Can Spherical Symmetry Techniques Extend Plane Symmetries?

[mnb96]

Hello,

it is known that the symmetry groups on the 2d Euclidean plane are given by the point-groups (n-fold and dihedral symmetries) and the wallpaper groups.

However we can create more symmetries on the plane than just those.
For example we can stereographically project the 2d plane onto the unit sphere, and consider all the spherical symmetry groups (that are much more than those on the plane), and stereographically re-project the sphere onto the plane to obtain new symmetries.

Has this idea been explored already? I bet it was, but I can't find information on this.
And ultimately, why do people say that the symmetries of the plane are just the point-groups and the wallpaper groups?

 

[micromass]

This might be interesting: http://en.wikipedia.org/wiki/Mobius_transformation

 

[mnb96]

Ok...so if I understand correctly, the idea is to use the Möbius transformations, which form the group of isometries of the Riemann sphere.

However I was thinking about the following alternative way of constructing new symmetries on the plane: that is, we express the surface of the Riemann sphere in spherical coordinate angles $(\phi,\theta)$ and we apply the wallpaper group on the curvilinear coordinates $(\phi,\theta)$ ?

Can this be expressed as Möbius transformation?