  Representation theory, for the classical binary polyhedral groups is encoded
by the affine Dynkin diagrams E6^{(1)}, E7^{(1)} and E8^{(1)} (McKay
correspondance). The quantum versions of these classical geometries are
associated with representation theories described by the usual Dynkin diagrams
E6, E7 and E8. The purpose of these notes is to compare several chosen aspects
of the classical and quantum geometries by using the study of spaces of paths
and spaces of essential paths (Ocneanu theory) on these diagrams. To keep the
size of this contribution small enough, most of our discussion will be limited
to the cases of diagrams E6 and E6^{(1)}, i.e. to the quantum and classical
tetrahedra. We shall in particular interpret the A11 labelling of the vertices
of E6 diagram as a quantum analogue of the usual decomposition of spaces of
sections for vector bundles above homogeneous spaces. We also show how to
recover Klein invariants of polyhedra by paths algebra techniques and discuss
their quantum generalizations.
