What a treat to read this! Shows that us mathematicians are often fifty years ahead of the times ;)
Any book by Ian Stewart on mathematics is a good read.
The original paper, Turings last can be read here ~ http://www.dna.caltech.edu/courses/cs191/paperscs191/turing....
But how much cooler is it that this can be observed by a visit to a zoo! He (or perhaps biologists who studied his work?) conjectured that some biological process behaved according to his equations, and therefore produced the behavior they model. However, until now, no one had ever found such a process.
In fact, they had shown it not to be true for some other striped animal (can't recall which one), which I think makes this discovery even more interesting.
We present a method for texture synthesis based on the simulation of a process of local nonlinear interaction, called
reaction-diffusion, which has been proposed as a model
of biological pattern formation. We extend traditional
reaction-diffusion systems by allowing anisotropic and
spatially non-uniform diffusion, as well as multiple competing directions of diffusion. We adapt reaction-diffusion
systems to the needs of computer graphics by presenting
a method to synthesize patterns which compensate for the
effects of non-uniform surface parameterization. Finally,
we develop efﬁcient algorithms for simulating reactiondiffusion systems and display a collection of resulting textures using standard texture- and displacement-mapping techniques.
The why seems simpler: camouflage.
for what? for what reason, cause, or purpose? 
I am talking about cause and you're talking about purpose. I think we can both ask why