Indra's Pearls
Cambridge University Press, 4/25/2002
EAN 9780521352536, ISBN10: 0521352533
Hardcover, 416 pages, 25 x 20 x 3 cm
Language: English
Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: 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 coexisting symmetries. For a century, these images 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
Introduction
1. The language of symmetry
2. A delightful fiction
3. Double spirals and Möbius maps
4. The Schottky dance
5. Fractal dust and infinite words
6. Indra's necklace
7. The glowing gasket
8. Playing with parameters
9. Accidents will happen
10. Between the cracks
11. Crossing boundaries
12. Epilogue
Index
Road map.
'I rarely feel a certain kind of euphoria by just looking at the cover of a mathematics book. But that happened with Indra's Pearls: The Vision of Felix Klien ... fantastic illustrations together with apparently well founded mathematical explanations. and that presented in an accessible way, which dares to prioritize general comprehension above a strict theoretical approach ... As far as I know this book is one of the most beautiful examples of the illustration of the inherent aesthetic beauty (which exists)within mathematics ... the images are of the highest quality obtainable at present for mathematical structures. Everyone, who ever tried to create something comparable, knows how difficult it is.' Juergen Richter-Gebert, Technische Universitaet Muenchen 'I truly enjoyed reading Indra's Pearls. I am sure that the book will have a major impact on the way we teach geometry and dynamics ... a jewel that will more than repay the persistent reader's efforts.' Science 'The production of the book leaves nothing to be desired. It is splendid. Printed entirely on glossy paper, with practically all of the many figures in glorious color, the book has a number of admirable design features: large type and wide margins wherein references are given and occasional comments (often quite talky) are made. CU Press has done a beautiful job, and David Tranah of the CU Press deserves special commendation for his role in pulling out all the stops.' SIAM News 'This unique book can serve as a pedagogical and visual introduction to group theory for school children, and yet is just as suitable for professional mathematicians: I believe that both of them would read the book from the beginning to the end. Finally, it can be used as a book for popularising science, but is very different from most fashionable books on strings, black holes, etc.: it gives you the joy of seeing, thinking and understanding.' EMS 'It has been a great please to read such a gracefully written, original book of mathematics. In the academic world the book could be used as the basis of a college course on 'chaos and fractals', or read as an introduction to group actions, or read just for enjoyment by graduate students and, not least, by professors. The three authors, with the support of Cambridge University press, have produced a book that it as handsome in physical appearance as its content is stimulating and accessible. The book is an exemplar of its genre and a singular contribution to the contemporary mathematics literature.' AMS 'This is a beautifully presented book, rich in mathematical gems.' The Mathematical Gazette 'One can browse through the numerous beautiful and fascinating pictures and marvel at them. ... Readers with widely different backgrounds will find something enjoyable in this unique book.' Acta Scientiarum Mathematicarum