One floret, then turn, then the next
There is no plan and no map. A seed head grows from the centre outward, laying down one floret at a time. Each new floret starts at the middle and is pushed out as the ones behind it crowd in, so it lands a fixed divergence angle around from the floret before it. That single angle decides everything — how tightly the head packs, and whether you see spirals at all.
This is Vogel's model, and it is the whole engine on this page: floret n sits at angle n × θ and radius c × √n. Nothing else. Every seed you see was placed by that one line, in order, exactly as a real head fills.
The most irrational turn there is
Set the angle to a nice round fraction of a circle — a half, a third, a tenth — and florets stack on top of the ones a few rows back, wasting the light and leaving bald radial spokes. To fill a disc evenly, each floret must land where none has landed before, forever. That demands an angle that can never be written as a neat fraction.
The golden ratio φ is the hardest number in the world to approximate with fractions, so the golden angle — 360° × (2 − φ) = 137.50776° — is the turn that most stubbornly refuses to repeat. It is not chosen for beauty. The beauty is what falls out.
Nobody drew the 34 and 55
Look at a sunflower and you count two families of spirals winding opposite ways — most often 34 one way and 55 the other, sometimes 55 and 89. Those are consecutive Fibonacci numbers, and no cell was ever told to make a spiral. They are parastichies: an illusion of alignment your eye assembles from florets that were only ever following the turn.
Flip Illuminate the spirals and the two families are traced in gold and leaf. The counts in the readout are computed live from the angle itself — the denominators of its best fractional approximations. Nudge off golden and watch them jump to numbers no sunflower would keep.
Keep turning
One angle, held to five decimals, and the sunflower does the rest.