Structure of the truncated icosahedron (such as fullerene or viral coatings) and a 60-element conjugacy class in PSl(2, 11)

Abstract

The proper symmetry group of a truncated icosahedron P is the icosahedral group PSl(2, 5). However, knowing the symmetry group is not enough to specify the graph structure (e.g., the carbon bonds for fullerene, C60) of P. The group PSl(2, 5) is a subgroup of the 660-element group PSl(2, 11). The latter contains a 60-element conjugacy class, say M, of elements of order 11. I show here that M exhibits a model for P where the graph structure is expressed group-theoretically. For example, the 12 pentagons are the maximal commuting subsets of M. Such a model creates the opportunity of using group-based harmonic analysis (e.g., convolution calculus) to deal with problems concerning the truncated icosahedron.