VW  CVM 1.1 
We have found that the eigenfunctions of the Laplacian invariant under translation are
where K and L are the lengths of the two translations and m and n are integers . But we have not yet accomplished invariance under every element of pmg. This group also includes halfturns.
Functions invariant under halfturns through the origin must satisfy the
equation
At this point we pause to note that any infinite sum of terms of the form
is formally a wallpaper function with group at least as big as p2. If the series converges pointwise everywhere, we have an actual wallpaper function periodic relative to a rectangular lattice, invariant under halfturns.

