The Tickle Trunk

Turing's Reaction-Diffusion Model of Morphogenesis

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This is a small companion piece to my page on L-systems. As mentioned there, in 1952 Alan Turing wrote a paper [2] proposing a reaction-diffusion model as the basis of the development of patterns such as the spots and stripes seen in animal skin. Inspired by the methods described in [1], I wrote the small applet that appears on this page. By entering different constants for the equations, it is possible to produce a variety of natural-looking textures. Since the state of the system is initially random noise, repeating the computation with the same constants will produce a different image with a similar pattern. The algorithm behaves as if the "cells" were arranged on the surface of a torus, which results in textures that can be tiled seamlessly.

Many animals develop their coat patterns in stages. Typically, a secondary pattern will emerge as the animal transitions to adulthood. The following examples all use multiple stages:

To create a multi-stage texture, uncheck the *Randomize Cells at the Start of Each Run*
box and drop the number of iterations to a low value (between 100–400) to
give you better control over the results.

**The applet requires that your browser support at least the Java 6 runtime.**
If the applet doesn't work properly, this is almost certainly the problem.
Click this button to install the latest version of Java:

Some combinations of constant values will not reach a stable state. Typically, trying to solve such a system will eventually underflow the floating point arithmetic used by the applet, producing a blank image. Just pick a different set of values and try again.

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**[1]**
Rafael Collantes. **Algorithm Alley.** *Dr. Dobb's Journal*, December
1996.

**[2]** Alan M. Turing. **
The
chemical basis of morphogenesis.** *Philosophical Transactions of
the Royal Society of London**. B* **327**, 37–72
(1952)

*April 14, 2002 — Updated
January 01, 2011*