This is, both literally and figuratively, a beautiful book. It contains lots of interesting mathematics (hence the “figurative” beauty) but it also contains many full-color, and visually quite striking, pictures. In fact, in both its oversized shape and contents, the book has the appearance of, and could function as, a slim coffee-table book, albeit one with much more sophisticated print content than the average such book.
This book was also, I must confess, a bit of a surprise to me. Because I came to it familiar, as most readers of this column are, with dihedral groups, and had also worked my way, years ago, through the proof that there are 17 wallpaper groups and 7 frieze groups, I assumed that I would already be pretty familiar with the topics covered here; in fact, however, I was wrong. I learned quite a bit from this book, including relationships between symmetry and areas of mathematics that I previously had no idea existed. Indeed, in addition to group theory (which is expected to appear), we see used here such topics as complex analysis, Fourier series, partial differential equations (specifically the wave equation), the modular group, and quadratic number fields.
The creation of beautiful symmetric patterns is basically what this book is all about. The author describes techniques, original to him, for creating symmetric patterns, principally but not exclusively wallpaper patterns. These techniques are a consequence of the author’s original way of looking at wallpaper patterns.
In a typical wallpaper pattern, a design is repeated endlessly left-to-right and up-and-down. The symmetries of any such pattern form a group, and it is a theorem that there are 17 possible groups that arise in this way. See, for example, this webpage, which gives illustrations of the various possible patterns. Typically when one sees these patterns in print (as in the previous link), it looks as though a certain design has been, to use the author’s phrase, stamped out with a carved potato. The author’s techniques, however, give much more interesting results; to put it very generally, he views wallpaper patterns as being created not from discrete blocks but from continuous waves.
The techniques described in this book rely on a variety of interesting ideas. One such idea is the fact that a function f from the set of complex numbers to itself can be visualized via the process of domain coloring: i.e., to every complex number z we assign a color, and we visualize our mapping f by coloring every point z with the color assigned to f(z). Another idea is that we can use a photograph as a “color wheel” from which we get the colors to use domain coloring. Still another idea is that certain nice functions, wallpaper waves, can be used to create other functions, called wallpaper functions, that leave lattices in the plane (such as the lattices defined by certain algebraic structures such as the Eisenstein integers) invariant. The precise details of all this are beyond the scope of this review, but suffice it to say that these ideas, and others, combine to create wallpaper patterns that are interesting and very attractive.
A number of these ideas have previously been discussed by the author in published articles. His review in the Monthly (June-July 1998, pp. 570–576; if you have access, read it on JSTOR; see a summary here) of Needham’s book Visual Complex Analysis, for example, introduces the domain-coloring concept; that review is complemented by a short article by him that is published on the MAA website, complete with some nice pictures.
The author describes all these ideas, step by step and in relatively easy increments, in considerable detail as the book progresses. The text starts with the simpler case of curves that exhibit symmetry properties and then progresses from curves (real valued functions of a real variable) to the more interesting situation of complex-valued functions of a complex variable. The bulk of the book is devoted to wallpaper groups in the plane, but, after doing these, the author turns his attention to interesting offshoots: in one chapter, for example, he talks about symmetry groups of the regular polyhedra (which are, of course, three dimensional, but which can be used to create plane symmetry figures); in another chapter, he discusses symmetry patterns in hyperbolic plane geometry, modeled here by the Poincaré half-plane.
Here and there throughout the book, the author discusses aesthetic considerations as well as mathematical ones; he devotes an entire paragraph on page 76, for example, to describing in purely artistic terms why the aesthetics of a certain image please him. So somebody reading this book is likely to learn something about art as well as mathematics.
The preface to this book identifies three groups of people as target audiences for it: working mathematicians, advanced undergraduates, and the less experienced “mathematical adventurers”. I agree without hesitation that people in the first two groups have much to gain by looking at this book. People in either of these two groups will see a lot of the mathematics they have previously learned put to surprising and interesting uses, thereby making this book an interesting choice as a text for some sort of senior seminar or capstone course. The author has certainly written the book so as to be accessible to advanced undergraduates, and a fortiori to professional mathematicians. The writing is generally quite clear and rather refreshingly personal; sentences like “But then I found myself staring at apparent local mirror symmetries…” appear throughout the book. Farris frequently writes as though he is talking to a friend or colleague, and describes in first-person terms how he came upon some of the more interesting ideas described here; I found, and I suspect many other readers will too, that this makes for effective communication.
I am much less convinced, however, that this book will prove to be as useful for people in the third group. Of course, anybody can enjoy this book by just looking at and admiring the many pictures in it, but to really get something out of the discussion, I suspect that a fairly decent background in mathematics is necessary. There is a lot of serious mathematics in this book, and it is discussed seriously as well; there is not a lot of the usual vastly oversimplified discussions that one often finds in math-for-laypeople books. Terms are defined precisely, and also, to answer the perennial question asked by students everywhere: yes, there are proofs. There are also exercises, some of which call for proofs. (Solutions to some, but not all, are provided.) At a minimum, a reader should have an understanding of basic calculus, as well as a degree of mathematical maturity sufficient to allow him or her to read about more advanced concepts with some degree of comfort, and to apply ideas that he or she has just been exposed to.
On some occasions, the author invites mathematical novices to simply skip some of the more technical passages and take some things on faith; at the beginning of chapter 5 on Fourier series, for example, the author states that any person who is willing to accept the basic idea that “every periodic curve in the plane can indeed be written as a superposition of waves, though infinitely many are required in general” can simply skip the entire chapter. There is, however, a limit to how much skipping a reader can do, and I should also note that some fairly nontrivial mathematics appears quite early in the book where it can’t be reasonably skipped: on the first two pages, for example, the author considers the situation where two quarters are placed on a table, one on top of another, and the top one rolls without slipping around the other; he asks for a parametrization of the curve formed by following the point that was originally at the bottom edge of the top circle. The solution (an epicycloid) involves some understanding of, and ability to deal with, vectors and trigonometry and would, I think, prove baffling, and perhaps frustrating, to many novices.
My belief that the author may be overly optimistic in his assessment of one segment of the potential audience for this book does not in any way detract from its quality. This is a thoughtful, innovative and interesting piece of work, discussing material that the author is obviously very enthusiastic about; such enthusiasm is, as is often the case, contagious. And even an old dog can learn some new tricks from it.
Mark Hunacek (email@example.com) teaches mathematics at Iowa State University.