linear recurrence relation yields sequences with many properties in common with the Fibonacci numbers—though with GoldenRatio replaced by other algebraic numbers.

(The average difference of successive values is maximized for h = GoldenRatio , as mentioned on page 891 .)