The problem, Farris realized, is that while the perpendicular plane in the three-dimensional case cuts through space nicely and contains an infinite wallpaper array of points with whole-number coordinates, the two perpendicular planes in five-dimensional space are irrational, containing no whole-number points at all (except the origin). Since the wallpaper pattern from the superimposed sine waves repeats itself over whole-number shifts, these planes fail to inherit a wallpaper pattern from 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 interplay between the so-called golden ratio, an irrational number which describes the directions of the two planes, and the Fibonacci numbers.
Thanks to these relationships, Farris was able to show that although the two planes contain no whole-number points, each plane comes extremely close to an infinite scattering of whole-number points whose coordinates are Fibonacci numbers. Each time the plane comes close to one of these Fibonacci points, the design looks almost exactly the same as it does at the origin, creating the illusion of an exact copy.
Farris has figured out ways to meld the colors and forms of a nature photograph with the wave functions 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 visible in the fake wallpaper.
Illustration and Photo: Frank Farris