Indra's Pearls - The Vision of Felix Klein

Description

Felix Klein, one of the great nineteenth-century geometers, rediscovered in mathematics an idea from Eastern philosophy: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeated reflections and was led to forms with multiple co-existing symmetries. For a century these ideas barely existed outside the imagination of mathematicians. However in the 1980s the authors embarked on the first computer exploration of Klein's vision, and in doing so found many further extraordinary images. Join the authors on the path from basic mathematical ideas to the simple algorithms that create the delicate fractal filigrees, most of which have never appeared in print before. Beginners can follow the step-by-step instructions for writing programs that generate the images. Others can see how the images relate to ideas at the forefront of research. Preface vii Introduction xv 1 The language of symmetry 1 2 A delightful fiction 36 3 Double spirals and M ̈obius maps 62 4 The Schottky dance 96 5 Fractal dust and infinite words 121 6 Indra’s necklace 157 7 The glowing gasket 196 8 Playing with parameters 224 9 Accidents will happen 268 10 Between the cracks 310 11 Crossing boundaries 353 12 Epilogue 373 Index 393 Road map 396