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Constructing Wallpaper Functions

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
[pmm example wallpaper]
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 [Link], and this led naturally to a complete method for constructing every possible wallpaper function [Skip].

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 [Skip].

[Next] Wallpaper Drums
[Skip] Example in a Rectangular Lattice
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[Prev] Abstracts

Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 12 Aug 1998 --- Last modified: Sep 30, 2003 9:34:51 AM
Comments to: CVM@maa.org