The Golden Angle
Every seed in this flower head is placed by one rule: turn by a fixed angle, step out a little, plant the next. At 137.508° the seeds pack edge to edge all the way to the rim, in 55 spirals one way and 89 the other. Move the angle a tenth of a degree either side and gaps open, spokes appear and the packing measured underneath falls off a cliff.
Count the spirals in a sunflower head and you will almost always find two Fibonacci numbers: 34 running clockwise and 55 counter-clockwise, or 55 and 89 on a big one. Pinecones show 5 and 8 or 8 and 13, pineapples 8, 13 and 21, daisies 21 and 34. Botanists have counted this for two hundred years. The pattern is so reliable that the interesting question stopped being whether it happens and became why a plant would care about a number sequence.
The plant doesn't. It does something simpler: each new seed forms at the growing tip, turned by a fixed angle from the one before, and the older seeds are pushed outward. In 1979 Helmut Vogel wrote the rule down as two lines — seed k at angle k times θ, at a distance proportional to the square root of k, so every seed gets the same area — and showed that one angle makes it work. At 137.508°, the golden angle, the head fills evenly from centre to rim. The spirals you see are not built in; they are just which neighbours happen to sit closest, and at that angle the counts come out as consecutive Fibonacci numbers by themselves.
The surprise is how intolerant the rule is. 137.5° looks the same as 137.508° to anyone, and for the first few hundred seeds it is, but further out it starts to leave gaps. 137° makes curved spokes. 144° — two fifths of a turn — puts every seed on five straight rays and wastes nearly the whole disc. The reason is number theory: any angle that is close to a simple fraction of a turn eventually lines its seeds up along that fraction's spokes. The golden ratio φ is the number that is hardest of all to approximate by fractions, so the golden angle is the one that never lines up. Its continued fraction is 1s forever, and the tiles below show it.
The same arithmetic places the leaves around a stem so that no leaf sits directly above another, the scales on a pinecone, the florets of a cauliflower and the spiralled buds of Romanesco, the leaves of a succulent rosette and the hexagonal eyes of a pineapple. Plant physiologists now think the angle emerges from hormone-driven spacing at the growing tip — each new bud forms where the inhibition from its neighbours is weakest — rather than being encoded anywhere. The flower does not know about φ. It just keeps putting the next seed in the biggest gap.
A storm grown one cell at a time from a single rule. Plant a lightning rod and watch the strikes bend toward it.