The angle that never comes home

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Plant seeds in a sunflower's way. Put the first one down. Turn a little, put the next one down a step further from the centre, and turn by the same amount again. Every seed the same turn. The only question is: how far do you turn?

Try 135°, three-eighths of a circle. It looks like a fine, irregular-enough number. But after eight turns you're back where you started — the ninth seed lands on the first one's ray — and every seed from then on lands on one of the same eight rays. You get a starfish of eight straight spokes with empty wedges between them, wasted ground. Every turn that is a tidy fraction of a circle does this sooner or later: it comes home, and coming home makes spokes.

Now turn by 137.5077...°, the golden angle: the circle cut in the golden ratio. This turn never comes home. No matter how many seeds you plant, no seed ever lands exactly on the ray of an earlier one.

And here is the click. Call the first seed seed 0, so the seed planted after k turns is seed k. Ask, as you plant, which seed has come closest to seed 0's direction. The record changes hands only at these seeds:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 ...

The Fibonacci numbers, and nothing else. Each new record-holder comes closer than the last by the same factor, the golden ratio itself: 85.0°, then 52.5°, then 32.5°, 20.1°, 12.4°, each about 1.618 times nearer. The near-misses arrive in Fibonacci order because the golden ratio is the number that fractions approximate worst, and the fractions that come closest to it are ratios of Fibonacci numbers. That is why a sunflower head shows 34 spirals one way and 55 the other, or 55 and 89. You aren't seeing the seeds. You're seeing the near-misses.

Two smaller clicks, for whoever wants them. Round the golden angle to 137.5° and you get exactly 144 spokes, because 137.5° is 55/144 of a turn, and 55 and 144 are Fibonacci numbers. And 135° itself is 3/8, also Fibonacci, just a coarser one. Every best approximation of the golden angle — each fraction that beats all simpler fractions at the job — is a Fibonacci fraction in disguise, and each one comes home eventually. Only the real thing never does.

Check it: plant a few hundred seeds at the golden angle, on paper or in a few lines of code, and keep track of who comes closest to seed 0's direction. Two thousand seeds were planted before this page went up, and the record-holders were exactly the Fibonacci numbers, as far as the planting ran.

The next flower is a few steps on.

Take a crown on your way out, whichever way out this is.