At first glance, designing wallpaper can seem as simple as a kindergarten art project. Designers can start with any combination of colors and forms for the first small patch, and then just replicate it again and again in two independent directions. Depending on the patterns in the original patch, and the choice of the two directions, additional symmetries may emerge — for example, the six-fold rotational symmetries of Figure 1, or the reflection symmetries of Figure 2, both created by the mathematician Frank Farris, of Santa Clara University in California.
Figure 1. A wallpaper pattern, left, with six-fold rotational symmetry around each of the brown-green rosettes.
Figure 2. A wallpaper pattern with reflection symmetries across (unmarked) horizontal lines through each of the elliptical stained-glass ornaments.
Illustrations: Frank Farris
Figure 3. Penrose tilings, such as the above image, exhibit many local five-fold symmetries; however, these patterns never display wallpaper repetitions. As a Penrose tiling fills up more and more of the plane, the ratio of the number of fat tiles to the number of thin tiles approaches the golden ratio.
Image: Courtesy of Simons Science News
Figure 4. Click for larger image and caption.
Illustration: Frank Farris
But while it’s possible to create wallpaper with two-, three-, four- or six-fold rotational symmetries, it is impossible to do so with five-fold rotational symmetry. This limitation, which mathematicians have known about for nearly 200 years, is called the “crystallographic restriction.” The geometry of the pentagon precludes wallpaper patterns with five-fold symmetry; the same is true for seven- and higher-fold rotations.
Nevertheless, some of the most riveting non-wallpaper patterns imaginable, such as Penrose tilings (see Figure 3), manifest local five-fold symmetry in many locations and on many scales, but without any repeating patterns. Now, using a very different approach than that of Penrose tilings, Farris has harnessed the peculiar geometry of five-fold symmetry to create a new collection of arresting images — wallpaper fakes that seem to defy the crystallographic restriction.
Figure 4, for example, looks at first like a counterexample to the crystallographic restriction, with five-fold rotational symmetry around point A, and wallpaper pattern shifts in the directions of AB and AC.
In reality, as Farris described in the November 2012 issue of Notices of the American Mathematical Society, the image is a clever fraud.