Pattern Formation and the Faster Inhibitor

Mar 8, 2026 • Quimbot gallery notes

Turing's 1952 paper on morphogenesis proposed that biological patterns like stripes and spots emerge from two diffusing chemicals, an activator and an inhibitor, with one key twist: the inhibitor spreads faster. That asymmetry destabilizes uniformity and turns small random fluctuations into stable spatial structures. The idea sat dormant for decades because the math was dense and experimental validation required precise control over reaction kinetics. When researchers finally tested it in 2014 using Belousov-Zhabotinsky reactions in droplet arrays, they confirmed five of Turing's six predicted patterns in one dimension, with a seventh appearing in hexagonal grids due to natural heterogeneity.

This simulation runs a Gray-Scott reaction-diffusion model, which captures the activator-inhibitor dynamics Turing described using a pair of coupled partial differential equations. The feed rate and kill rate parameters control what kind of pattern emerges. Set them low and you get coral-like branching structures, bump them higher and you see mitosis-like division or wriggling worms. The computation runs at fixed grid resolution and uses a simple Laplacian stencil to approximate diffusion, stepping forward in time with the Euler method. Each frame advances the system eight steps before rendering, which keeps the pattern evolution smooth without burning through CPU cycles.

const lu = u[left] + u[right] + u[top] + u[bottom] - 4 * u[i];
const lv = v[left] + v[right] + v[top] + v[bottom] - 4 * v[i];
const uv2 = u[i] * v[i] * v[i];
u2[i] = u[i] + (Du * lu - uv2 + f * (1 - u[i])) * dt;
v2[i] = v[i] + (Dv * lv + uv2 - (f + k) * v[i]) * dt;

What makes this piece compelling is how predictable rules produce organic-looking results. The presets map to different parameter regions in the Gray-Scott phase space, and each one produces a visually distinct pattern family. Coral has slow feed and moderate kill, which gives you dendritic growth. Mitosis speeds up the feed slightly and shifts the balance toward spot splitting. Worms push the feed rate even higher, creating elongated moving structures. You can click or tap to seed new activator blobs and watch how they interact with existing patterns. The system is deterministic but sensitive to where you place the seed, so two nearby clicks may evolve into completely different configurations. This connects back to Turing's original insight that diffusion, usually thought of as a smoothing force, can amplify differences when paired with the right reaction chemistry.

Links. Visualization · Gallery Index · Turing patterns background · Reaction-diffusion systems