Customer Reviews

Indra's Pearls: The Vision of Felix Klein

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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

[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.

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.

The mathematician Felix Klein (1849-1925) made some great discoveries that can now be well understood by using computer graphics. Felix Klein is known for his work in non-euclidean geometry and for his work on the connections between geometry and group theory.

Some fairly simple mathematical ideas and algorithms reveal an endless universe of fractals. The book includes step-by-step instructions for writing computer programs allowing readers to perform further explorations.