Science and Culture: Of waves and wallpaper

To most people, wallpaper is little more than a decorative background on a flat surface: Pleasing designs, pasted in vertical strips from wall to wall. But in the minds of mathematicians, the repetition of figures—coupled with the idea of covering 2D space—suggests a way to study the fundamental nature of symmetry.

Applied mathematician Frank Farris, at Santa Clara University in California, began thinking about symmetry and wallpaper in the mid-1990s when he found his undergraduate geometry teaching materials wanting. “I didn't like what I was seeing,” he says. The textbook used simple drawings to illustrate exquisite and powerful mathematical statements about symmetry. “I didn't like the images being held up to illustrate these beautiful math theories about patterns, and I didn’t like the vocabulary people were using. They were describing patterns as being broken into blocks, as if that were the only way to make a pattern.”

Anyone can make a symmetric wallpaper pattern: Start with a pattern, add colors and ink, then repeat side to side and up and down. Farris doesn’t have a problem with that approach. But he suspected that simple method—or simply repeating a shape over and over—wasn’t the only way to create a wallpaper design.

That suspicion, he says, launched him on an excursion into the intricacies of wallpaper and symmetries. Over the decades, Farris has developed novel ways to construct wallpaper symmetries by using a variety of tools from a mathematician’s toolbox. Prints of his patterns are on display as part of the Mathematical Art Gallery of the 2015 Bridges Conference, held this year in early August in Baltimore, Maryland, which showcases work at the intersection of math and culture. In June came Farris’s first book on the subject, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns (1). The book offers a mathematically robust tutorial on Farris’s method of using sine and cosine waves, rather than blocks, to create symmetries in the guise of wallpaper patterns.

As an example, start with a wallpaper pattern that covers a plane with nonoverlapping equilateral triangles. Then, Farris says, suppose waves pass through those triangles, and assign a color to each point on the triangle based on where the point falls on the wave. Such an approach would yield a pattern like the one shown in Fig. 1. The same method could be used to generate a wallpaper pattern from a photograph, as in Fig. 2.

Fig. 1.

One mathematically generated wallpaper pattern entails starting with a plane covered with nonoverlapping equilateral triangles. Image courtesy of Frank A. Farris.

Fig. 2.

Using a photo of a peach and its negative, Frank Farris created this hyperbolic pattern titled “Peaches to the Edge of the Universe.” Image courtesy of Frank A. Farris and Princeton University Press.

In his work, Farris also travels beyond the Euclidean world, which describes the flat geometry we observe: for example, the angles of a triangle add up to 180°. “In my original work, I was making Euclidean wallpaper,” says Farris. “But it’s natural for a mathematician to say, what about non-Euclidean geometry?”

Euclidean geometry depends on Euclid’s fifth postulate, which says that given a line and a point not on the line, only one other line passes through the point that is parallel to the original line. This works on a 2D plane. Non-Euclidean geometry, which invokes other spaces, throws this requirement out the window. Geometry might be done on a sphere. Or more exotically, geometry might be done in hyperbolic space in which a plane looks like the seat of a horse saddle.

So, Farris wondered, what do wallpaper patterns looks like in non-Euclidean space? He created colorful new patterns that broke the rules of repeating-pattern wallpaper. (Other examples of his wallpaper and fabrics are available at www.spoonflower.com/profiles/frankfarris.) One of his prints, displayed at the Bridges conference, for example, uses a photo of a peach and its negative, as source material to create a hyperbolic pattern titled “Peaches to the Edge of the Universe” (Fig. 2)

“We live in Euclidean geometry and see things in Euclidean geometry,” says Daina Taimina, a mathematician at Cornell University who for nearly 20 years has been crocheting hyperbolic models. She says non-Euclidean spaces automatically lend themselves to interesting applications, such as Farris’s wallpaper. Such visualizations, Taimina asserts, can pique people’s curiosity, even those with an aversion to math.

John Stillwell, a mathematician at the University of San Francisco in California, notes that non-Euclidean sketches first appeared in math texts of the 1890s but weren’t exploited for art until Dutch artist M. C. Escher began using them in his drawings. For example, Escher crafted circular wood-cuts known as “circle limit” patterns, in which figures become smaller as they near the circumference; these follow the same rules as many of Farris’s hyperbolic designs. In those works, says Stillwell, Escher’s objective was “to encompass an infinite world in a finite space.” Farris, says Stillwell, takes an alternative but attractive way of displaying the same geometry.

Farris spent the last two years writing his book and is ready to step back from mathematical wallpaper, although he’s continued making symmetric patterns on spheres that are often mistaken for Christmas ornaments. This may lead to his next venture. “I’m dying to figure out how to use 3D printers in my work,” he says.