|VW - CVM 1.1|
This image was produced from a function invariant under one of the 17 wallpaper groups, which are groups of isometries whose translational subgroup contains two linearly independent translations.
|First example wallpaper, titled "Bats".|
We are seeing a coarse coloring of the plane according to the values of this function, so that the boundaries of a colored region are level curves of the function.
In this article, we show how to construct every possible function invariant under one of the wallpaper groups. This is done with Fourier analysis on the fundamental region of the translational subgroup. We view every wallpaper function as a superposition of wallpaper waves, each with its fundamental frequency of vibration. Several movies of the vibrations of infinite wallpaper drums are provided.
Our approach amounts to constructing representations of the wallpaper groups on function spaces whose domains are the toric quotients of the plane by the wallpaper lattice. In many cases, these function spaces split nicely into two pieces, one with invariant functions, the other containing functions with anti-symmetries. This combination of algebra and analysis gives a new proof of the fact that there are 46 types of two-color patterns, for a total of 63 types of one- or two-color patterns, as classified by H. J. Woods in 1936 [Wo].
When complex-valued wallpaper functions are allowed, the method leads us to produce the various possible multicolored patterns, which we call color-turning wallpaper functions.