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.