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Plate VIII of IX

Seed

Why sunflowers count in Fibonacci

Plate VIII. A dense disc of small dots arranged in interlocking clockwise and counter-clockwise spiral arms, like a sunflower head. Every control writes itself into the page address, so a figure you like has a link of its own.

What you are looking at

Place seeds one at a time. Each new seed goes a fixed angle around from the last, and a little further out — specifically at distance proportional to √n, so that each seed gets the same area and the head stays evenly packed. That is the whole model, published by Helmut Vogel in 1979, and it produces a convincing sunflower.

Everything depends on the angle. Choose 90° and you get four straight arms. Choose 120° and you get three. Choose any rational fraction of a turn, p/q, and you get exactly q arms and enormous wasted space between them, because after q seeds you are back where you started and every subsequent seed lands on a ray you have already used.

So the angle needs to be irrational. But irrational is not enough — a number that is nearly rational is nearly as bad. 3.14159 turns is very close to 22/7, so an angle of 0.14159 turns gives you seven slightly curved arms and a lot of empty space.

The best possible angle is the one that is hardest to approximate by any fraction. That number is the golden ratio, because its continued fraction is [1; 1, 1, 1, …] — all ones, the slowest-converging expansion there is. The corresponding angle is 360° / φ² = 137.50776…°, the golden angle. At that angle no seed ever lands on a ray used by an earlier seed, no arms form, and the packing is as even as it can be.

Then the strange part. Even though there are no arms, your eye insists on seeing spirals — two families of them, curving opposite ways. Count them. The numbers you get are consecutive Fibonacci numbers: 21 and 34, or 34 and 55, or 55 and 89, depending on how far out you look. Those spirals are not in the model. They are an artefact of the golden angle's best rational approximations, which are ratios of consecutive Fibonacci numbers, and they are exactly what botanists find when they count the spirals on a real sunflower.

Try this

  • Press Golden angle, then drag the Nudge slider slowly across its whole range. The pattern holds, shears open into arms, and closes again, several times over two degrees. Small changes have violent effects — a strong argument that real plants are not landing on this angle by accident.
  • Type 137.5 into the angle box, then 137.508. Three extra digits visibly tighten the packing at the rim.
  • Try 149.117°, which is 360° × (√2 − 1). The silver ratio is the second-hardest number to approximate, so this packs almost as well — and the arm counts become 2, 5, 12, 29, 70. Those are the Pell numbers, the √2 analogue of Fibonacci.
  • Watch the Predicted arms readout, which is the denominator of the angle's best rational approximation. Then set Arms to colour by to that number and tick Trace one arm: the highlighted seeds fall on one smooth curve. Set it to a neighbouring number and they scatter.
  • Press 90° for four hard rays, or type 40 for exactly nine — 360/9, for the house.

Notes

The pattern is old news to botanists — Charles Bonnet was describing spiral leaf arrangements in 1754, and the Bravais brothers identified the golden angle in 1837. What took longer was an account of the mechanism. The modern answer is that a growing shoot tip lays down each new primordium in the largest gap available, where inhibitory chemical signals from existing primordia are weakest. Simulate that rule with no mention of φ anywhere and the golden angle emerges on its own. In 1992 Stéphane Douady and Yves Couder demonstrated it physically, dropping magnetised ferrofluid into a dish of silicone oil: the drops repelled each other, and they spontaneously arranged themselves at 137.5°.

So the golden angle is not a design choice, and the plant is not doing arithmetic. It is the fixed point of a very simple packing rule, and it would be hard for a growing tip to avoid.

Further reading

Helmut Vogel, "A better way to construct the sunflower head," Mathematical Biosciences (1979). S. Douady and Y. Couder, "Phyllotaxis as a physical self-organized growth process," Physical Review Letters (1992).