The Regular Polyhedra
(of index two)
Click here to see these in stereo.
Here are all five regular polyhedra (of index two). By name, going from left to right, they are the ditrigonal dodecadodecahedron, dodecadodecahedron, Ef_{1}g_{1}, medial rhombic triacontahedron, and De_{1}f_{1}. The first two are uniform polyhedra, the third and fifth are stellations of the regular icosahedron, and the fourth is a stellation of the rhombic triacontahedron. The labels for the icosahedra are from The Fifty-Nine Icosahedra by Coxeter, et al.
These polyhedra have index two because, for each polyhedron, the symmetry group of the immersed polyhedron is a subgroup of the full symmetry group of the polyhedron, and the index is two. As it turns out, the direct product of C_{2} and S_{5} acts on each of these polyhedra, but the symmetry group of each immerseed polyhedron is isomorphic to the symmetry group of the regular icosahedron, which is isomorphic to the direct product of C_{2} and A_{5}. Note that C_{2} is the two-element group, S_{5} is the group of all permutations on five letters, and A_{5} is the group of even permutations on five letters. One can find a proof that there are only five regular polyhedra (of index two) in the last reference cited below.
"Regular" means about what one would expect it to mean. It means the same thing for the five Platonic solids and for the four Kepler-Poinsot "star" polyhedra. Technically, "regular" means that the group of symmetries acts transitively on the flags consisting of triples {v,e,f} where v is a vertex, e is an edge, f is a face, and all three of these objects are incident with one another. Naturally, one should observe that all the faces are regular polygons, although they may be considerably distorted. (What makes polygon "regular"?)
The following table gives some data for these polyhedra. Notice that each of these polyhedra has 60 edges. The "valence" is the number of edges emanating from each vertex. All these polyhedra represent closed, connected, and orientable surfaces, so the only remaining invariant is the genus, and this is also given for each polyhedron.
Polyhedron | Vertices | Valence | Faces | Genus |
Ditrigonal Dodecadodecahedron | 20 | 6 | 24 pentagons | 9 |
Dodecadodecahedron | 30 | 4 | 24 pentagons | 4 |
Ef_{1}g_{1} | 20 | 6 | 20 hexagons | 11 |
Medial Rhombic Triacontahedron | 24 | 5 | 30 squares | 4 |
De_{1}f_{1} | 24 | 5 | 20 hexagons | 9 |
The polyhedra are arranged in the photo so that a reflection about a vertical line yields a duality. Thus, the ditrigonal dodecadodecahedron and the icosahedron De_{1}f_{1} are duals, the dodecadodecahedron and the medial rhombic triacontahedron, and the icosahedron Ef_{1}g_{1}, remarkably, is self-dual. This should also be evident from looking at the data in the table.
References
H. S. M. Coxeter. Regular Polytopes.
3rd ed. Dover Publications Inc., New York, 1973.
H. S. M. Coxeter, P. Du Val, H. T. Flather, J. F. Petrie.
The Fifty-Nine Icosahedra.
University of Toronto Press, Toronto, Ontario (Canada), 1938.
J. M. Wills.
The combinatorially regular polyhedra of index 2.
Aequationes Mathematicae
34 (1987) 206-220.