# Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

At first glance, by the format of the book and the many colorful pictures on glossy paper with wide margins, one would expect here a coffee-table book. And to some extent, it is. The pictures are certainly abundantly present, but the text that accompanies them is seriously discussing mathematics. Farris starts with nice curves generated by a point of a circle that is roling along a circle that is roling along another circle. A kind of epicycloidal epicycloid. In complex notation, a point on a circle with radius $r_j$ is described as $r_je^{in_jt}$ so that the curve is a sum of three such exponentials. To make the curve 2π-periodic, one needs the three frequency numbers $n_j$ to be equivalent modulo some number $m$. This simple fact suffices for Farris to make a link with (rotation) groups, vector spaces (combination of exponentials), Laurent polynomials, and Fourier series. Next, periodicity of the circle and periodicity along the line with period 2π (translation group) is immediate and that leads to the combination of groups to generate the seven possible frieze patterns. This is how periodic curves lead to friezes, but one may go back by mapping the upper half plane to the unit disk, and this results in rosettes. Add to this the possibility of playing with color patterns and you get already a lot of possibilities to create nice pictures. The color patterns used by Farris are provided by ordinary pictures. A complex value corresponds to a point in the plane, and thus identifies a pixel in the picture and its color defines the color to be used in the pattern. Sometimes the mathematics are interupted to discuss the artistic aspect about choosing the proper parameters and the appropriate picture to get a nice result. This usually requires some experimentation with parameters and pictures defining the color pattern.

The next step is to require periodicity in different directions in the plane. Thus one gets lattices and wallpaper patterns. Although the original graph that started the book had 5-fold symmetry, here the choice of symmetry is mostly 3-fold and sometimes 4-fold. The remarkable fact that 5-fold symmetry is not possible here is proved. This is an essential step to show that there are only seventeen different wallpaper patterns. Meanwhile one learns how to link symmetry with eigenvalue problems, with waves defined on lattices, the differential equations that define waves, and the point group.

Although 3D symmetry is not really the subject of this book, it is possible by stereographic projection to catch some of polyhedral symmetry in a planar picture. When mapping back to the sphere, which we observe as a disk, we see the spherical patterns somewhat distorted near the boundary of the disk due to the curvature of the sphere. That reminds a bit of hyperbolic geometry, which is the next design pattern used, both in the half plane and in the disk. This, and also the previous patterns, reminds one of the artwork of M.C. Escher. However, Escher added something, which was not so much the color used here by Farris. Esscher's playful element avoiding the plain repetitivity of the pattern is a gradual transform from one form to another. This idea of morphing wallpaper patterns is discussed in the last chapter.

This is a marvelous book that brings groups, and along the way many other mathematical concepts, to the reader in an unconventional way. Rather than by an axiomatic approach, the concepts are introduced as required by our urge to understand the mathematical rules that guide and restrict an otherwise unbound creativity in designing the patterns. It is rather specific for the application in mind, but it is very stimulating as a first contact to abstract algebra. There are also many (literally) marginal exercises marked with 2, 1 or no star to indicate the difficulty. There is only one drawback. Although there is a chapter on the C++ code that Farris used to produce his pictures, he says it is not really polished enough to make it available to a broad public. A hands-on accompanying software tool would have made this exploration of symmetries so much more exciting.

**Submitted by Adhemar Bultheel |

**21 / Jul / 2015