What is on screen

The canvas is a map of one quantity: the concentration of chemical V in each cell.

Dark cells hold no V. They are pure U, the other chemical, which fills the lattice at the start. Orange is a moderate amount of V, and pale yellow is the most V any cell reaches. U is not drawn separately, because it is close to the complement of what is shown: dark regions are full of U, bright regions have had their U consumed.

The bright edges around each shape are where the reaction is actually happening. The interior of a large blob is spent — the U there is gone — so growth only ever occurs at a boundary. That is why the patterns read as fronts, rings and filaments rather than solid areas.

How it works

Every cell holds an amount of U and an amount of V, and three things happen to them each step.

The reaction. U + 2V → 3V. V converts U into more V, so V is self-amplifying: where there is a little, more appears, as long as there is U left to convert. The rate goes as $uv^2$, so it is effectively zero where V is absent. This is why a lattice of pure U sits still forever, and why drawing is what starts anything.

The feed and the drain. U is replenished everywhere at rate F, fastest where it has been most depleted. V is removed everywhere at rate F + k. Without the drain, V would eventually convert the whole lattice; without the feed, it would exhaust it. The pattern is the balance.

Diffusion. Both chemicals spread into neighbouring cells, U at twice the rate of V. That difference is what makes patterns instead of blur: V builds up locally while the U it needs is drawn from a wider area, so each structure suppresses others nearby and the pattern acquires a spacing. Turing showed this is enough for a uniform mixture to break up on its own.

Together:

$$\frac{\partial u}{\partial t} = D_u \nabla^2 u - uv^2 + F(1-u), \qquad \frac{\partial v}{\partial t} = D_v \nabla^2 v + uv^2 - (F+k)v$$

with $D_u = 0.16$, $D_v = 0.08$. The lattice wraps at every edge, so a shape leaving one side returns on the opposite one.

Controls:

Draw deposits V under the cursor; Erase returns cells to pure U. Both work while running, so a new seed can be dropped into a finished pattern. Reset scatters a few random seeds, Clear empties the lattice to draw on from scratch, and Random seed adds more blobs without removing what is there.

The seed sets where structure appears, not what kind: a dot, a line and a scribble all settle into the same pattern for given F and k, because the spacing comes from the diffusion rates. A symmetric seed does give a symmetric result.

F and k are the only difference between every pattern here. The preset buttons jump to seven regions of that plane — spots, worms, chaos, maze, solitons, fingerprint, holes — each named for what it produces. Switching preset does not clear the lattice, so an existing pattern reorganises into the new regime. Outside a narrow band the pattern either floods until V fills the lattice or dies back to pure U.

Graphs:

(a) mean concentration $\langle v \rangle$ against time. Rising means the pattern is still spreading; flat means the state on screen is its final one. Under chaos it never settles.
(b) distribution of v over all cells, log scaled. Two peaks, one for the dark background and one for the bright structures, with few cells between them — which is why the pattern reads as objects on a ground rather than a gradient.
(c) the current (k, F) as a cross, with the presets as open circles. Moving either slider moves the cross; leaving the band stops the pattern.