## On Mobius transformations

It should be noted that the noncompactness of the group
of conformal transformations of S^n is a nontrivial phenomenon
which contradicts everybody's geometric intuition.
It is not clear at all why there exists a single conformal transformation
of S^n, which is not a rigid rotation.
Similarly, one cannot see by a plain eye not equipped with
mathematical machinery any non-trivial conformal transformation of
R^n (which as we know maps round spheres to round spheres) where
"trivial" refers to the similarity transformation.

Even geometrically minded artists, designers of symmetric
patterns, could not overcome this limitation of human imagination.
If we look at the incredible number of ornaments designed through the
centuries all over the world, we see all kinds of translational and rotational
symmetries but never a conformal symmetry. Yet, in recent times
conformal symmetries were displayed in many beautiful drawings of
Escher. However, the idea of those was communicated to the artist by
a mathematician, namely Coxeter.

(G. d'Ambra and M. Gromov, Lectures on transformation groups: Geometry and
Dynamics, in: Surveys in Differential Geometry, 1, 1991.)