The goal of this book is to present to an audience with minimal (say, high-school) background in mathematics the basic mathematics underlying the concept of symmetry. The book is a substantially rewritten version of the first edition, which we reviewed

here.

The first edition was addressed largely to non-majors in mathematics. In this second edition, the author, while not increasing the minimal prerequisites, has, in several respects, made the book a bit more “mathematical” and suitable for reading by math majors as well as non-majors. First, he has added a section to each chapter titled “Elements of Mathematics” that “highlights a specific nuts-and-bolts aspect of mathematics that arose in the chapter.” (Examples include, for example, the converse of a statement, indirect proofs, inductive proofs, and well-defined definitions.) Second, he has added several optional sections (marked as such in the table of contents) that address somewhat more sophisticated topics and proofs.

The appearance of the book has also changed. In the first edition, the typography and large font size combined with the numerous full-color pictures to give the book a rather elementary appearance, one that some might think was inappropriate for a college-level audience. This new edition retains the many full-color illustrations, but has a much more adult appearance, as befits a book that one might want math majors to look at.

Much of this book deals with rigid motions of the plane and three-dimensional space and the topics that naturally accompany this study, including quite a lot of group theory. (In addition to dihedral, symmetry, permutation, wallpaper and matrix groups, abstract groups and isomorphism are also studied, as are product groups.) There is a chapter on the platonic solids. Another chapter uses symmetry ideas to sketch a proof of what I refer to as Queen Dido’s theorem (though that name is not used in the book): of all closed curves with a given perimeter, the circle has the greatest area.

Several chapters, however, address topics whose relevance to symmetry is somewhat less clear. For example, chapter 10 discusses the real numbers and chapter 11 looks at some issues in number theory (including the prime number theorem) and set theory (including a first look at cardinality, and a proof that the set of real numbers is uncountable). Actually, these chapters struck me as being a bit out of place, and I would have preferred that the author had omitted them in favor of, say, discussing other ideas involving symmetry—e.g., the use of rigid motions to prove theorems in geometry.

This book has much to recommend it. The material is presented clearly and enthusiastically, with great pains taken to motivate the material and make the abstract ideas as accessible as possible. In appropriate cases, proofs are sometimes omitted, and, when they are presented, their technical details are sometimes replaced by intuitive plausibility arguments. (The proof that a circle maximizes area is a good example of this.) When this is done, the author says so, so the student is not fooled into thinking that a completely rigorous argument is presented. In addition, technical terms are sometimes avoided so as to explain things more intuitively. For example, the author defines an isomorphism between two groups as a “one-to-one matching (dictionary) between their members that converts each true equation in one group into a true equation in the other group.” (Another mild quibble: perhaps, because one of the goals of this new edition is to make things a bit more mathematical, the author could have added an optional section explaining how this definition segues into the more precise one using the familiar equation \( f(xy) = f(x)f(y) \).)

In addition, there are a lot of well-chosen exercises, ranging from very easy to more challenging. Solutions are not provided and there is, to my knowledge, no solutions manual available.

Another good feature of the book is the use of online supplements. The author has, for example, provided a link to a

webpage in which can be found powerpoint slides for use by instructors. Brief teaching suggestions also appear here. In addition, several chapters of the book, if accessed electronically, offer narrated videos instead of the still pictures that appear in the book.

To summarize and conclude: this is a well-written book, covering interesting material, that can serve a number of possible audiences. While it may be a bit too sophisticated as a text for a freshman qualitative literacy course (my experience at Iowa State with such courses is that the students require more computation and less abstract reasoning), it can be a good text for an honors seminar for non-majors or a useful supplement to courses in abstract algebra or upper-level geometry. Also, given that many math majors will never have seen many of the topics discussed in this book, the book might, perhaps with some supplementation of topics and technical details by the instructor, serve as a useful text for a capstone course for majors.

Mark Hunacek (

[email protected]) is a Teaching Professor Emeritus at Iowa State University.