Catalog of All Fullerenes with Ten or More Symmetries

By a fullerene, we mean a trivalent plane graph Γ = (V,E, F ) with only hexagonal and pentagonal faces. It follows easily from Euler’s Formula that each fullerene has exactly 12 pentagonal faces. The simplest fullerene is the graph of the dodecahedron with 12 pentagonal faces and no hexagonal faces. It is frequently easier to work with the duals to the fullerenes: geodesic domes, i.e. triangulations of the sphere with vertices of degree 5 and 6. It is in this context that Goldberg[5], Caspar and Klug[1] and Coxeter[2] parameterized the geodesic domes/fullerenes that include the full rotational group of the icosahedron among their symmetries. In this catalog we extend their work by giving a complete parameterization of all fullerenes with ten or more symmetries. In their model, the faces of the icosahedron are filled in with equilateral triangles from the hexagonal tessellation of the plane. In [6], the author generalized their method to other plane graphs filling in the faces with other polygonal regions from the hexagonal tessellation of the plane. Assigned to each fullerene is a 12-vertex plane graphs with edge and angle labels called its signatures. The fullerene can then be reconstituted from its signature by gluing together the regions from the hexagonal tessellation of the plane corresponding to the faces of its signature. To distinguish between edges in the graph model of a fullerene and the edges of its signature, the latter are referred to as segments. The region corresponding to a face is completely determined by the “Coxeter coordinates” of its segments and the “types” of the angles between segments. In Figure 1, we have drawn several segments. Referring to that figure the left hand segments have Coxeter coordinates (4, 2) and (1, 5), respectively, and subtend an angle of type 2. The thinner lines show how the Coxeter coordinates are determined. The type of an angle is the number of centers of edges of the central hexagon between the segments. Segments which run through successive centers are assigned a single Coxeter coordinate and contribute 1 2 to each of the angle types on either side. This is illustrated by the isosceles triangle at the right in the figure. One may think of the segments of a signature as the shortest segments joining the centers of pentagonal faces in a spherical representation of a fullerene. Since only the shortest segments necessary to connect all pentagons are included, there are tight restrictions on the polygons in the hexagonal tessellation that can be faces of a signature. For example there are only six types of triangles that can occur. These are pictured in the next figure. We present these triangles with variable Coxeter coordinates. The variables are independent positive integers; any assignment