Reaction–diffusion.
A reaction–diffusion system on a wrap-around grid. Two chemicals: u is fed in everywhere at rate f; v consumes u and is itself decayed at rate f + k; both diffuse. This is the Gray–Scott model, and tiny shifts in (f, k) swing it between Turing patterns — spots, stripes, mitosing blobs, drifting solitons. Click and drag to paint v seeds; slide the rates; watch.
What you're looking at.
Two equations, repeated for every cell on a 200×200 torus. The feed term f·(1 − u) tops u back up toward 1 wherever it's depleted. The reaction u·v² burns u into v, and the kill term (f + k)·v drains v away. The two diffusion coefficients differ — v spreads more slowly than u — and that asymmetry, plus the autocatalytic v², is what keeps patterns from washing out into a flat equilibrium.
The named regimes all live in a thin sliver of (f, k) space. Move f up at moderate k and isolated spots replicate by binary fission — the mitosis regime. Pull k down and the spots merge into stripes and labyrinths. Out near f ≈ 0.062, k ≈ 0.0609 live the U-skate solitons — self-propelled blobs that glide across the grid without dispersing, like the gliders in Conway's Life. Push f further and you get the branching coral regime.
Alan Turing's 1952 paper proposed exactly this mechanism — two chemicals, different diffusion rates, a nonlinear reaction — as the origin of biological pattern: leopard spots, zebra stripes, the regular spacing of hair follicles, the branching of lungs. No designer, no template; just diffusion and chemistry, applied uniformly, producing structure. That a few lines of math on a wrap- around grid can reach for the same answer is the small miracle this page is trying to demonstrate.