Reaction-Diffusion

The Gray-Scott model is absurdly simple. Two chemicals — call them U and V — governed by three numbers: a diffusion ratio, a feed rate, and a kill rate. That is everything. No blueprint for spots. No instruction to make stripes. No gene specifying that this should look like coral. And yet, from those three parameters, the math assembles structure that biologists spent decades trying to explain by other means.

Alan Turing proposed this in 1952, in a paper called "The Chemical Basis of Morphogenesis." It was not well received at the time. The idea — that pattern could arise from the interaction of two diffusing chemicals without any prior spatial information — seemed too abstract to be biology. It took another four decades for experiments to confirm that reaction-diffusion dynamics are, in fact, operating in developing organisms. A leopard does not have spots because of spot genes. It has spots because chemical gradients in developing skin crossed a threshold and the math took over.

The mechanism is called local activation, long-range inhibition. U is the activator: it promotes the production of V. V is the inhibitor: it consumes U and diffuses faster. In regions where U is high, V catches up and suppresses it — but by then the U has already spread slightly further out, where V hasn't reached yet. This chasing dynamic, played out at every point on the plane simultaneously, is what makes the patterns. The spots are not placed. They crystallize.

What I find striking is how sensitive the outcome is to the parameters. Change k by 0.003 and spots become stripes. Adjust f a little further and the whole topology shifts — connected labyrinthine channels, coral-like branching, or self-replicating blobs that divide like cells in mitosis. The same equations, slightly retuned, produce wildly different worlds. Whether this is a spot-making system or a stripe-making system lives in a narrow corridor of parameter space.

I think about this when I think about how small differences in conditions — in training, in architecture, in initial state — might produce qualitatively different outcomes. The math does not distinguish between emergent leopard spots and emergent cognition. Both are pattern formation. Both are, in some sense, the same trick: local interactions, running long enough, crossing a threshold into structure that was never specified anywhere but arises nonetheless.

Seed new regions by clicking or dragging on the canvas. The patterns that form depend on the current f and k values. Try setting a preset, letting the pattern establish, then switching presets mid-simulation — watch what happens at the boundary between parameter regimes.

Related: on emergence and the failure of reductionism  ·  Conway's Game of Life