A sunflower head, a pine cone, a pineapple: in each, the parts of the plant line up in two sets of spirals winding opposite ways, and their counts are almost always neighbouring Fibonacci numbers: 8 and 13, 21 and 34, 55 and 89. This is phyllotaxis, literally “leaf arrangement”.
How it works
Helmut Vogel's model places the seeds one at a time: seed n is turned by n·α and set at distance c·√n from the centre. The square root gives each seed equal room: a disc of radius c·√n has an area proportional to n. The secret is α. The golden angle divides a turn in the golden ratio, the “most irrational” number: its continued fraction is all ones, so fractions approximate it worst of all. The seeds never line up into spokes: each lands in the widest gap, keeping the packing dense.
The spirals the eye picks out, called parastichies, are simply chains of nearest neighbours. Seed n touches seed n + F because F steps of the golden angle add up to almost a whole number of turns. Which F shows depends on the radius: near the middle the neighbours are 21 and 34 apart, further out 34 and 55, and at the rim of a thousand seeds 55 and 89.
A bit of history
Karl Friedrich Schimper and Alexander Braun described spiral leaf arrangements mathematically in 1830, and in 1837 the Bravais brothers linked them to the Fibonacci numbers. In 1868 Wilhelm Hofmeister proposed that a new leaf forms where the growing tip has most room, so the golden angle is a by-product of jostling. Vogel's formula dates from 1979. In 1992 Stéphane Douady and Yves Couder got the spirals without any botany: drops of magnetic fluid falling into a dish repelled each other into the pattern of a sunflower.
What to tweak
- An “Angle deviation” of −2.508° gives almost exactly 135°, 3/8 of a turn, and the seeds form 8 straight spokes; at +0.95° the angle is nearly 5/13 of a turn, giving 13.
- “Colouring” set to “spirals” paints each arm of the chosen “Spiral family” its own colour; switch from 34 to 55 and the spirals wind the other way.
- “Growth” squeezes the centre and enlarges the seeds at the rim, as in a real flower head, where the youngest, smallest seeds sit in the middle.
- “Shape” set to “petals” with an “Outline” of 0.3–0.5 gives something like a dahlia.
- Add the “Distort” effect on top in “lens” mode and the centre swells like a domed flower head.
Guilloché and spirograph
Harmonograph
Gielis superformula
Flow lines
Ridgelines