We think the classification of these will not be too difficult. But we
are not currently working on this, waiting for a more abstract formulation
of this entire subject in terms of group representations.
Vibrating wallpaper on the sphere:
There aren't many groups of symmetries of the sphere, so one ought to
be able to find a basis for the spherical eigenfunctions invariant under
each of these groups.
Vibrating wallpaper in the Poincare disk:
Here, of course, there are infinitely many groups of isometries to play
with. We are in the progress of using Eisenstein series (from the field of
modular forms) and the technique of domain-coloring to make pictures of
With our methodology for constructing repeat patterns, we think it
should be possible to deform the functions to introduce stereogram images.
We have experimented just a bit with this, but don't have any results.