Two chemicals diffusing at different rates, one activating the other and the other inhibiting back. From this minimal recipe — Alan Turing's 1952 paper, two years before he died — emerge the stripes of a zebra, the spots of a leopard, the patterns on a tropical fish. Click to seed; watch the system find its pattern.
Two chemicals u (substrate) and v (activator). The activator v consumes the substrate (the u·v² term), and the substrate replenishes at rate F. v dies at rate F + k. Crucially, the substrate u diffuses about twice as fast as the activator v — local activation, long-range inhibition. That difference in diffusion rates is the essential ingredient.
Turing's insight (1952): in a uniform initial state, any tiny perturbation grows because the activator spreads faster than its own substrate replenishes. Spatial frequencies near the "Turing wavelength" amplify; everything else damps. The result is a pattern whose size scale is set by the diffusion ratio, completely independent of the system's size. That's why a small fish and a large fish have similar-sized spots.
Biology really does use it. Kondo & Asai (1995) showed the stripes of the angelfish Pomacanthus rearrange as they grow exactly as a Turing model predicts; in zebrafish the pigment cells themselves interact like an activator–inhibitor pair (Nakamasu et al. 2009); and Müller et al. (2012) measured the signaling proteins Nodal and Lefty diffusing at different rates in the early zebrafish embryo — a short-range activator and a long-range inhibitor. Decades before the experiments, Turing had the mechanism from first principles.