One number, four different disguises: a self-repeating rectangle, a hidden triangle inside a star, a ladder of fractions, and the reason a sunflower's leaves never block each other's sun.
Every idea on this page comes from one number. It has no simple fraction, but it has a very simple personality: it's the only positive number where squaring it, and just adding 1 to it, land you in exactly the same place.
Draw a regular five-pointed star by connecting every point of a pentagon to every other point. Hidden inside it are triangles where the long side, divided by the short side, always gives you φ.
Take the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13...) and divide each one by the one before it. The answers zig-zag past φ, a little closer every single time.
A plant must grow its leaves without them overlapping, or it will starve in its own shadow. Test different angles to see how φ dictates survival.