Turing Patterns

In 1952, two years before his death, Alan Turing published a paper that had nothing to do with codebreaking or computers. He asked how an embryo, which starts as a ball of nearly identical cells, decides where the stripes go. His answer was chemistry: two substances that react with each other and spread at different speeds. One makes more of itself; the other holds it back and travels faster. An even mixture of the two can be unstable, breaking up by itself into spots or stripes with a natural spacing — no blueprint required.

The surprise is how little it takes. The dish on this page runs one such pair, the Gray–Scott reaction, in which a chemical V eats a feed chemical U to make more of itself. Every preset uses the same two equations, the same diffusion rates, the same time step and the same grid. The only things that change are two numbers: how fast U is poured in (the feed, F) and how fast V is drained away (the kill, k). Move them a few thousandths and the leopard becomes a zebra, the zebra becomes coral, and coral becomes a culture of cells that grow, pinch in the middle and split in two.

The map beside the dish is the whole space of those two numbers. The dashed line is a fact about the reaction on its own: to its left an even, living mix of both chemicals can exist; to its right the only even state is an empty dish. Every pattern here lives within a hair of that line, and the spots and dividing cells live just past it, where no even mixture can survive and life persists only in separate islands. Drag the point across the line and watch what the dish does.

Real biology is messier than two equations, but Turing's mechanism has turned up where he guessed it might: in the spacing of hair follicles in mice, the ridges on the roof of the mouth, the digits of a developing limb and the stripes of zebrafish. Similar arithmetic describes vegetation bands in dry grassland and the patterns on some seashells. A simple local rule, applied everywhere at once, is enough to draw a leopard.