Indra's Pearls: The Vision of Felix Klein [Hardcover]

David Mumford (Author), Caroline Series (Author), David Wright (Author)

Book Description

ISBN-10: 0521352533 | ISBN-13: 978-0521352536 | Publication Date: May 6, 2002

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.

Editorial Reviews

Review

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

"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

"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

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

"...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

"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

Book Description

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.

Product Details

• Hardcover: 416 pages
• Publisher: Cambridge University Press (May 6, 2002)
• Language: English
• ISBN-10: 0521352533
• ISBN-13: 978-0521352536
• Product Dimensions: 10 x 7.8 x 1 inches
• Shipping Weight: 2.6 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #305,680 in Books (See Top 100 in Books)

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15 of 15 people found the following review helpful:

5.0 out of 5 stars How mathematics can be used to create physical beauty, August 10, 2003

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
(TOP 500 REVIEWER)    (VINE VOICE)    (HALL OF FAME REVIEWER)   

This review is from: Indra's Pearls: The Vision of Felix Klein (Hardcover)

As a long-time reviewer of mathematics books, there was a time when I grew very bored with books written for the general mathematical audience. For years, it seemed mandatory that all contain a section on basic fractals and the Mandelbrot and Julia sets. It was not that the topics were not interesting, I found them fascinating, it was just that the explanations were all so similar that it became tedious to read them. Therefore, when I looked at the coverage of this book, I felt a pang of negative nostalgia, thinking that what I would find would be a repeat of what I had read so many times.
Well, I am happy to report that my pang was unfounded. The first chapter covers the language of symmetry, and some of the enormous number of forms in which it appears, which sets the stage for the fractal operations. A large part of the book is devoted to the patterns that are simultaneously symmetrical under two Mobius maps, which makes the analysis of fractals in this book different from what I have seen in others. Indra's necklace is a limit set formed by a chain of tangent circles, and is quite beautiful.
Very high quality figures are heavily used throughout the book to demonstrate the results of the operations. They are also beautiful, and in my opinion, some are works of art. Other mathematical operations that are used in the generation of the results are: matrix operations, group theory, non-Euclidean geometry, continued fractions, formal language theory, tiling of surfaces and function theory. The incorporation of so many different areas of mathematics really spices up the book, and makes it more enjoyable for a wider audience of mathematicians. It cannot be said that it is written for a general audience, the level of mathematics is beyond the non-mathematician, and one probably has to have the skill set of a junior or senior undergraduate math major in order to understand the explanations.
Mathematical results are very beautiful in their internal consistency and the power of the ideas. In this book, you also see some of the physical beauty that can be created by applying mathematics in the appropriate way

13 of 14 people found the following review helpful:

5.0 out of 5 stars Discrete groups made easy, October 4, 2002

By 

alessandro rosa (Brindisi, Italy) - See all my reviews

This review is from: Indra's Pearls: The Vision of Felix Klein (Hardcover)

[this review shall replace the already existing one]

Indras pearls provides a very well-made introduction to the basics of the theory of discrete groups acting on the complex plane. The whole discussion on the related limit sets had been accomplished in such a hand-by-hand method.
The reader starts from complex numbers and after he is led into the deepest concepts: Möbius trasformations, limit sets of discrete groups (Schottky, Fuchsian, ...).
These limit sets are related to another interesting topic in today maths: complex dynamics on the Riemann sphere (Julia sets, ...).
As known, computer experiments had been fundamental for supporting complex dynamics and the successive success of this latter topic helped to promote and increase the interests for discrete groups too: in fact this book evinces already strong interest in the visualization and in the study of the properties of such limit sets since '80s, due to the efforts of the same authors.
One of the major points of attraction in Indra pearls is that all the theory had been helped by displaying a lot of detailed and colorful pictures which, aside the historical biography of the mathematicians that contributed to this theory, set this book as one of the masterpieces in this topic, for his lucid
and fresh approach to basic concepts.
In addition, the presence of amusing comic-strips, explaining some topological concepts on manifolds (for example), guarantees the easy-learning for the reader and also the approach, as imaginaed and completely accomplished by the authors. In this direction, it is clear how passion had been squandered by authors.
The goal has been reached: finding an easy way to introduce the harsch theory of discrete groups.
Interested readers will be rewarded and also excited.
No doubts: this book strikes and it will be a corner-stone for present and future.

6 of 8 people found the following review helpful:

5.0 out of 5 stars Discontinuous Groups now made easy !, September 24, 2002

By 

Alessandro Rosa (Brindisi, Italy) - See all my reviews

This review is from: Indra's Pearls: The Vision of Felix Klein (Hardcover)

Indras pearls provides a very well-made introduction to the basics of the theory of discontinuous groups acting on the complex plane. The whole discussion about limit sets had been accomplished in such a hand-by-hand method.
That is, the reader starts from complex numbers and, after, he is taken into deepest concepts as Möbius trasformations and so to discontinuous groups (Schottky, Fuchsian, ...).
Limit sets of kleinian groups are related to another interesting topic in today maths: complex dynamics on the Riemann sphere (Julia sets, ...). The success of this latter topic helped to increase the interests for discontinuous groups too. Indra pearls also witnesses and resumes the last twenty years of efforts spent for studying the properties of the limit sets.
One of the major points of attraction in Indra pearls is that all the theory had been helped by displaying a lot of detailed and colorful pictures which, aside the historical biography of the mathematicians that contributed to this theory, set this book as one of the masterpieces in this topic, for his lucid
and fresh approach to basic concepts.
In addition, the presence of amusing comic-strips, explaining some topological concepts on manifolds, guarantees the easy-learning of the approach, achieved by the authors. In this direction, it could be evinced that authors were really enjoyed while writing.
The goal has been reached: finding an easy way to introduce the harsch theory of discontinuous groups.
Interested readers will be rewarded about their choice and also excited.