**How the Leopard Got His Spots**

In one of his celebrated just-so stories, Rudyard Kipling recounted how the leopard got his spots. But taking this approach to its logical conclusion, we would need distinct stories for every animal's pattern: the leopard's spots, the cow's splotches, the panther's solid colors. And we would have to add even more stories for the complex patterning of everything from molluscs to tropical fish.

But far from these different animals requiring separate and distinct explanations, there is a single underlying explanation that shows how we can get all of these varied and different patterns using a single unified theory.

Beginning in 1952, with Alan Turing's publication of a paper entitled "The Chemical Basis of Morphogenesis", scientists recognized a simple set of mathematical formulas could dictate the variety of how patterns and colorings form in animals. This model is known as a reaction-diffusion model and works in a simple way: imagine you have multiple chemicals, which diffuse over a surface at different rates and can interact. While in most cases, diffusion simply creates a uniformity of a given chemicalâ€”think how pouring cream into coffee will eventually spread and dissolve and create a lighter brownâ€”when multiple chemicals diffuse and interact, this can give rise to non-uniformity. Even though this sounds somewhat counterintuitive, not only can it occur, but it can be generated using only a simple set of equations, and in turn explain the exquisite variety of patterns seen in the animal world.

Mathematical biologists have been exploring the properties of reaction-diffusion equations ever since Turing's paper. They've found that varying the parameters can generate the animal patterns we see. Some mathematicians have even examined the ways in which the size and shape of the surface can dictate the patterns that we see. As the size parameter is modified, we can easily go from such patterns as giraffe-like to those seen on Holstein cows.

This elegant model can even yield simple predictions. For example, while a spotted animal can have a striped tail (and very often does) according to the model, a striped animal will never have a spotted tail. And this is exactly what we see! These equations can generate the endless variation seen in Nature, but can also show the limitations inherent in biology. The just-so of Kipling may be safely exchanged for the elegance and generality of reaction-diffusion equations.