Quanta Magazine

https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/ March 5, 2013

How to Make Impossible WallpaperA tale of forbidden symmetries.

By Erica Klarreich

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 througheach of the elliptical stained-glass ornaments.

Figure 3. Penrose tilings, such as the above image, exhibit manylocal five-fold symmetries; however, these patterns never display wallpaper repetitions. As a Penrosetiling fills up more and more of the plane, the ratio of the number of fat tiles to the number of thintiles approaches the golden ratio.

At first glance, designing wallpaper can seem as simple as a kindergarten art project. Designers canstart with any combination of colors and forms for the first small patch, and then just replicate itagain and again in two independent directions. Depending on the patterns in the original patch, and

Quanta Magazine

https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/ March 5, 2013

the choice of the two directions, additional symmetries may emerge — for example, the six-foldrotational symmetries of Figure 1, or the reflection symmetries of Figure 2, both created by themathematician Frank Farris, of Santa Clara University in California.

But while it’s possible to create wallpaper with two-, three-, four- or six-fold rotational symmetries, itis impossible to do so with five-fold rotational symmetry. This limitation, which mathematicians haveknown about for nearly 200 years, is called the “crystallographic restriction.” The geometry of thepentagon precludes wallpaper patterns with five-fold symmetry; the same is true for seven- andhigher-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 withoutany repeating patterns. Now, using a very different approach than that of Penrose tilings, Farris hasharnessed 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. Click for larger image and caption.

Figure 4, for example, looks at first like a counterexample to the crystallographic restriction, with

Quanta Magazine

https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/ March 5, 2013

five-fold rotational symmetry around point A, and wallpaper pattern shifts in the directions of AB andAC.

In reality, as Farris described in the November 2012 issue of Notices of the American MathematicalSociety, the image is a clever fraud.

“You know the symmetry you’re seeing is impossible,” said Stephen Kennedy, of Carleton College inNorthfield, Minnesota.

The five-fold rotational symmetry around point A is valid enough. But if you look closely, you’ll seethat the pinwheels at B and C in fact have minor differences from the one at A. If we were to zoomout to see more of the pattern, the apparent wallpaper repetitions would grow less and less similarto the design at point A, even as new and even more convincing copies of A would simultaneouslyspring up in other locations, as in Figure 5. In fact, Farris has shown, it’s possible to produce newillusions at ever larger scales by zooming out by specific amounts — namely, in increments ofFibonacci numbers (the number sequence 1, 1, 2, 3, 5, 8, 13, 21, … in which each number is the sumof the preceding two), which also play a role in the geometry of Penrose tilings.

Figure 5. Click for larger image andcaption.

“We know intellectually that this has to be a cheat,” Farris said. Nevertheless, he wrote in Notices,these images “invite our eye to wander and enjoy the near repeats.”

Farris came upon these fakes by modifying a technique he had developed for making genuinewallpaper designs with three-fold rotational symmetry, such as the pattern in Figure 6.

Quanta Magazine

https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/ March 5, 2013

To create a three-fold wallpaper design, Farris started out in three-dimensional space, which has oneparticularly natural three-fold rotation that simply cycles the three coordinates, spinning points inspace 120 degrees around a diagonal line. Farris then created three-dimensional wallpaper patternsby superimposing carefully chosen sine waves and using a preselected palette to color pointsdepending on their position on the superimposed waves. Finally, Farris derived a flat wallpaperpattern by restricting this coloring to the two-dimensional plane that cuts perpendicularly throughthe rotation axis at the origin.

This smooth, sinusoidal approach to creating wallpaper patterns is a departure from the traditionalcut-and-paste method, Kennedy said. “It’s a very novel way to make symmetric patterns.”

Figure 6. A wallpaperpattern with three-fold rotational symmetry, created through Farris’s sine-wave method.

A similar procedure in five-dimensional space might be expected to produce wallpaper patterns withfive-fold symmetry, if we didn’t know this to be impossible. Where, Farris wondered, does everythingfall apart?

Five-dimensional space exists — at least in theoretical terms, although it’s hard to visualize — andhas a natural five-fold rotation analogous to the three-fold rotation in three-dimensional space. Infive-dimensional space, there are two natural flat planes to look at, each one perpendicular to theaxis of rotation and to each other. On each of those planes, the rotation acts by spinning the planearound the origin by 72 or 144 degrees – five-fold rotations. (It might seem counterintuitive toimagine two planes and a line that are all perpendicular to each other, but in dimension five, there’splenty of room for all these objects.)

Quanta Magazine

https://www.quantamagazine.org/how-to-make-impossible-wallpaper-20130305/ March 5, 2013

The problem, Farris realized, is that while the perpendicular plane in the three-dimensional casecuts through space nicely and contains an infinite wallpaper array of points with whole-numbercoordinates, the two perpendicular planes in five-dimensional space are irrational, containing nowhole-number points at all (except the origin). Since the wallpaper pattern from the superimposedsine waves repeats itself over whole-number shifts, these planes fail to inherit a wallpaper patternfrom the higher-dimensional design.

“This throws a fly into the ointment,” Farris wrote in Notices.

Nevertheless, the two planes each inherit the illusion of wallpaper structure, owing to the interplaybetween the so-called golden ratio, an irrational number which describes the directions of the twoplanes, and the Fibonacci numbers.

Thanks to these relationships, Farris was able to show that although the two planes contain nowhole-number points, each plane comes extremely close to an infinite scattering of whole-numberpoints whose coordinates are Fibonacci numbers. Each time the plane comes close to one of theseFibonacci points, the design looks almost exactly the same as it does at the origin, creating theillusion of an exact copy.

Farris has figured out ways to meld the colors and forms of a nature photograph with the wavefunctions that go into his wallpaper designs, thereby creating a dazzling range of wallpaper frauds,such as this one derived from the adjacent meadow view. Some of the tree branches are still visiblein the fake wallpaper.

This article was reprinted on ScientificAmerican.com.