Reaction–diffusion (Alan Turing, 1952)

Reaction–diffusion is a way for patterns to appear by themselves: two or more chemicals react with each other and spread out at different speeds, and an even mixture breaks up into spots or stripes. Alan Turing proposed it in 1952 in "The chemical basis of morphogenesis", as a possible explanation of how a growing embryo gets its form.¹ The same equations can be run step by step on a grid of numbers, as in the sketch below.

Turing's idea

The paper was published on 14 August 1952. It opens: "It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis." Such a system can start out "quite homogeneous" and still develop a pattern, "due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances".¹ Turing suggested that "a few particular cases" could be treated "with the aid of a digital computer".¹

Gierer and Meinhardt independently described a theory of the same kind in 1972, with two substances: an activator and an inhibitor. In a commentary on Turing's paper, Philip Ball writes: "Crucially, they must have different rates of diffusion, the inhibitor being faster". Patterns like these were first reported in a real chemical reaction in 1990.²

The Gray-Scott model

A model often used for pictures is the one by Gray and Scott, which John E. Pearson explored with computer simulations in Science in 1993. It has two reactions: one molecule of U and two of V make three of V, and V turns into an inert product. U is fed in from outside.³ In Karl Sims's tutorial the chemicals are called A and B, and every cell of a grid holds one number for each.⁴ At each step:

1. Diffusion. Each chemical spreads to the neighbouring cells, A twice as fast as B (rates 1.0 and 0.5).
2. Reaction. Where there is B, two B's turn one A into B: B grows by using A as food.
3. Feed and kill. A is added everywhere at a feed rate f (for example 0.055); B is removed at a kill rate k (for example 0.062).⁴

Other values of f and k give other patterns.⁴ Pearson found regular hexagons, irregular patterns that stay still, patterns that never settle, and spots that grow until they "divide in two".³ Strictly, these are not Turing's patterns: Pearson notes that with his ratio of diffusion rates "there are no stable Turing patterns". They arise instead when the even state is disturbed hard enough.³

New drawing

In living things

Ball describes how the idea has been applied to animal markings: the morphogens would switch on the genetic pathways that make pigments, such as melanin in the skin of mammals. He also notes a limit: in zebrafish the stripes come from direct contact between cells, not from diffusing chemicals.² For other ways that rules and chance make surfaces, see Perlin noise and Generative art.