Enhanced descriptions from Syndetics:

Chaotic dynamics (known popularly as chaos theory or, more simply, chaos) is among the most fascinating new fields in modern science, revolutionizing our understanding of order and pattern in nature. Symmetry, a traditional and highly developed area of mathematics, would seem to lie at the opposite end of the spectrum. From the branching of trees to the rose windows of great cathedrals, symmetric patterns seem the antithesis of such chaotic systems as weather patterns. And yet, scientists are now finding connections between these two areas, connections which could have profound consequences for our understanding of the physical world. In Symmetry in Chaos, mathematicians Michael Field and Martin Golubitsky offer an engaging look at where these two fields meet. In the process, they have generated mathematically a series of stunning computer images linking symmetry and chaos.
Field and Golubitsky describe how a chaotic process eventually can lead to symmetric patterns (in a river, for instance, photographs of the turbulent movement of eddies, taken over time, often reveal patterns on average) and they provide clear explanations of the science that lies behind the generation of these pictures. And the images they generate are spectacular. Because of the symmetry, these full-color and black-and-white images--some chaotic and some fractal--have a surprisingly classical appearance. Indeed, through comparisons with pictures from nature, such as sea shells and flowers, and decorative designs ranging from Islamic motifs to contemporary graphic logos to ceramic tiles, the authors highlight the familiar yet unusual nature of these mysterious pictures. Finally, the book features an appendix containing several BASIC programs, which will enable home computer owners to experiment with similar images.
This lavishly illustrated, oversized volume offers both a fascinating glimpse of the frontier of modern science and a stunning collection of remarkable images. Symmetry in Chaos will intrigue science buffs as well as anyone interested in decorative art and pattern design.

Table of contents provided by Syndetics

• Preface
• Introduction
• Planar symmetries
• Patterns everywhere
• Chaos in symmetry creation
• Symmetric icons
• Quilts
• Symmetric fractals
• Appendix I Picture parameters
• Appendix II Basic programs
• Appendix III Icon mappings

Reviews provided by Syndetics

CHOICE Review

Barely a decade since the publication of Benoit B. Mandelbrot's The Fractal Geometry of Nature (1982), there are now books on fractals, chaos, and nonlinear processes numbering in the hundreds, some for the general reader, others targeted at special audiences ranging from mathematicians, natural scientists, and engineers, to economists, philosophers, and artists. The four volumes under review are indicative of this diversity. Moon's book is, perhaps, best suited for practicing engineers who wish to decide whether learning chaos theory will be relevant and profitable to their work. The many examples of engineering applications where chaos has been observed form the soul of the book. The noteworthy Chapter 2 offers broad heuristics for identifying chaotic phenomena in the laboratory. Actually a rewrite of the author's Chaotic Vibrations (CH, Jun'88), the book has grown to nearly twice its former size from the addition of exercises, new physical applications, and more mathematical explication. The mathematics remains vague, even though the reader must have some mathematical sophistication. As such, this book cannot be recommended for those who need to learn the details of the subject. The editing is haphazard. For example, one meets Feigenbaum's number, as though for the first time, on four occasions, the index noting only the first three. By contrast, Peitgen, Jurgens, and Saupe's book serves the mathematical neophyte who craves a deep knowledge of the mathematical foundations of chaos. This is an extremely leisurely, careful, detailed, copiously illustrated exposition. Not a popularization, it nevertheless should find the wide audience that is usually served only by diluted accounts. It is hard to think of another book like it, but then no other mathematical subject has ever so captured the public imagination. There is also much here for mathematically sophisticated readers and it is easy to find what interests one and to dig in. With its beautiful design, superb organization, and clear style, it is not premature to declare this book a classic. Some overlap notwithstanding, this volume does not supercede The Science of Fractal Images ed. by Peitgen and Saupe (CH, Mar'89) or Peitgen and P.H. Richter's The Beauty of Fractals (CH, Dec'86). Given the nearly one thousand pages, the low price is worthy of note. The raison d'etre of Field and Golubitsky's work must be the stunningly beautiful color plates of some new types of fractal images. The mathematical genesis of these images is the question that asks when the orbits of a dynamical system statistically exhibit the symmetries of the system as a whole. Remarkably, the answer may change from ^D" to ^D" as one varies a parameter-describing system; a real world example (that the authors leave underdeveloped) is wobbly train wheels that wear unevenly at slow speeds, evenly at high speeds. The abstract ideas, together with the basics of chaos theory and group theory, do receive a passable treatment, some in the text, some in technical appendixes that also give full details on the construction of the images. Unfortunately, the text is also full of uninspired generalities about symmetry and chaos and unconvincing comparisons with images from art and nature that bare superficial resemblance to these fractals. Finally, Briggs's is a popular account full of the grandiose ^D" posturing that gives chaos theory a bad name in some circles. Nevertheless, it does collate some interesting information and many fascinating and beautifully produced images. This book is not appropriate for academic libraries; this is the first book about fractals this reviewer has seen that does not contain a single equation. For coffee tables only. D. V. Feldman University of New Hampshire

Booklist Review

In this stunningly visual and cogently textual tour of the structure of symmetry, patterns, and chaos, the authors examine quilts, the Rose window of Chartes, diatoms, shells, and elaborate computer graphics, including excellent examples of symmetrical fractals.

Author notes provided by Syndetics

About the Authors:
Michael Field is Reader in Pure Mathematics at the University of Sydney. He has published several books and papers in advanced mathematics. In 1990, he was Visiting Scientist at the Mathematical Sciences Institute, Cornell University.
Martin Golubitsky is Cullen Professor of Mathematics at the University of Houston. He is co-author with Ian Stewart of Fearful Symmetry: Is God a Geometer.