It should be noted that the noncompactness of the group of conformal transformations of is a nontrivial phenomenon which contradicts everybody’s geometric intuition. It is not clear at all why there exists a single conformal transformation of , 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 (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. Quoted from Eremenko, with thanks.)