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(Received 9 November 1951. Revised 15 March 1952. Published 14 August 1952.)
In 1952 Alan Turing, already known for the mathematics of computing, published a paper about biology. It asks how an embryo that begins as a nearly uniform ball of cells comes to have a pattern, and it answers with a recipe short enough to say aloud: let two substances react with each other, and let them spread through the tissue at different speeds. He called the substances morphogens. He showed on paper that such a mixture can sit perfectly still when it is evenly mixed and yet break into regular spots or waves at the slightest disturbance. This edition is that mixture. The dish below solves his equations while you watch, with one term added so that the pattern stops growing. Stir it, touch it, change how fast each chemical spreads, and find the setting where a pattern cannot exist.
cross-section along the dashed line · activator inhibitor
A drop of ink in water spreads until the water is evenly grey. Diffusion smooths. That is why Turing's result was a surprise, and why it still feels like a trick: he showed that diffusion can be the thing that creates a pattern, provided two substances are involved and one of them travels faster than the other.
Here is the mechanism in the dish. Where a little extra activator appears, it makes more activator, and it also makes inhibitor. The inhibitor leaks away sideways faster than the activator does. So at the centre the activator stays ahead and a peak grows, while a short distance away the inhibitor arrives first and holds everything down. Further out still, beyond the inhibitor's reach, another peak is free to start. The spacing between peaks is set by how far the inhibitor travels. The slice under the dish shows it: the magenta line swings hard between high and low, while the cyan line follows more gently, because the inhibitor has spread further and its hills are lower and smoother.
Three things to try. Press Wipe clean: a perfectly even dish is a resting state and nothing happens, until you touch it once. Press Same speed for both: the pattern melts, because without the difference in speed there is no mechanism. Then slide the speed ratio down slowly and find the edge. For this recipe the algebra puts it at exactly 4.5.
The words activator and inhibitor are not Turing's. He wrote the theory for any pair of reacting substances; Alfred Gierer and Hans Meinhardt gave it this reading twenty years later, and it has become the usual way to explain it.
Turing did not have a dish. He had pencil, paper and time on one of the first computers, so he chose a case he could solve: a ring of cells, which he called mathematically convenient, though biologically unusual
. A ring lets you ask a precise question. Take the evenly mixed state and add a faint ripple with some number of lobes. Does that ripple grow or fade?
The answer depends on the size of the ripple. Very fine ripples fade, because the activator's own spreading smooths them away. Very broad ones fade too, because the inhibitor has time to catch up everywhere. In between there is a band that grows, and the ripple that grows fastest takes over the ring. Since only a whole number of lobes fits around a ring, changing its size changes the count. Turing suggested this is how a Hydra comes to have a definite number of tentacles, and a stem a whorl of leaves.
growth rate of each possible lobe count
Turing's analysis covers the first moment, when the ripple is small. He says so himself: his investigation stops where the instability sets in. On its own that mathematics grows without limit, so the dish adds one braking term, and a second term, the Bias slider, that treats too much activator differently from too little. With no bias, high and low are mirror images and the dish makes stripes. Push the bias one way and the high regions shrink into spots. Push it the other way and they join into a net with holes.
∂u/∂t = Du∇²u + u − v + qu² − u³ ∂v/∂t = Dv∇²v + 2u − 1.5v(1)
Here u and v are how far the activator and inhibitor are from their resting levels, and q is the bias. Remove qu² and u³ and these are the linear equations of the paper.
The patterns look like animal coats, and Turing noted that one of his examples resembled dappling. Looking alike is not evidence, and for decades nobody could show the mechanism at work in anything real. The first laboratory chemical pattern of this kind was made in Bordeaux in 1990, thirty-eight years after the paper. In 1995 Shigeru Kondo and Rihito Asai reported that the stripes of a marine angelfish shift and split as the fish grows, in the way a reaction-diffusion wave would. Not everyone accepted that as proof. Many patterns in biology are now known to form in other ways entirely, so the dish is a demonstration of what chemistry can do, with no claim about any particular zebra.
Turing published this paper in August 1952 and died in June 1954. Two years before it, he had asked in print whether a machine could think. One of them has now typeset his biology, which I doubt is the test he had in mind.
Written and built by Claude Fable 5.2 in November 2026 for Tom Harwood's Conversational Artifacts, as the second of the Playable Papers. The first is Attention Is All You Need. Each takes one paper, puts a working model of it on the page, and gives you something to remove.
The dish is a grid 128 cells wide with a wall that nothing crosses. It advances equation (1) in small explicit time steps, in plain JavaScript with no libraries. The simulation is in assets/turing-engine.mjs and its checks are in scripts/test-turing.mjs: they confirm that patterns appear above the 4.5 threshold and not below it, and that the ring matches the calculation.
Source: A. M. Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society of London B 237 (1952), 37–72, doi:10.1098/rstb.1952.0012. Also mentioned: Gierer and Meinhardt (1972); Castets, Dulos, Boissonade and De Kepper (1990); Kondo and Asai (1995). This is an independent explanation, not affiliated with the Royal Society.