VW - CVM 1.1

Invariance under a Reflection

Our candidates for eigenfunctions of the Laplacian are

cos(nX) cos(mY)   or   sin(nX) sin(mY) .

To complete invariance under the entire group pmg, it suffices to achieve invariance under a horizontal mirror whose axis is a quarter of the way up the cell. The functions will have to satisfy:

Working out the trigonometry (just apply the angle addition identity), we find that we require m to be odd in the sine terms, with m even in the cosine terms.

Recipe for Invariance under pmg Lattice Waves -- Translational Invariance Example in a Rectangular Lattice Constructing Wallpaper Functions Invariance under Half-Turns