Reaction-Diffusion
How a leopard gets its spots
Alan Turing's 1952 paper "The Chemical Basis of Morphogenesis" is one of the strangest documents in science. Its author was the world's foremost expert on computation, and he was asking: how does an embryo — a sphere of genetically identical cells — produce structured form? Spots and stripes, left and right, head and tail?
His answer involved two hypothetical chemicals: an activator that promotes its own production and stimulates pigmentation, and an inhibitor that suppresses the activator. The key: the inhibitor diffuses faster than the activator. This simple asymmetry creates a feedback instability. Where activator is high, it produces more of itself — but also sends inhibitor racing outward. The result: local high-activator peaks surrounded by inhibitor moats, separated by roughly equal distances. Spots. Stripes. Labyrinths. The geometry depends on the ratio of diffusion rates and reaction speeds.
The beautiful thing is that this mechanism is substrate-independent. It works in chemicals, in neural activation, even in predator-prey dynamics. Wherever you have local activation and long-range inhibition, Turing patterns emerge.
"We may hope to 'explain' some of the striking differences between species in terms of the physical and chemical properties of their skins."
— Alan Turing, The Chemical Basis of Morphogenesis (1952)Gray-Scott System · Interactive
The chemistry of form
The Gray-Scott model is a two-chemical reaction-diffusion system (U feeds V; V self-replicates; both decay). The four parameter presets below produce qualitatively different pattern regimes — from coral growth to cell division to fingerprints. Click or drag on the simulation to inject activator chemical.
Coral Growth
Branching, labyrinthine channels. Resembles reef structures and neural dendrites.
Mitosis
Spots that grow and split, mimicking cell division under a microscope.
Zebrafish Stripes
Alternating bands with irregular edges, matching pigmentation seen in fish and big cats.
Worms
Long, meandering filaments that slowly rearrange, like slow-motion wriggling.
Phase Transitions
The magnet that explains everything
In 1925, Ernst Ising solved a simplified model of a ferromagnet: a line of atoms, each with a magnetic "spin" pointing up or down, each trying to align with its neighbors. He found no phase transition in one dimension and concluded the model was uninteresting. He was wrong about the physics, but right about the math — and the model he dismissed became one of the most studied objects in theoretical physics.
Lars Onsager's 1944 exact solution for the two-dimensional Ising model revealed something remarkable: at a critical temperature Tc ≈ 2.27, the system undergoes a continuous phase transition. Below Tc, spins spontaneously align into large domains — the magnet chooses up or down, breaking the symmetry of the equations that govern it. Above Tc, thermal noise destroys order. At exactly Tc, magnetic domains exist at every scale simultaneously — a fractal, scale-invariant state where no length scale is privileged.
This critical behavior is universal: the same mathematics governs phase transitions in fluids, superconductors, ferroelectrics, neural networks, and stock markets. The renormalization group explains why: at the critical point, the system looks the same at every scale, and only the general symmetry of the equations matters, not their specific form.
Ising Model · Interactive
Watch symmetry break
Each pixel is a magnetic spin — pink for up, dark for down. Drag the temperature slider through the critical point at T ≈ 2.27 and watch the system spontaneously choose a majority spin, forming fractal domains that look the same at every zoom level.
T < 2.27
T > 2.27