Reaction-Diffusion
Two chemicals, simple math, infinite patterns.
How reaction-diffusion works
What you're seeing
Two chemicals — call them U and V — spread across a grid. U is the substrate (food), V is the catalyst. V consumes U and produces more of itself, but V also decays. The color shows the concentration of V: more V means more color.
Why patterns form
Alan Turing figured this out in 1952. If V diffuses slower than U, something strange happens: a small patch of V eats the local U, then U diffuses back in from the surroundings, feeding the V further. But the V can't spread as fast, so it stays concentrated. This "activator-inhibitor" dynamic creates stable structures from nothing — spots, stripes, labyrinths.
The parameter space
Two numbers control everything: feed rate (f) and kill rate (k). f controls how fast U is replenished. k controls how fast V decays. Small changes in these numbers produce wildly different patterns — mitosis (splitting blobs), coral (branching growth), stable spots, winding stripes, pulsing waves. There's a whole phase diagram of behaviors in the f-k plane.
Where you'll find this
Leopard spots, giraffe patches, zebrafish stripes, coral growth, fingerprints, the Belousov-Zhabotinsky chemical reaction, seashell pigmentation patterns — all driven by the same activator-inhibitor mechanism. The math doesn't know it's making a leopard. It just diffuses and reacts.