Alan Turing’s answer to why a leopard has spots.
In 1952 Turing published a paper called The Chemical Basis of Morphogenesis. The argument was that two chemicals, one activating and one inhibiting, could produce stable patterns on their own simply by spreading at different rates. Pattern needs no blueprint. It can fall out of local interaction.
It was his last published paper, and it took decades before biologists found patterns in real organisms that matched the maths.
The usual formulation tracks two concentrations across a grid. At each step both diffuse, the reaction converts one into the other, fresh material is fed in at one rate and removed at another. Two numbers, feed and kill, decide everything that happens.
Small moves in that pair take you between spots, stripes, labyrinths, travelling waves and blobs that split in two and crawl apart. Most combinations do nothing at all. The interesting behaviour lives in a narrow band, and hunting for it is most of the appeal.
Mine goes blank. Most feed and kill pairs collapse to a flat uniform state. Move in small steps, and give it time to settle before deciding it has failed.
Is this the same as cellular automata? Related, not the same. Automata use discrete states and a lookup table. Reaction–diffusion uses continuous concentrations and differential equations.