Grow a sunflower
Each new seed is placed a fixed angle around from the last one, and pushed outward as the flower grows. Drag the angle slider slowly, then press golden. Most angles give ugly spokes or gaps. One angle fills the disc evenly, with no spokes and no gaps opening up.
What's going on
If the angle is a simple fraction of a full turn, like 90° (a quarter) or 120° (a third), seeds line up in a few straight spokes, wasting all the space between them. Angles close to a simple fraction give spokes that slowly curl.
The golden angle, about 137.508°, is the "most irrational" fraction of a turn: it is as far as possible from every simple fraction. So seeds never line up, and the disc fills evenly no matter how many seeds there are. Even 137.4° or 137.6° eventually falls apart into spokes if you add enough seeds.
Count the spirals you can see curving one way, then the other. You will get neighbouring Fibonacci numbers: 13 and 21, or 21 and 34. Many real sunflower heads show the same counts. Real flowers are not perfect machines, though: a study of hundreds of heads found some with other counts and some with no clear spiral count at all. The golden angle is an idealised model that nature follows closely, not exactly.
Sheet: Fibonacci and the golden ratio.