Regular patterns are found in abundance in nature, from the spots on a leopard's back to the ripples on a sandy beach or desert dune. There has a been a flurry of recent research activity seeking to explain their appearance and evolution, and the selection of one pattern over another has turned out to be an inherently nonlinear phenomenon. My aim in writing this book has been to provide an introduction to the range of methods used to analyse natural patterns, at a level suitable for final year undergraduates and beginning graduate students in UK universities.

The book brings together several different approaches used in describing pattern formation, from group theoretic methods to envelope equations and the theory of patterns in large-aspect-ratio systems. The emphasis is on using symmetries to describe universal classes of pattern rather than restricting attention to physical systems with well-known governing equations, though connections with particular systems are also explored. I have taken a wholeheartedly nonpartisan approach, unifying for perhaps the first time in a textbook a multiplicity of methods used by active researchers in the field.

It was David Crighton who originally suggested I should write this book. I had been lecturing a Cambridge Part III course on pattern formation, and David mentioned in passing that it might be a nice idea to turn my lecture notes into a book. Of course I had no idea what I was letting myself in for, but David was always persuasive and inspirational so naturally I said yes. Several years of sweat and toil later I have finally produced the book, though it bears little resemblance to my Part III course, which is probably just as well.