New Book by John H. Conway and Colleagues Reveals Symmetry and the World

June 9, 2008

A new book by three mathematicians offers a novel way to understand symmetry and a fresh way to see the world. Written by John H. Conway of Princeton University, Chaim Goodman-Strauss of the University of Arkansas, and Heidi Burgiel of Bridgewater State College, The Symmetries of Things provides a comprehensive mathematical theory of symmetry.

For the last century, mathematicians have relied on a notational system, developed by crystallographers to describe symmetries, that couldn't easily be applied elsewhere. Conway came up with a new one that is "much more than a naming system," Goodman-Strauss says.

"A good notation is amazing because it's not just a way of naming things," he remarks. "It's a way of making the structure of things transparent and simultaneously providing a way of enumerating them, classifying them, and proving that's what the classification is—all at once."

While mathematics has been called "a descriptive art," Goodman-Strauss says, mathematicians are not simply trying to describe things. Rather, he notes, "we're trying to understand what inherently can be described in a quantitative, analytical way."

Orbifolds, for example, are formed when symmetrical patterns on a surface are folded or rolled so that every distinct feature, every point on a pattern, is brought together with its corresponding point. The result is a geometrical shape—a sphere, a cone, or a cylinder, for example—that shows one example of the design element that is repeated to make the symmetrical pattern. As a tool, the orbifold approach yields an efficient way to understand patterns.

The first section of the book, which focuses on symmetries of finite objects and plane repeating patterns, is accessible to anyone interested in symmetry. The second section, which concerns color symmetry, group theory, and tilings, and the third section, which discusses repeating patterns in hyperbolic and other spaces, are aimed at mathematicians and other experts.

The entire book, Goodman-Strauss says, "is meant to be engaging and reveal itself visually as well."

Source: University of Arkansas, May 21, 2008.