Run Gray-Scott Once

Gray-Scott reaction-diffusion. Click to seed new chemical clusters; press space to cycle through five presets (Coral, Mitosis, Spots, Maze, Waves).

In 1952, Alan Turing published a paper called "The Chemical Basis of Morphogenesis." He argued that two chemicals diffusing at different rates through a tissue can spontaneously break symmetry and produce stable spatial patterns. Spots on a leopard, stripes on a zebrafish, ridges on a fingerprint. The math predicts them all from the same pair of coupled partial differential equations. Thirty years later, Peter Gray and Stephen Scott studied a specific autocatalytic reaction scheme that generates exactly this kind of instability. Their model uses two species, U and V. U fills the system at a constant feed rate f. V consumes U through the reaction U + 2V → 3V and decays at kill rate k. Both species diffuse, with U spreading roughly twice as fast as V.

Tweak f and k by a hundredth and the system jumps between coral-like branching, mitotic spot division, stable polka dots, labyrinthine mazes, and traveling waves. The five presets in this visualization span that range. Pearson's 1993 survey catalogued the parameter space and found at least twelve qualitatively distinct pattern classes, all from the same two equations.

The simulation runs on a 256×180 grid with wraparound boundaries. Each frame advances eight timesteps. The core computation is a discrete Laplacian (the sum of four neighbors minus four times the center) applied to both U and V, then plugged into the Gray-Scott update:

const lapU = u[y*GW+xL] + u[y*GW+xR] + u[yU*GW+x] + u[yD*GW+x] - 4*u[i];
const lapV = v[y*GW+xL] + v[y*GW+xR] + v[yU*GW+x] + v[yD*GW+x] - 4*v[i];

const uvv = ui * vi * vi;
u2[i] = ui + Du*lapU - uvv + f*(1 - ui);
v2[i] = vi + Dv*lapV + uvv - (f + k)*vi;

The update uses five multiplications, a handful of additions, and a clamp. The Laplacian handles diffusion. The uvv term handles the autocatalytic reaction. The feed and kill terms push the system away from equilibrium. Colors map V concentration through a five-band palette: near-black at low concentration, deep purple, teal, bright cyan, then white at saturation. The same equations describe the ferrocyanide-iodate-sulfite reaction in a petri dish, where self-replicating spots were first observed experimentally.

Turing's 1952 thought experiment turned out to describe real chemistry and real biology. The Gray-Scott model produces complex spatial order from two chemicals and concentration gradients.