What kind of a book is this?

This is a book about serious mathematics, but one which we hope will be enjoyed by as wide an audience as possible. It is the story of our computer aided explorations of a family of unusually symmetrical shapes, which arise when two spiral motions of a very special kind are allowed to interact. These shapes display intricate ‘fractal’ complexity on every scale from very large to very small. Their visualisation forms part of a century-old dream conceived by the great German geometer Felix Klein. Sometimes the interaction of the two spiral motions is quite regular and harmonious, sometimes it is total disorder and sometimes – and this is the most intriguing case – it has layer upon layer of structure teetering on the very brink of chaos.

As we progressed in our explorations, the pictures that our computer programs produced were so striking that we wanted to tell our tale in a manner which could be appreciated beyond the narrow confines of a small circle of specialists. You can get a foretaste of their variety by taking a look at the Road Map on the final page. Mathematicians often use the word ‘beautiful’ in talking about their proofs and ideas, but in this case our judgment has been confirmed by a number of unbiassed and definitely non-mathematical people. The visual beauty of the pictures is a veneer which covers a core of important and elegant mathematical ideas; it has been our aspiration to convey some of this inner aesthetics as well.