Acta Cryst. (1997). A53, 251

Symmetry in chaos

By M. Field and M. Golubitsky

Pp. 230. Oxford University Press, 1996
Price £15.00 (Paper). ISBN 0-19-853688-7

In this book, the authors present the kinds of pictures that may be produced when the seemingly contradictory ideas of symmetry and chaos are combined, and they provide in as elementary a way as possible the mathematical ideas that lead to those pictures. Both symmetry and chaos are topics about which much has been written lately at all levels, from the supposedly introductory to the very sophisticated, but this book takes nothing for granted. Each topic is carefully defined so that it is placed in the proper context. Symmetry is first defined, then demonstrated, with simple examples that lead into a nice introduction to groups. The same is done for chaos; first the definition, then carefully explained examples, and then a more general treatment of chaotic systems. The two concepts are then brought together very neatly with an excellent explanation of what is meant by symmetry in chaos.

Since chaos has become a topic of interest with very little general understanding of what it means precisely, it is good to have a book by authors who understand this state of affairs and who set out to explain clearly what is meant by a chaotic system. They develop the logistic equation from the initial premise inherent in the special population problem it is meant to describe. They next explain in detail how chaos can arise from such a simple system and how it comes about. This simple equation, which is not symmetric, is transformed into a symmetric equation and the first step to symmetric chaos is developed. With these fundamentals out of the way, the authors then work out in detail the interesting cases of symmetric icons - symmetric chaotic systems having either rotational or reflection symmetry, or both - and of systems having translational symmetry, so-called quilts. The final chapter deals with symmetric fractals, and is again a clearly written exposition of an interesting topic.

In 54 color plates, the authors illustrate the outcome of the formalism. These pictures are not only aesthetically pleasing but are the real message of the book. That being so, the authors realize that, unless the way in which the pictures are generated is clearly explained, they will not fully convey the meaning of symmetry in chaos. They explain the method by which the images are created by first using a very simple model of a screen, then developing the introduction of color, and then showing in detail how they are completed. The process is brought to closure by including computer programs for the use of the reader.

We need more books of this type. In the best sense of the word, this is a pedagogical book that lays the groundwork for the further work it is sure to generate on the part of the reader. A certain level of mathematics is expected, but it is not very high, and each mathematical topic is developed from a very elementary level. The reader will benefit from the authors' clear grasp of what they wish to say and the great care they have taken to make sure that the essence of this interesting topic is presented in a clear and concise fashion. I found the book an excellent introduction to a field of very general interest.