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, such as 21, 34 and 55.

Looking at the seeds of a sunflower, one sees spirals winding to the left and to the right. Count them and you mostly 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:

\frac{360^\circ}{\varphi^2} \approx 137.508^\circ

It divides the full circle in the golden ratio. The physicist 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 $\sqrt{i}$ from the centre, turned by i times 137.5°. The slider shows how sensitive this is.

Angle

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.

Fibonacci and Kepler

The sequence bears the name of the merchant Leonardo of Pisa, called Fibonacci, who presented it in 1202 in his arithmetic book Liber abaci in a problem about the breeding of rabbits: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and each number is the sum of the two before. If one divides two consecutive numbers by each other, the result approaches the golden ratio more and more, 89 divided by 55 is already 1.61818.

The first to connect this sequence with plants was Johannes Kepler. In a little treatise on the snowflake, which he gave a friend as a New Year’s present in 1611, he remarked that the ratio of the numbers approaches the golden ratio and linked this with five-petalled flowers (see Johannes Kepler). Why Fibonacci numbers in particular appear as spirals is shown by the model above: the fractions of consecutive Fibonacci numbers are the best approximations to the golden ratio, and so the nearest neighbours of a seed always lie at distances that are Fibonacci numbers.

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.

How often is it true?

A beautiful pattern deserves an honest test. In 2012, for Alan Turing’s 100th birthday, researchers in Manchester called on people in Britain and beyond to grow sunflowers and count their spirals. The evaluation of 2016 found: about three quarters of the spirals counted were indeed Fibonacci numbers, but about one in four was not. Some followed a related sequence, others were irregular. And many plants arrange their leaves quite differently, for instance mint, with pairs of leaves each turned by 90°.

So the pattern is frequent but no law without exceptions, and the mechanism of Douady and Couder explains why: it produces the golden angle when the conditions of growth are right, and other orders when they are not. Caution is also called for with other often-cited examples. The spiral of the nautilus, for instance, does not follow the golden ratio, its turns grow by a clearly smaller ratio (see Chance and pattern).

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 one can see a counterpart to it in the growth of plants: the living fills its space with the golden angle, the number of the pentagram, because it produces the most even order without repetition. That is an image, not a reason, but an apt one.

Sources and further reading

Sources

Helmut Vogel, A better way to construct the sunflower head, Mathematical Biosciences 44, 1979
Stéphane Douady and Yves Couder, Phyllotaxis as a physical self-organized growth process, Physical Review Letters 68, 1992
Leonardo of Pisa, Liber abaci, Pisa 1202
Johannes Kepler, Strena seu de nive sexangula (The six-cornered snowflake), Frankfurt 1611
Jonathan Swinton, Erinma Ochu et al., Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment, Royal Society Open Science 3, 2016