The Graph of the Truncated Icosahedron and the Last Letter of Galois

SO(3) it is the proper symmetry group of the icosahedron (and the dodecahedron). It is the unique finite subgroup of SO(3) which equals its own commutator subgroup—in fact, it is the unique non-abelian simple finite subgroup of SO(3). It can be thought of as the beginning or smallest member of the family of non-abelian finite simple groups. From the point of view of the McKay correspondence it (or more exactly its “double cover”) corresponds to the exceptional Lie group E8. A group isomorphic to A5 will be referred to as an icosahedral group and any structure admitting such a group as a symmetry group is said to have icosahedral symmetry. Given the unique role of the icosahedral group in group theory, any natural structure having icosahedral symmetry surely deserves special attention. In this paper we will be concerned with one such structure—a structure which seems to be appearing with increasing frequency in the scientific literature. Prior to the discovery of the Fullerenes, around ten years ago, the only known form of pure solid carbon was graphite and diamonds. These two forms are crystalline materials where the bonds between the carbon atoms exhibit hexagonal and tetrahedral structures, respectively. In neither of these two substances, however, are there isolated molecules of pure carbon. On the other hand, in