Indra's Pearls: The Vision of Felix Klein

by David Mumford, Caroline Series, David Wright

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.

Genres: Mathematics, Nonfiction, Science, Reference

418 pages, Hardcover

First published April 24, 1998

About the author

David Mumford is a Professor Emeritus of Mathematics at Brown University.

(David B. Mumford)

Reviews

[Chris · Rating 5 out of 5]

I would give this book 6/5 stars if I could.

This is the one math book that non-mathematicians should read. One of the few opportunities non-mathematicians have to feel one thread of an exciting tapestry they don't even know exists.

If you think mathematics is work and not play, you're luckily wrong.



If I could force one book into high schools, this would probably be it.

[Alun Williams · Rating 5 out of 5]

This is a wonderful book, beautifully produced,illustrated and edited, and, considering its subject-matter, requiring amazingly little in the way of previous mathematical knowledge. It will be most enjoyed by readers prepared to undertake the "projects" that occur through-out the book, which means that access to a computer, and some means of programming will be needed. However, it is beautiful just to look at, and has many short standalone sections (such as potted histories and biographies), which make this an excellent book for a mathematician to have on his or her coffee table. A teacher running a maths-club or a computer-programming club for bright secondary school students ought to be able to find plenty of interesting projects in this book.

Early chapters of the book explain the concepts of symmetry groups, complex numbers, and transformations. Then the reader is introduced to Möbius transformations and various families of Kleinian groups with each new family giving rise to more intricate fractal patterns than the last, and by the end of the book has learned some of the mysteries of cusp and degenerate groups, and the connections with hyperbolic geometry.

I cannot commend this book too highly. Even casual readers can enjoy dipping into it, or simply enjoy the pictures, but it is a book to be studied and worked at over a long period. 7 years after buying it there are still ideas within it that I am still exploring and learning more about.

[R Ramachandran · Rating 5 out of 5]

Quite revealing.
Not much difficult to follow even for a person not trained in mathematics.

[Bob Woodley · Rating 4 out of 5]

Clear explanation of Kleinian limit sets with tons of pretty pictures. One reason this book is so popular is that it has examples of algorithms you can use to create pretty pictures yourself. Wish more math-books had an algorithmic focus instead of a proof-based focus.

So really a great book, but a couple of nit-pics:
1 - They spend a little too much time on the pseudo-code. One senses that the authors were comfortable with the math, but not so comfortable with the code. I have the reverse affinities, so would have preferred a little more detail on the maths, a little less of the obvious (for me) details of the pseudo-code.

2 - We live in a WebGL world now where we have to use pixel shaders to draw graphs. In other words we are given a pixel and have to determine how to color it. This involves inverting all the algorithms in the book in order to use them. The book's algos use a top-down approach instead. (Given a starting point, apply rules Ab, AAb, AAbA, etc to get to the point you have to color). So I couldn't use any of the book's code as I was working with shaders.

[Mikhail Ignatev · Rating 2 out of 5]

Да ладно, это такая, знаете, хорошая книга в подарок: очень красиво о простом, без претензий, всё БЕЗУПРЕЧНО правильно и довольно бессмысленно. Но уж поосмысленней, конечно, чем какая-нибудь "Фрактальная геометрия природы"!!))

[Sally]

available here:
http://books.google.com/books?id=XFE3...