VW  CVM 1.1 
Suppose G is one of the 17 wallpaper groups. A wallpaper function with group G is a real (or later, complex) valued function on the plane, such that
A function invariant under the group pmm.  
f(gx) = f(x)for every x in the plane, and every g in G.
For example, here is one view of a wallpaper function with group pmm. That group has translations, along with horizontal and vertical mirrors.
Our original purpose was to find a way to construct wallpaper functions. Surprisingly, this led to the consideration of vibrating wallpaper drums , and this led naturally to a complete method for constructing every possible wallpaper function .
It may surprise you that our plan to construct wallpaper patterns led us to
think of setting the wallpaper in motion, but that's what worked. And that
naturally led us to PDEs. To skip the PDEs, you may continue with an
example where the group is pmg .

