It has been said by many people that mathematics is the science of patterns and symmetry. Whether or not you agree with this definition, it is certainly the case that many mathematicians have analyzed symmetry from any number of angles. In fact, there are already 25 books in the MAA Reviews database with the word 'symmetry' in the title, and a keyword search for 'symmetry' in Amazon.com's Mathematics section turns up nearly 15,000 hits, with books ranging from Herman Weyl's classic rumination on Symmetry to Mario Livio's The Equation That Could Not Be Solved, a book for general audiences about Galois and his theories to research monographs on Mirror Symmetry. With all of these books that have already been written, it may seem as though there is no need for yet another book on symmetry, but this new book by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss, simply entitled The Symmetries of Things, proves to be a worthy addition to the literature.
The first thing one notices when one picks up a copy of The Symmetries of Things is that it is a beautiful book, with glossy pages and filled with gorgeous color pictures, some of which are reproductions of famous artwork (including the requisite MC Escher prints which you thought of when you saw the title of this book) or photographs, but many of which were generated by Goodman-Strauss using software called Kaleidesign — many examples of this artwork can be found at his website, http://mathbun.com. Unlike some books which add in illustrations to keep the reader's attention, the pictures are genuinely essential to the topic of this book.
While one can notice symmetries showing up in all kinds of settings, ranging from classical music to physical chemistry, the authors choose to focus, as the title of their book suggests, on symmetries of actual things, by which they mean geometric objects and artistic patterns. The simplest examples of this type of symmetry are the bilateral symmetry that the letter A has, or the 180o rotational symmetry of the letter S. But as anyone who has looked at the art of MC Escher, the architecture of the Alhambra, or even many sidewalks knows, the artistic properties of symmetries can become quite intricate and quite aesthetically appealing. Readers of The Symmetries of Things will be convinced that the mathematical properties have a similar intricacy and appeal.
The Symmetries of Things is divided into three parts, each of which has a different target audience, although I think that any reader will gain something from each part of the book, even if it is only an appreciation of the included pictures. The first part of the book introduces the very notion of symmetries of geometric objects, and is intended for a very general audience. Through the use of copious illustrations, the authors show many of the possible symmetries of finite objects, of planar patterns, and of frieze patterns. Along the way they introduce a new "signature notation" to keep track of the different symmetries that an object might have. They then go on to prove that they have come up with a complete classification, and their notation lends itself particularly nicely to both the statement of this classification, which is a cleaner version of results that many mathematicians have previously seen, as well as its proof.
One interesting decision made by the authors is to prove these "magic theorems" in reverse, by first stating the results and showing examples, and then consistently deferring one piece of the proof to the next chapter, where they prove that fact except for one piece and so on, until they have finally reduced the problem to one that is easily proven. This structure may feel a bit odd to most mathematicians, but I think it will work particularly well for readers who are not accustomed to reading dense proofs, such as students in a "math for humanities majors" course or high school students or even artists who know very little mathematics and initially pick up the book because of the aforementioned pretty pictures. The authors also make the reader's job easier by concluding each chapter with a summary entitled "Where are we?" which both summarize what has been done in the chapter as well as start to prepare the reader for what is to come.
The second part of the book is intended for readers with more mathematical sophistication, although all that is needed is a little previous exposure to group theory. This part of the book is dedicated towards symmetries of colored objects, and how the introduction of color changes the classifications given in the first part. Along the way, the authors discuss group presentations and introduce some new wrinkles in the signature notation to deal with these differences. Orbifolds, the geometric objects one gets by identifying points on the object which can be mapped to one another via the symmetries of the object, play a key role in this classification, as the authors quote what they call William Thurston's commandment that "Thou shalt know no geometrical group save by understanding its orbifold."
The final chapter in Part II is dedicated to abstract groups, and the authors introduce the reader to groups such as the dihedral, quaternionic, and p-groups, as well as introducing tools such as extensions, which they then use to classify all groups whose order is at most 2009. This discussion stands on its own from the rest of the book, and would provide interesting side reading for abstract algebra students.
While the early parts of the book are dedicated to symmetries of finite objects as well as two-dimensional crystallographic groups, the third and final part of the book is dedicated to symmetries in other kinds of spaces. The authors begin by introducing the hyperbolic plane and hyperbolic groups, and then move on to higher dimensional analogues of the classification of symmetries. Using generalized Schlafli symbols they are able to classify the 35 "prime" crystallographic groups in three dimensions (those that don't fix any families of parallel lines) as well as discuss the other 184 groups which occur.
A final chapter considers symmetries of four-dimensional objects and gives the complete categorization of four-dimensional Archimendean polytopes which has never appeared in print, despite being discovered by Conway and Guy forty years ago. These chapters are more technical than what has come before, and they are intended primarily for researchers in the related fields, but I think that most readers will gain something from discussions about topics such as the distortions that are inherent in making two dimensional maps of the Earth's surface, what it would be like to play Asteroids on a Klein bottle, or the hints about the mathematics behind Thurston's Geometrization Conjecture.
Conway, Burgiel, and Goodman-Strauss have written a wonderful book which can be appreciated on many levels. As discussed above, mathematicians and math-enthusiasts at a wide variety of levels will be able to learn some new mathematics. Even better, the exposition is lively and engaging, and the authors find interesting ways of telling you the things you already know in addition to the things you don't.
The book can also be viewed as an ode to good notation — the new notations for symmetries which are introduced make the statements and proofs of several previously-known theorems much cleaner and easier to understand. While some mathematicians may argue that there is a bit of a fetishization of notation in our profession, the authors of this book give evidence to the contrary, and I expect that the book will make many mathematicians think a little harder about coming up with a better notation than merely adding a third subscript to some behemoth of an equation.
The book also contains just enough exercises that one can imagine using parts of it as a textbook for certain kinds of courses. And finally, The Symmetries of Things would make an excellent coffee table book that people could flip through merely to look at the pictures, and how often can you say that about a math book?
Darren Glass is an assistant professor of mathematics at Gettysburg College whose research interests include Galois Theory, Number Theory, and Cryptography. He can be reached at firstname.lastname@example.org.