Watch spots, stripes and coral grow themselves out of nothing but two chemicals spreading and reacting. This is the Gray-Scott model — the classic demonstration of Turing patterns, where a fast diffuser and a slow one break their own symmetry into structure. Like our Game of Life and falling sand, it's rich global order emerging from nothing but simple local rules. Tune the feed and kill rates, or paint your own reaction fronts onto the canvas.
Click or drag on the canvas to paint reaction fronts
PatternCoral
f, k0.0545, 0.0620
Iterations0
In 1952 Alan Turing showed that two reacting, diffusing chemicals can break their own symmetry into spots and stripes. His paper on morphogenesis proposed that a slow-spreading activator and a fast-spreading inhibitor are enough to turn a featureless start into an ordered pattern — a mechanism now widely invoked for leopard spots, zebra stripes and seashell markings. The Gray-Scott model shown here is one of the simplest systems that reproduces them: change nothing but the feed and kill rates and the same two equations grow coral, dividing cells, mazes or moving waves.
How It Works
Two chemicals, a reaction, and two speeds of spreading — nothing more.
1
React and diffuse
Chemical U is fed in everywhere; the reaction U + 2V → 3V turns it into V; and V is steadily removed. Both chemicals also diffuse, but U spreads about twice as fast as V. Each cell updates from just its neighbours — a local rule applied everywhere at once.
2
The feed/kill balance picks the pattern
Two dials decide everything: the feed rate f (how fast fresh U arrives) and the kill rate k (how fast V is removed). A low kill lets V spread into mazes and coral; a higher kill pinches it into discrete spots; special balances make cells that keep dividing or fronts that travel as waves.
3
Local rules become a global Turing pattern
No cell knows the shape it belongs to, yet the fast inhibitor and slow activator settle into stable, evenly spaced marks — a Turing pattern. Paint a fresh blob of V and watch a new front nucleate and organise, the same way structure is thought to emerge on a growing animal's skin.
What is reaction-diffusion?
Reaction-diffusion describes what happens when substances both spread out (diffuse) and transform into one another (react) at the same time. When two chemicals diffuse at different speeds and feed back on each other, the smooth mixture can spontaneously organise into stable spots, stripes and labyrinths instead of blurring into uniform grey. It is a standard model for pattern formation in chemistry, biology and physics.
What is the Gray-Scott model?
The Gray-Scott model is a simple two-chemical reaction-diffusion system. A chemical U is steadily fed in, a reaction U + 2V → 3V converts it into V, and V is removed at a set kill rate. Each chemical also diffuses, with U spreading roughly twice as fast as V. Despite only two parameters — the feed rate f and the kill rate k — it reproduces a remarkable zoo of patterns: dividing cells, coral, moving spots, mazes and travelling waves.
What are Turing patterns, and how do they relate to animal coats?
Turing patterns are the spots and stripes that emerge when a slow-diffusing activator and a fast-diffusing inhibitor react together — a mechanism Alan Turing proposed in 1952 to explain how a featureless embryo can develop structure. The same short-range-activation, long-range-inhibition idea is widely used to explain leopard spots, zebra and tiger stripes, the ridges on seashells and the markings on tropical fish, where the balance of the two signals sets the size and spacing of the marks.
What do the feed and kill rates do?
The feed rate f sets how fast fresh U is supplied, and the kill rate k sets how fast V is removed. Together they decide which pattern the system settles into. Low kill relative to feed lets V spread into mazes and coral; higher kill isolates it into discrete spots; particular balances produce cells that keep dividing (mitosis) or fronts that travel as waves. Small changes to f and k can switch the pattern entirely, which is why they are the two dials that matter most.