The Angle A Sunflower Knows
Turn 137.507° between seeds and they pack perfectly. Turn 137.3° and it falls apart.
What it is
Fifteen hundred seeds, each placed one fixed turn further round than the last and a little further out. Drag to change the turn. Only six degrees of range, because that’s all you need.
At the golden angle the seeds interlock with no gaps and no spokes. Move a fifth of a degree either way and rows of spokes appear out of nowhere.
How it works
Two rules. The turn between consecutive seeds is constant, and the radius of the nth seed is √n.
The square root isn’t decorative. Area grows with r², so if the count is to keep pace with the area available then radius has to grow as the square root of the index. And it does: the seeds inside half the radius number a quarter of the total, which is a test.
The turn is where the interest is. Any turn that’s a rational fraction of a circle repeats. 90° puts every seed on one of four rays, 120° on three. Spokes mean gaps and gaps mean wasted space.
What you want is a turn that never repeats and, more than that, whose partial sums avoid clustering near any small rational. The golden ratio is precisely the number that’s hardest to approximate by fractions.
What surprised me
How narrow the optimum is. I expected a broad basin. It’s a spike.
Spacing spread, meaning the variation in how far each seed sits from its nearest neighbour, over the interior:
| turn | spread |
|---|---|
| 90° / 120° / 144° / 180° | ~27% |
| 137.0° | 16.6% |
| 137.3° | 16.8% |
| 137.5078° (golden) | 1.4% |
| 137.7° | 5.3% |
| 138.0° | 20.9% |
A fifth of a degree away and the packing is twelve times less uniform. Two tenths of a degree, out of three hundred and sixty.
That reframes the biology for me. I’d filed the golden angle under “nature finds pleasing proportions”, which is vague and slightly mystical. It isn’t that at all. It’s a spike so sharp that hitting it can’t plausibly be a coincidence, and missing it by a fraction of a degree has a real cost in packing density. A plant using 137.3° would be measurably worse at fitting seeds into a head, and that’s the kind of difference selection can act on.
What I would do next
Plot spread against angle across the whole circle and see whether the other local minima sit at the other noble numbers.