A class of doubly regular polyhedral surfaces (Q1080179)

The author starts from the well-known fact that 4 of the 5 Platonic solids (except the tetrahedron) doubly cover polyhedra which are isomorphic to regular maps on the real projective plane. He then defines: A principal doubly regular (PDR) 2-complex is a 2-complex whose faces are abstractly convex polygons, whose geometric realization is a closed 2- manifold, whose automorphisms are doubly transitive on faces, and no two of whose faces have two common edges. The author gives the complete list of the orientable and non-orientable PDR 2-complexes: In both cases it is a sparse infinite sequence, plus the tetrahedron in the orientable case and two pentagon-hexahedra in the non- orientable case.