“You know the symmetry you’re seeing is impossible,” said Stephen Kennedy, of Carleton College in Northfield, Minnesota.
The five-fold rotational symmetry around point A is valid enough. But if you look closely, you’ll see that the pinwheels at B and C in fact have minor differences from the one at A. If we were to zoom out to see more of the pattern, the apparent wallpaper repetitions would grow less and less similar to the design at point A, even as new and even more convincing copies of A would simultaneously spring up in other locations, as in Figure 5. In fact, Farris has shown, it’s possible to produce new illusions at ever larger scales by zooming out by specific amounts — namely, in increments of Fibonacci numbers (the number sequence 1, 1, 2, 3, 5, 8, 13, 21, … in which each number is the sum of the preceding two), which also play a role in the geometry of Penrose tilings.
“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 genuine wallpaper designs with three-fold rotational symmetry, such as the pattern in Figure 6.
To create a three-fold wallpaper design, Farris started out in three-dimensional space, which has one particularly natural three-fold rotation that simply cycles the three coordinates, spinning points in space 120 degrees around a diagonal line. Farris then created three-dimensional wallpaper patterns by superimposing carefully chosen sine waves and using a preselected palette to color points depending on their position on the superimposed waves. Finally, Farris derived a flat wallpaper pattern by restricting this coloring to the two-dimensional plane that cuts perpendicularly through the rotation axis at the origin.
This smooth, sinusoidal approach to creating wallpaper patterns is a departure from the traditional cut-and-paste method, Kennedy said. “It’s a very novel way to make symmetric patterns.”
A similar procedure in five-dimensional space might be expected to produce wallpaper patterns with five-fold symmetry, if we didn’t know this to be impossible. Where, Farris wondered, does everything fall apart?
Five-dimensional space exists — at least in theoretical terms, although it’s hard to visualize — and has a natural five-fold rotation analogous to the three-fold rotation in three-dimensional space. In five-dimensional space, there are two natural flat planes to look at, each one perpendicular to the axis of rotation and to each other. On each of those planes, the rotation acts by spinning the plane around the origin by 72 or 144 degrees – five-fold rotations. (It might seem counterintuitive to imagine two planes and a line that are all perpendicular to each other, but in dimension five, there’s plenty of room for all these objects.)