Look at the middle of a sunflower and you see two sets of curving spirals, one winding clockwise and one anticlockwise, crossing each other. Count them and the numbers are often neighbours in the Fibonacci sequence: 34 one way and 55 the other, or 55 and 89 on a big head. Nobody told the flower to count. The spirals are a side effect of one very fussy rule about turning.
A sunflower head grows from the centre outwards. New seeds form one after another at the edge of a small growing middle, and as the head expands each seed is pushed outwards. A simple model of this, published by Helmut Vogel in 1979, needs only two lines. Seed number n sits at a distance proportional to the square root of n from the centre, and at an angle of n times a fixed turning angle. The square root keeps every seed the same area to itself. The turning angle is the interesting part.
Drag the angle slider slowly. Most settings leave bare lanes: the seeds fall into a handful of straight arms with empty gaps between them. Only a few narrow angles fill the disc evenly, and the best known sits at about 137.508 degrees.
Suppose each seed turns exactly a quarter of a circle (90 degrees). Seed 4 lands exactly on top of seed 0's direction, seed 5 on seed 1's, and so on, so you get four straight arms and a lot of wasted space. A turn of 2/5 of a circle gives five arms. Any angle that is a fraction of a full turn, p over q, ends up with q arms, because after q seeds it returns to where it started. Try the 90, 120 and 144 degree buttons to see it.
The cure is an angle that never lines up with anything. Numbers differ in how well fractions can approximate them, and the golden ratio, about 1.618, is the number that fractions approximate worst. Its fractional part, 0.618, gives the golden angle: 360 degrees times (1 minus 0.618…), which is 137.508 degrees. Every seed lands in the widest gap the previous ones left behind, which is why the disc fills so evenly.
The golden angle is picky. Press the 137.3 and 137.6 degree buttons: a difference of about a fifth of a degree is enough to open clear spokes in the pattern, and the biggest-hole bar jumps. A sunflower has no ruler, yet its growth process settles on this angle with great precision. In 1992 Stéphane Douady and Yves Couder reproduced the effect with a physical experiment in which drops of magnetic fluid repel each other, and the pattern settled onto the golden angle by itself.
Rather than looking at seed 5 or 6, look at the seeds that are almost in line with seed 0. After n turns of the golden angle, seed n is a bit off a whole number of turns. For n equal to 13 it is 0.034 of a turn away, for 21 it is 0.021, for 34 it is 0.013 and for 55 it is 0.008. Those are Fibonacci numbers, each a better near-return than the one before. Seeds 34 apart sit almost directly outward from each other, so lines joining every 34th seed trace out 34 slow spirals, and every 55th gives 55. Use the join buttons to draw them. The spiral counts you see are the near-returns of the golden angle, not a separate design.
Pinecones, pineapples, cauliflower florets and the leaves round some stems follow the same recipe: new parts appear one at a time and turn by nearly the golden angle. Plants do not do arithmetic. They only have to place each new part in the biggest gap, and the golden angle is what that rule produces. Not every plant follows it, and some species sit at other angles, but the sunflower is the classic case.