Polyhedral and Dihedral Caustics in the $\mathbb R^3$

The role of the symmetries in the topology of sets of Lagrangian singularities is studied in a simple physical model: the envelope of the rays emanating from a convex wave front invariant under the action of discrete subgroups of $O(3)$. New point-singularities of integer index are found. They are located at the vertices of the polyhedron or of its dual. For the dihedral subgroups, we have found a remarkable property of stability of umbilics. These properties result from the interplay between the symmetries of the singularities and the topology of the wave front. An application to fine-particle magnetic systems is given.