Symmetry

Most undergraduate mathematics majors learn at some point in their studies that there are connections between group theory and geometric symmetry — if nothing else, for example, they study the Dihedral groups of symmetries of a regular polygon. However, the connections between geometric symmetry and group theory go far beyond that. Groups can, for example, be used to classify the symmetries of frieze patterns (think of a picture or motif being repeated infinitely often along a line) or wallpaper patterns (think of the motif being repeated infinitely often not only to the left and right, but also up and down). It turns out that there are 7 geometrically distinct frieze patterns and 17 distinct wallpaper patterns, each with an associated group. These results, though available in the undergraduate textbook literature (see, for example, Martin’s Transformation Geometry) are, however, not a part of the typical undergraduate’s mathematics training.

The book under review won’t really change that; in fact, it’s not even a mathematics book at all. The author is not a mathematician; he is an emeritus professor at the University of Copenhagen in the geology department, with strong research interests in crystallography. His book contains no theorems labeled as such, and certainly no proofs; there are also no mathematical definitions — the word “group” gets used, but the technical definition of one is not given. What this book does do is illustrate, with dozens and dozens of beautiful photographs (all of them taken by the author), how symmetry is found in all sorts of examples of classical art from around the world: mosaics, marble floors, windows, vases, etc. The objects, the author tells us, were photographed in what he calls their “natural” state, “neither modified nor manicured”, so “you have to put up with chipped tiles and discolored glazes in this book”. The author also tells us that there is a strong emphasis on Islamic art in the text, because that “is the art with the most developed symmetries and often also with the most beautiful ornaments.”

I’d call this a “coffee table book”, but that term brings to mind a thick book of oversized dimensions; this one, by contrast, is normal-sized and, at 240 pages, relatively slim.

Frieze and wallpaper patterns occupy a lot of the author’s attention (the wallpaper chapter is the largest in the text), but there are other kinds of symmetries discussed as well. Several chapters, for example, are devoted to the introduction of colors in art, so that the arrangement of the colors must be considered in addition to the underlying symmetry of the object. There is also a discussion of “layer symmetry”, where objects have a “front” and “back” and we consider reflections transforming one side to the other. (It turns out that there are 80 different groups of symmetries here.)

Even fractals are discussed, and examples given. Of course, the examples here are not the computer-generated fractal images that we all have seen elsewhere, but examples of art that illustrate the kind of “self-similarity” that typifies fractal images. Floor mosaics in a church in Rome, for example, illustrate patterns resembling the Sierpinski triangle.

This book brought to mind Frank Farris’ Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, but, although both books discuss symmetry and both are very visually appealing, they are very different indeed. Farris is a mathematician, and his book is very mathematical: one can find in it not only group theory but also, among other things, partial differential equations, Fourier series and algebraic numbers. Also, the art in Farris’ book was created by him using mathematics, whereas Makovicky investigates symmetry in objects and designs that were created by artists. Makovicky states an isolated mathematical fact or two (that there are 17 wallpaper patterns, for example) but mostly his book is valuable for actually seeing applications of mathematics in classical art. I can’t imagine it being used as a text in a mathematics course, but I can certainly see a professor showing slides from it to illustrate the mathematics that is learned from another source. And I can also see a professor just flipping through the pages of this book for fun, enjoying one beautiful photograph after another.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.