Great Dirhombicosidodecahedron

Introduction. This is a paper model of the great dirhombicosidodecahedron. This polyhedron is a singular example of a uniform polyhedron. It is "uniform" because it's a polyhedral approximation of a compact surface such that (a) the faces are all regular polygons, (b) all the vertices are congruent to one another, and (c) the symmetry group is a finite Coxeter group of rank 3. There are many uniform polyhedra in existence and the great dirhombicosidodecahedron is only one of them. Another, more accessible example is the "cube". A cube has 6 square faces that meet in sets of three at every vertex. The cube represents a polyhedral decomposition of the sphere. A more advanced example is the octahemioctahedron:

The octahemioctahedron has 8 triangles and 4 hexagons with two of each meeting at each of the 12 vertices. It has octahedral symmetry and the polyhedron presents a decomposition of the torus. The uniform polyhedra have been classified, and the paper by Coxeter et al is an excellent reference.

To further describe the great dirhombicosidodecahedron, the faces consist of 40 equilateral triangles, 24 regular pentagrams (the edges intersect one another), and 60 squares. In the pictured model, the triangles are yellow, goldenrod, red, pink, and orange, the pentagrams are blue, green, violet, greenish blue, lavender, and baby blue, and the squares, being the most difficult to identify, are all black. There is a good reason that the squares are difficult to identify. They all pass through the center of the polyhedron, so it is difficult to see a square "head-on". Inspecting a vertex, one discovers a feature not shared by any other uniform polyhedron in existence: As many as eight faces meet at every vertex, these being two triangles, two pentagons, and four squares. The great dirhombicosidodecahedron has 60 such vertices, all congruent to each other.

Another uniform polyhedron closely related to the great dirhombicosidodecahedron is the quasirhombicosidodecahedron:

One can arduously check that the underlying surface of the great dirhombicosidodecahedron is orientable, so, upon computing the Euler characteristic,

c=v-e+f=60-240+(40+24+60)=-56,

one finds that the surface is homeomorphic to a sphere with 29 handles. Now carefully study the data for the quasirhombicosidodecahedron. The faces of this polyhedron are 20 equilateral triangles, 12 regular pentagrams, and 30 squares. Also, every vertex of the quasirhombicosidodecahedron has two squares, one triangle, and one pentagon. The data for the great dirhombicosidodecahedron are obtained by doubling all these numbers. What about the vertices? As it turns out, both polyhedra have precisely 60 vertices, so this doubling phenomenon is not all-pervasive. The problem that's prohibiting the continuation of this "doubling" map is the genus of the surface underlying the quasirhombicosidodecahedron. Again, one uses the Euler characteristic,
c=v-e+f=60-120+(20+12+30)=2.

Thus, in this case, the polyhedron is homeomorphic to the sphere.

This phenomenon is an example of a branched cover. There is a function p which maps points of the great dirhombicosidodecahedron to the quasirhombicosidodecahedron, and this function is "2-to-1" nearly everywhere. That is, the pre-image under p of almost every point of the quasirhombicosidodecahedron has two elements. If one wants to know why the phrases "nearly everywhere" and "almost every" appear, one needs to see what happens at the vertices of the great dirhombicosidodecahedron. Because the 60 vertices are the only points that are not "doubled", they are called branch points. Also, since the map is 2-to-1 (as opposed to n-to-1 where n>1) everywhere except at the vertices, the map may be described as "hyperelliptic".

References.

H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller. Uniform Polyhedra. Phil. Trans. 246 A, (1954), 401-450.

Magnus Wenninger. Polyhedron Models. Cambridge University Press, 1970.