Sequences · golden ratio
Start with 1 and 1. Every next number is the sum of the two before it. From this one rule grow the golden ratio φ ≈ 1.618, a spiral of squares, and the pattern a sunflower uses to arrange its seeds.
F₁ = 1
←→ angle / step (Shift 0.01°)+− terms / seedsSpace animation
Drawings generated from the same golden angle — on the surface of a cylinder, a cone, and in buds that repeat the shape of the whole.
Same rule, different start: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76… These are the Lucas numbers. They also approach φ, and their formula is even simpler:
For example, F10 = 55 = 5 · 11 = F5·L5.
A rectangle with sides in the proportion 1 : φ. Cut a square off it and a smaller rectangle with the same proportion remains, because φ − 1 = 1/φ. The Fibonacci squares from the “Spiral” mode approximate it better and better.
In a regular pentagon, diagonal / side = φ. The diagonals form a pentagram, and each one is cut by the others in the golden ratio. The triangles with angles of 36° and 72° inside it are “golden” triangles too.
As a continued fraction, φ is made of nothing but ones — the smallest possible terms. That's why the fractions Fn+1/Fn approximate it more slowly than for any other irrational number (Hurwitz's theorem). And that's exactly why the golden angle never “hits” an old direction and the seeds don't form spokes.
The numbers of spirals in sunflowers, pine cones and pineapples are usually consecutive Fibonacci numbers, and the golden angle shows up in many plants, because new buds grow where there is the most room.
But it's not a law without exceptions: in a large citizen-science study from 2016 (Swinton and Ochu), roughly one sunflower in five had spirals that didn't match the Fibonacci sequence or were irregular.
The nautilus shell is a beautiful logarithmic spiral, but not a golden one. A golden spiral grows by φ every quarter turn, which means φ4 ≈ 6.85 times per full turn. Measurements of real shells show clearly slower growth, and it differs from one specimen to another.
The Fibonacci sequence starts with two ones, and each next term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34… The ratio of neighboring terms approaches the number φ. If the ratio has a limit x, then the rule Fn+1 = Fn + Fn−1 gives x = 1 + 1/x, that is x² = x + 1.
Why the golden angle? If the angle is a simple fraction of a turn p/q, then every q-th seed lands in the same direction — you get q straight spokes with empty wedges between them. The golden angle is the “most irrational” fraction of a turn, so no direction ever repeats and the seeds fill the disk evenly. The eye then sees two families of spirals, and their counts are successive approximations of φ: 13 and 21, 21 and 34, 34 and 55…
Simplifications: Vogel's model is an idealized description. In a real plant, new buds form where there is the most room (controlled partly by plant hormones), and the golden angle emerges from this process only approximately. The spiral made of quarter circles in squares approximates the true golden logarithmic spiral.