Parallelogram-faced isohedra with edges in mirror-planes

Regular (or Platonic) polyhedra have been studied since antiquity, and many other kinds of polyhedra with various symmetry properties have been investigated since then. These include the traditional Archimedean polyhedra (regular-faced with congruent vertex gures), isohedra (polyhedra with faces all equivalent under symmetries of the polyhedron), isogonal polyhedra (all vertices of which are equivalent), uniform polyhedra (regular-faced isogonal polyhedra), Kepler–Poinsot regular polyhedra, and others. The study of rhombohedra (isohedra with rhombic faces) started with Kepler, and continued in more recent times. However, there has been no complete enumeration, or classi cation, of such polyhedra. In the present paper we shall make a beginning of such a systematic investigation. For reasons of time and space we restrict the consideration to polyhedra with the octahedral symmetry group, but admit any parallelograms as faces; the analogous but much more numerous polyhedra with icosahedral symmetry will be discussed elsewhere. Another restriction imposed in the present investigation concerns the kinds of objects we shall accept as ‘polyhedra’. As is quite generally accepted, we treat polyhedra as collections of planar polygons (parallelograms in the present case), such that edges are shared by two faces, faces incident with any one vertex form a single circuit, and the set of faces is strongly connected. We do not insist that distinct faces determine distinct planes, and we do not object to overlaps among faces. We also admit the possibility of several vertices being represented by the same point, as long as the circuits of faces incident with each are disjoint. (We shall comment on this permissiveness in the concluding section.) It seems that this class of polyhedra is wide enough to encompass all the traditional families mentioned above, without leading to some of the ‘strange’ possibilities described in [7,11] for polyhedra of more general kinds. A nal restriction on the polyhedra considered here is that every edge of the polyhedron is to be contained in a mirror, that is, a plane of re ective symmetry of the