Topic · Reaction–diffusion

Turing patterns

Two chemicals, two diffusion speeds — and a leopard's coat appears.

In 1952 Alan Turing showed that two interacting substances, one diffusing faster than the other, can spontaneously break a uniform mixture into spots, stripes, and branching networks. The same equations explain leopard skin, zebra stripes, angelfish bars, and a startling amount of how an embryo decides where to put things.

Discover the story Open the Explorer

First encounter

Two chemicals. Different speeds. A coat appears.

The big idea

Mix two invisible chemicals on a flat sheet. One spreads slowly and amplifies itself. The other spreads quickly and shuts the first one down. Out of a perfectly uniform soup, dots, stripes, and branching networks slowly resolve. The very same equations explain the spots on a leopard, the stripes on a zebra, the bars on an angelfish — a single rule with two knobs, painted across every surface that needed a pattern.

A concrete example

Call them activator and inhibitor. The activator amplifies itself locally, like a small fire kindling more fire next to it. The inhibitor diffuses out faster and snuffs the activator at a distance. Where the activator wins, you get a spot. Around it, the inhibitor's halo prevents another spot from forming too close. Tile that competition across a surface and you get an even sea of dots — or stripes, if the geometry tilts that way.

Why it matters

Turing wrote this down in 1952, in 'The Chemical Basis of Morphogenesis', before any biology supported it. Modern molecular biology has since found the activator–inhibitor pairs he predicted, in zebrafish skin, mouse hair follicles, and digit spacing in vertebrate limbs. The same mathematics also appears in seashell pigmentation and sand-ripple physics.

Below: run a small Gray-Scott simulator and tour four of the regimes Pearson catalogued in 1993.

Section 01 · The equation

Two coupled PDEs, two diffusion constants

Let u and v be the concentrations of two interacting substances on a flat surface. Each one diffuses at its own rate, and each one reacts with the other. The system reads ∂u/∂t = D_u ∇²u + f(u, v) and ∂v/∂t = D_v ∇²v + g(u, v). The Laplacian terms describe spread; the f and g terms describe the chemistry. That's it — a uniform initial state plus the reaction terms plus a difference in D_u and D_v is enough to make patterns appear out of nothing.

Section 02 · Activator and inhibitor

Local activation, lateral inhibition

The cleanest intuition is the Gierer–Meinhardt picture: u is an activator that catalyses both itself and its own brake, v. The activator wants to ignite into a hot spot wherever a fluctuation lifts its concentration. But it also produces v, which diffuses outward faster and suppresses u in the surrounding ring. Result: every fledgling spot fences off its neighbours. The spacing between spots is set by how far v travels before u can catch up.

activator · inhibitor

substance	role	diffusion
u · activator	Catalyses itself and produces the inhibitor.	Slow (D_u)
v · inhibitor	Suppresses the activator at distance.	Fast (D_v ≫ D_u)

Section 03 · Why diffusion ratios matter

The Turing instability needs the brake to be faster than the fire

Without diffusion, the uniform state is locally stable: small perturbations die out. Turing's surprise was that adding diffusion can destabilise it — but only if D_v ≫ D_u. The slower activator clumps; the faster inhibitor sweeps the clumps clean, except at a characteristic wavelength where the rates balance. That wavelength becomes the spacing of the spots, the period of the stripes. Change the ratio and the pattern's scale changes with it — the same chemistry, retuned by one number.

The Turing condition

D_v / D_u ≫ 1

The inhibitor must outrun the activator. Equal diffusion gives a uniform smear; a strong ratio gives a sharp wavelength — the very scale of the spots.

Interactive · turn the two knobs

Run the model in your browser

Watch real-time Gray-Scott reaction-diffusion on a small grid. The four presets correspond to the canonical regimes Pearson labelled in 1993; the sliders let you slip between them. The dynamics are sensitive: a change of 0.001 in F or k can take you from a leopard's spots to a maze of stripes.

Interactive · Gray-Scott reaction-diffusion

Pearson presets

F · feed0.0250

k · kill0.0600

Tiny 120×100 grid, toroidal wrap, 30 sub-steps per frame. Pick a Pearson preset or sweep the F and k sliders. Tip: small steps matter — most of the (F, k) plane gives empty fields; the interesting strip is razor-thin.

Section 04 · The Gray–Scott model

Two knobs: F (feed) and k (kill)

The most famous concrete realisation is Gray–Scott: ∂u/∂t = D_u ∇²u − uv² + F(1−u), ∂v/∂t = D_v ∇²v + uv² − (F+k)v. The reaction u + 2v → 3v autocatalyses v; the feed F replenishes u, and the kill (F+k) removes v. Sweeping F and k across a tiny rectangle of the (F, k) plane traverses an entire bestiary, first catalogued by John Pearson in 1993: spots, stripes, mazes, coral, mitosis, U-skate world, solitons. One pair of equations, a whole zoo of behaviours.

Interactive · pattern gallery

Four famous Gray-Scott regimes

From a single pair of equations comes an entire bestiary. These four cells show the canonical examples: discrete spots, winding mazes, branching coral, and self-replicating mitosis. They live on tiny grids so they fit on a phone — but each one runs the same equations as the explorer.

Interactive · the Pearson zoo

Spots

Round, isolated dots — the texture of a leopard's coat or a peacock's eye. Each spot fences off a halo around itself; the spacing is set by how far the inhibitor reaches before the activator catches up.

Four regimes, four colour maps. Hover or tap a cell to read what it is. Each cell is its own live simulation.

Section 05 · Biology and beyond

Embryos, fish skin, seashells, sand

Hans Meinhardt spent decades fitting reaction–diffusion models to morphogenesis — limb formation, leaf venation, seashell pigment lines drawn out as a wavefront on a growing edge. Kondo and Asai showed in 1995 that the stripes on Pomacanthus angelfish rearrange exactly as a reaction–diffusion model predicts when the fish grows. The same mathematics describes sand ripples in shallow water and density variations in convecting fluids.

Section 06 · What we still don't know

Open problems and surprising scaling

Which exact molecules play the activator and inhibitor in mammalian skin is still debated — candidates involve Wnt and Notch signalling — and how the system scales the pattern with body size remains a mystery: a mouse and an elephant grow the same kinds of patterns at very different scales, yet the underlying chemistry should fix one absolute wavelength. Mathematically the connection to amplitude equations and pattern selection is well understood; biologically, the surprise that 'Turing was right' keeps deepening rather than closing.

Take it further

Grow some spots.

The Explorer gives you the full 200×200 grid, every Pearson preset, control over the diffusion ratios D_u and D_v, multiple colour palettes, and the freedom to slow the simulation down and watch each step.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Developmental biology

Stripes on zebrafish, spacing of hair follicles in mice, fingerprint formation, and digit patterning in vertebrate embryos have all been measured to follow Turing dynamics.

Plant phyllotaxis

Spiral arrangements of sunflower seeds, pinecones and pineapple scales emerge from reaction-diffusion + golden-angle phyllotaxis — Turing chemistry on a growing surface.

Generative art & graphics

Reaction-diffusion patterns are widely used as procedural texture noise (skin, bark, coral) in 3D modelling tools like Substance Designer and Houdini.

Drug delivery & materials

Self-organising microstructures in polymer membranes and drug-release coatings are engineered using Turing-type instabilities.