VW - CVM 1.1

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 . Then we would determine which of these are also invariant under half-turns , and finally what conditions are required for those to be invariant under the mirror reflections that are in the group . This will provide us with a recipe for generating functions invariant under pmg .

Invariance under Translations Lattice Waves -- Translational Invariance Constructing Wallpaper Functions Wallpaper Drums