Edited by Nigel Lesmoir-Gordon

Springer, 2010, x+207 pp.

In 1980, at the IBM Thomas Watson Research Centre in upstate New York, the mathematician Benoit Mandelbrot was probably the first person to view a geometric picture of an unusual set of pixels on the computer screen and discover its enormous complexity and geometric richness. This set had its origins much earlier on at the beginning of the 20th century in the research of the French mathematicians in the field of complex dynamics, Pierre Fatou and Gaston Julia. In the eighties, the properties of this set were well investigated and it was named the Mandelbrot set (or commonly called the M-Set). In the following decades, applications and ramifications of the M-set began to emerge in other disciplines.

This book is an offshoot of the full-length 1995 television documentary with the same title, presented by the late science fiction writer and thinker Arthur C. Clarke and produced by the film and documentary maker Nigel Lesmoir-Gordon. While the film is a visually stunning display of computer graphics and animation that captures the imagination of its viewers, the book version is literally an intellectual eye-opener for those who have only vaguely heard of the M-Set or of fractals (a term coined by Mandelbrot himself).

There are not many mathematical equations in the book. There is basically only one important equation — presented in an iterative form \(Z^2Dz^2+c\) (which Arthur

Clarke himself likened to the Einstein equation \(E=mc^2\) in depth and simplicity). The emphasis is on pictures for visualising the endless process of “self-similarity” in nature. The psychedelic colours and dream-like patterns generated have a surrealistic, hypnotic effect. It is almost unbelievable that they are produced mechanically by a deterministic equation. Little wonder that the M-set is sometimes called the “thumbprint of God”.

It is not just the pictures that mesmerise the mind. A hint of the mathematics behind the apparent order emerging from chaotic processes is beautifully illustrated by an imaginary soccer game played in a definite and yet random way. The mathematically minded reader will enjoy the exposition of the chapter on “Fractal transformations” written by Michael Barnsley and Louisa Barnsley. In it you will find references to the Iterated Function System (IFS), the mathematics behind fractal image compression technology.

If you have seen the documentary, you would have heard Michael Barnsley recalling the recurrent nightmare he had for over 20 years which finally gave him the key to fractal image compression — the Collage Theorem. For the benefit of those who have not watched the film, it is an account that is reproduced in “The Film Script” which is appended at the end of the book. If you have seen the film, reading this complete film script is like watching it again in your mind. If you read it first and then see the film, you would appreciate the film even more.

Lesmoir-Gordon describes in the book how his first acquaintance with the M-Set in 1991 led him in an obsessive-compulsive and yet serendipitous way to a collaboration with Arthur Clarke on a project beyond his wildest imagination. It is an inspiring behind-thescenes account that is not usually revealed or widely known.

The serious reader will find it intellectually challenging and satisfying when digesting the 134 pages of exposition on the geometric visualisation of the M-Set, the tilings of Escher and the fractal view of financial markets — given by the masters and practitioners of the art and science of fractals like Ian Stewart, Arthur Clarke, Benoit Mandelbrot, Michael Barnsley, Will Rood, Gary Flake and David Pennock.

**Y. K. Leong.**

National University of Singapore, Singapore.

*Source:- Asia Pacific Mathematics Newsletter, Volume 1 No. 2 (April 2011).*

*It has been republished here with a special permission from World Scientific.*

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