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Example in a Rectangular Lattice

Suppose we wish to determine all eigenfunctions of the Laplacian that are invariant under an action of pmg on the plane. This group is a good place to start because it's neither too simple nor too complicated.

The first step would be to find eigenfunctions that are invariant with respect to horizontal and vertical translations [Link]. Then we would determine which of these are also invariant under half-turns [Skip], and finally what conditions are required for those to be invariant under the mirror reflections that are in the group [Link]. This will provide us with a recipe for generating functions invariant under pmg [Skip].

[Next] Invariance under Translations
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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 9:26:46 AM
Comments to: CVM@maa.org