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Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature

Michael Field and Martin Golubitsky
Publication Date: 
Number of Pages: 
[Reviewed by
Thomas Hagedorn
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This is a wonderful book filled with beautiful images, so beautiful that while I was reading it, my spouse (a historian) was intrigued by the images and wanted to understand something about the mathematics behind them. Other mathematicians also agree, as many of Field’s and Golubitsky’s images (from the book’s first edition) have been used in textbooks, and by conferences and professional societies for cover images and posters.

The mathematical subject of the “Symmetry in Chaos” is to “present the ideas of symmetry and chaos — as they are used by mathematicians — that are needed to understand how these pictures [of symmetric chaos] are formed.” The authors present many beautiful pictures of symmetric chaos and also “present the ideas that are needed to produce these pictures.”

The book has seven chapters. The first two chapters are focused on describing how the pictures are created. In Chapter One, the authors introduce the basic concepts of symmetry, chaos, sensitive dependence on initial conditions, statistics, and how individual pixels in a picture are colored based upon a mathematical formula. The second part of the chapter discusses the significance of the pictures and “why they might be interesting.” In particular, they discuss how symmetric equations need not have symmetric solutions, the presence of turbulence in many problems, the need to look at an averaged solution, and whether these averaged solutions need to be symmetric. Chapter Two is focused on planar symmetry and basic group theory is introduced. Tiling and wallpaper patterns, the wallpaper groups, and many beautiful pictures, some computer-created, others from real life are presented.

Chapter 3, “an intermezzo” in the words of the authors, presents a number of pictures of symmetry found in nature or in decorative designs paired with similar symmetric pictures created by computers using the symmetric chaos method. In Chapter Four, the authors present a detailed example of how to create these pictures. They introduce the logistic map and show how the odd logistic map gives rise to a symmetric pattern on average.

The last three chapters are concerned with three different mathematical methods are used to compute the symmetric icon, quilt, and symmetric fractals images. In Chapter 5, for the symmetric icons (those with only rotational and reflection symmetric), a symmetric icon with symmetry group D3 is constructed. To do this, polar coordinates and complex numbers are introduced and used. At the end of the chapter, formulas for creating icons with symmetry group Zn and Dn are described. Similarly, in Chapter 6, explicit methods for creating quilt patterns — patterns with translational symmetry in addition to rotational or reflectional symmetry — are described. Finally, in Chapter 7, one constructs symmetric fractals.

“Symmetry in Chaos” is well written and by profitably read by anyone, particularly undergraduates, who would like to understand how mathematics and computers can be used to produce these beautiful images. It is a testament to the accessibility of this material, and its interest, that one of the authors (Fried) has used the material in this book to teach courses to junior and senior art students at the University of Houston and to teach seminars for local teachers.

Reviewer’s Note: On page 107, the first line of the second paragraph should read “mixing in Figure 5.3 (left)” instead of “mixing in Figure 5.3 (right).”

Tom Hagedorn is Associate Professor of Mathematics at The College of New Jersey.