Why do sunflowers favour an angle of 137.5°?

New seeds keep appearing at the centre of a sunflower head and push older ones outwards. In 1979 the mathematician Vogel wrote this as a simple model: seed $n$ sits at

where $\alpha$ is the angle each seed is turned from the one before. The radius $\sqrt n$ gives every seed roughly the same area. The whole pattern depends on the single number $\alpha$.

Turn by $\frac{p}{q}$ of a circle each time and every $q$ seeds you are back facing the same way: the seeds line up in $q$ straight rows, and the space between them is wasted.

An angle that is merely close to a fraction, such as 137°, close to $\frac{8}{21}$ of a turn, gives 21 slightly curved spokes, with gaps that widen further out.

So the best angle should stay as far as possible from every simple fraction.

Written as a continued fraction, the golden ratio $\varphi=\frac{1+\sqrt5}{2}\approx1.618$ has a 1 at every level:

Large numbers in a continued fraction make a number easy to approximate by fractions; all ones make it the hardest of all irrational numbers to approximate.

The golden angle is the smaller part of a full turn split in the golden ratio:

Turn by it and the seeds never line up in spokes. The "minimum spacing" readout compares the two most crowded seeds at the current angle with the golden angle: at 137.6° only about 63% is left.

The spirals your eye picks out join each seed to its nearest neighbours. At the golden angle, the nearest neighbours of seed $n$ are seeds $n-55$ and $n-89$ (on the outer ring of 1000 seeds), so you see 55 and 89 spirals.

The reason is that the best fractional approximations of the golden ratio are ratios of consecutive Fibonacci numbers: $\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8},\dots,\frac{89}{55}$. Further out there are more seeds and the visible pair moves up: 13 and 21, 21 and 34 nearer the centre, 55 and 89 at the edge. Real sunflowers often show 34 and 55 or 55 and 89 spirals.