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on 1 May 2003
This is one of those rare books which achieves clarity and accessibility without over simplifying. Based around an explanation of techniques for creating images of Kleinian groups (groups of repeated Mobius transformations), this book is packed with amazing images, but it also describes the mathematics behind them. Very little previous mathematical knowledge is assumed - the book starts with an introduction to the basic concepts of symmetry and a crash course in complex numbers - but it does move forward at a rapid pace.
As well as covering the mathematics of Kleinian groups, this book also describes the techniques used in the computer programs which drew the images that it contains (no particular programming language is assumed) and gives enough details for a competent programmer to generate similar images of their own. In fact, one of the themes of the book is to encourage hands-on exploration, and each chapter ends with ideas for projects to explore related areas.
This book is essential reading for anyone who is interested in the use of computers to explore and illuminate complex mathematical concepts.
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VINE VOICEon 20 February 2009
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.
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VINE VOICEon 12 November 2007
Overview (I.M.H.O)

This book is carefully written, serious mathematical textbook that is written to slip `traditionally difficult' Mathematical skills under your `Radar' disguised as fun things to read and do. The book can be seen from several levels for a wider potential audience. To me, its starts as if its designed to increase the mathematical skills in Computer Science students, and ends as a book for the more mathematical inclined. In no time, the tougher supporting mathematical justifications are to help the reader conceptual grasp. I bought this with a short-term special offer from AND favorable exchange rates for a nice price. the book is printed on excellent paper and will last a long time.

Summary of theoretical methods (i.m.h.o)

The topic of this volume is the study of symmetry, when using complex numbers. The influence of `Klein' in shown that any group of transformations can represent symmetry. In the initial chapters, a brief reminder of concepts, such as transformations using translation, rotation, and reflections. To explore stereographic projections between spherical and Euclidean planes allows the inclusion of `Riemann Spheres', so we can see its effects using complex transforms as expansion, rotation, and expansion & rotation. So, this allows the additional operation of inversion. By using these spheres linking to methods based upon `Möbius transformations, these being the most general transformations of the extended complex plane that fully map circles to circles. The usage of Möbius techniques allows an almost mechanical linking between matrices that fully represent these transformations between circles.

Summary of described generalised theory, (i.m.h.o)

Encouragingly we are asked to consider what would occur in terms of symmetry if we link two `Möbius' transformations to each other? To represent this, a grouping of 4 circles with the edges of the non-overlapping circles that only just touch, known as `Schottky Disk Groups' or otherwise `kissing circles'. In this case, we can make each circle represent a function, and the physically opposing circle in the Schottky group to represent its inverse function. By next step is in accepting with further application of `Möbius transformations' / matrices to create circles within circles transformations. To create images we are quickly encouraged to play by programming in order to create a list of these combinations that represent the reflection, rotation, translation and inversion and store these results in a logical `wordtree'. At this point, the technique used to access these combinations (Breadth / Depth / Random searching) greatly influences the results.

Summary of theory as its put into practice for a wider audience (i.m.h.o)

Interestingly if a `random -first' method is used then its possible to generate Fractals! Also is an exploration of chaotic behaviour. To create in the creation of small application the computer is suited to a recursive approach. Hence, this means discrete methods are involved when using a computer. At it's most basic; a computer application can be constructed which uses Matrices representing Möbius transforms. So, in this way a method allowing linking of any three points representing any circle to any other three points in another circle. By placing the solutions of the matrix, polynomial equations into carefully constructed `generator matrices' that are then multiplied together allows colour-coded fractals to be created. These represent the transforms by repeating multiplying themselves and colour coding the answer values can be displayed. Some of the most detailed graphical displays can require huge `wordtree' often over 1.5 Gb picture files!

Interestingly, other methods are discussed that generate solutions of arbitrary complex function, such as using `Newton's Method' to find its roots. As Newton's requires an intermediate gradient calculation, the reasons why this can fail are explained.

Summary: This book is an exciting exploration of a mix of the dark arts of computing and mathematical understanding.
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