### Synopses & Reviews

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.

#### Review

"I truly love this book...a magnificent text." American Mathematical Monthly

#### Review

"It has been a great pleasure to read such a gracefully written, original book of mathematics. The three authors, with the support of Cambridge University Press, have produced a book that is 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." Notices of the AMS

#### Review

"The production of the book leaves nothing to be desired. It is spendid. 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

#### Review

"All of it is patiently explained...By the time you finish, you'll know your way around this complex plane." American Scientist

#### Review

"...raises a number of interesting issues... The book itself is a work of art... 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

#### Review

"This book is written as a guide to actually coding the algorithms which are used to generate the delicate fractal filigrees, most of which have never appeared in print before....Beginners can learn to understand what the images mean and follow the step-by-step instructions for writing computer programs that generate them. Experts in the geometry of discrete groups can see how the images relate to ideas that take them to the forefront of research." Mathematical Reviews

#### Synopsis

Highly illustrated realization of infinitely reflected images related to fractals, chaos and symmetry.

#### Synopsis

Felix Klein, a great geometer of the nineteenth century, rediscovered an idea from Hindu mythology in mathematics: the heaven of Indra in which the whole Universe was mirrored in each pearl in a net of pearls. Practically impossible to represent by hand, this idea barely existed outside the imagination, until the 1980s when the authors embarked on the first computer investigation of Klein's vision. In this extraordinary book they explore the path from some basic mathematical ideas to the simple algorithms that create delicate fractal filigrees, most appearing in print for the first time. Step-by-step instructions for writing computer programs allow beginners to generate the images.

#### Synopsis

Highly illustrated realization of infinitely reflected images related to fractals, chaos and symmetry.

#### Synopsis

Felix Klein 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. The whole Universe was mirrored in each pearl. For a century this idea, practically impossible to represent by hand, barely existed outside the imagination of mathematicians. In the 1980s the authors embarked on the first computer exploration of Kleinâs vision. Join them on the path from simple mathematics to computer programs that generate these images related to ideas at the forefront of knowledge.

#### Synopsis

For a century Klein's vision of infinately repeated reflections, practically impossible to represent by had, barely existed outside the imagination of mathematicians, In the 1980's the authors embarked on the first computer exploration of Klein's vision, here available for the first time in print, with the programs that generate them.

### About the Author

David Mumford has been University Professor in the Division of Applied Mathematics at Brown University since 1996. Before that he was at Harvard University for 35 years. He has received many honours for his mathematical work, including a Fields medal in 1974. He has been a contributor to the Calculus Textbook Project, led by Hughes-Hallet and Gleason.

Caroline Series is Professor of mathematics at Warwick University where she currently holds an EPSRC Senior Research Fellowship. She was Kennedy Scholar at Harvard University from 1972 to 1974. In addition to technical publications, she has contributed widely to the popularization of mathematics.

David Wright is Associate Professior at Oklahoma State University. He has a guest professorship at the University of Göttingen, and was Sloan Fellow at the Institute for Advanced study in Princeton from 1988 to 1990. From 1997 to 1999 he helped compose the famous William Lowell Putnam Mathematical Competition.

### Table of Contents

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. Indras 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