The golden angle – how plants count
Sunflower seeds, pine cones and leaf arrangements grow 137.5° apart, the golden angle. Spirals appear in Fibonacci numbers, 21, 34, 55.
Looking at the seeds of a sunflower, one sees spirals winding to the left and to the right. Count them and you almost always find two consecutive Fibonacci numbers: 21 and 34, 34 and 55, or in large flower heads 55 and 89. The same holds for pine cones, pineapples, artichokes and the arrangement of many leaves on a stem. Botany calls the arrangement of leaves and flowers phyllotaxis.
A single angle
The explanation is astonishingly simple. At the tip of a growing plant new buds form one after another, and each is set off from the previous one by the same angle. This angle is usually the golden angle:
360° / φ² ≈ 137.508°
It divides the full circle in the golden ratio. The mathematician Helmut Vogel showed in 1979 with a simple model that exactly this angle fills the area most evenly: the i-th seed sits at distance √i from the centre, turned by i times 137.5°. The slider shows how sensitive this is.
Mark spirals
Vogel's model: 600 seeds, each turned by the same angle against the previous one. Only at the golden angle do they fill the area without gaps. Already at 137.0° gaps appear, at 144° straight rays. The buttons mark the spirals the eye sees.
If the angle deviates by only half a degree, the seeds line up in rays with gaps between them. A rational angle, such as 144° = 360° × 2/5, produces five straight arms. The golden ratio, by contrast, is the number that is hardest to approximate by fractions (see The pentagram and the golden ratio). That is why at the golden angle no seed ever lies exactly above an earlier one, and each finds the largest free gap.
Why nature grows like this
Plants do not calculate, of course. The physicists Stéphane Douady and Yves Couder showed in 1992 with a lovely experiment that the golden angle arises by itself: they let magnetised droplets fall at a regular rate into the centre of a dish, where they repelled one another and drifted outwards. Each new droplet settled in the largest gap, and the droplets arranged themselves in spirals with the golden angle. In the plant, the distribution of a growth substance plays this role. Alan Turing, the pioneer of computing, had already studied Fibonacci numbers in plants in the last years of his life.
Five and life
The numbers that appear here are the Fibonacci numbers, whose ratios approach the golden ratio, and the golden ratio is the number of the pentagram. If hermeticism read the pentagram as the sign of life (see The tetractys at a glance), then this reading has a very tangible counterpart in the growth of plants: the living fills its space with the number of five, because it produces the most even order without repetition.