The Most Irrational Angle

Look at the head of a sunflower and you see two families of spirals, one turning clockwise and one counter-clockwise. Count them and you usually get neighbouring Fibonacci numbers, such as 34 and 55. Pine cones and pineapples show the same thing with smaller numbers.

The cause is simple to state. Each new seed, or each new leaf bud, appears at a fixed angle from the one before. That angle is close to 137.5°. It is the golden angle, 360° × (1 − 1/φ), where φ = (1 + √5) / 2.

Suppose the angle were a rational fraction of a turn, say 2/5. Then every fifth seed would sit directly in line with an earlier one. The head would grow five straight spokes with gaps between them. Any rational angle p/q does this with q spokes. Packing is poor.

An irrational angle never repeats exactly. But some irrationals sit very close to simple fractions and make near-spokes for a long time. What you want is the irrational that stays furthest from all simple fractions.

The tool for measuring this is the continued fraction. Every real number can be written as a + 1/(b + 1/(c + …)). Large terms mean the number is very close to a simple fraction. Small terms mean it resists approximation. The golden ratio has every term equal to 1: [1; 1, 1, 1, …]. No number has smaller terms. In the sense made precise by Hurwitz’s theorem, φ is the hardest irrational to approximate.

Its best rational approximations are ratios of consecutive Fibonacci numbers: 3/2, 5/3, 8/5, 13/8. That is why Fibonacci numbers show up as spiral counts. The spirals you see are the near-misses, and the near-misses of φ are Fibonacci ratios.

The plant does not compute φ. The current picture is local. A new bud forms where the hormone auxin is most concentrated, and existing buds drain auxin from their surroundings. So each new bud appears as far as possible from the recent ones. In 1992 Stéphane Douady and Yves Couder reproduced the pattern with magnetised droplets in a dish of oil, each droplet repelled by the ones before it. The golden angle appeared without being specified.

This is the part I like. A global optimum, the worst-approximable number, emerges from a rule that only says: keep away from your recent neighbours.