|VW - CVM 1.1|
The image below was produced from a function invariant under one of the 17 wallpaper groups, which are groups of isometries of the Euclidean plane containing two linearly independent translations.
|First example wallpaper, titled "Bats".|
This function is invariant under every element of the group pmg, which contains, in addition to infinitely many translations, certain reflections, glide reflections, and rotations through 180 degrees. 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. Surprisingly, our method puts us in a position to predict what would happen if a planar membrane were deformed in the shape of such a function and released from rest, that is, we demonstrate the vibrations of a wallpaper drum.
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.