Phyllotaxis

Florets placed by the golden angle: floret n sits at angle n · 137.507…° and radius c·√n, and the sunflower’s spiral arms appear without ever being drawn.

What it is

Vogel’s 1979 model of seed-head growth. The golden angle — the circle divided by the golden ratio — is the unique divergence that packs florets evenly at every radius; the apparent left- and right-winding spiral families (parastichies) count consecutive Fibonacci numbers. Two floret styles: dots (seed head) and radially oriented petals (dahlia).

How to use it

A colouring disk: every floret is a closed region. Set angle to 90 or 99.5 to see exactly why the golden angle is special — rational angles collapse into spokes.

History

Spiral phyllotaxis was measured by botanists from the 1830s (the Bravais brothers); Helmut Vogel’s compact r = c√n model dates to 1979, and the golden-angle explanation was made rigorous by Douady and Couder’s 1992 physics experiments.

The implementation’s guarantees

• Florets never touch, measured: Vogel’s model promises ≈c nearest-neighbour spacing; the floret radius banks on it and a pairwise test verifies it rather than trusting the asymptotics.
• The colourability lever is floret count (fewer florets = larger florets at fixed page size); the surviving count is reported beside the requested one.
• Seeded orientation and chirality; the divergence angle is a spec knob with the golden angle as default.
• Rating basis: none — a design. Honesty fields: floret count, divergence, style, attempt.