Structure of the truncated icosahedron (such as fullerene or viral coatings) and a 60-element conjugacy in \(\text{PSl}(2,11)\). (English) Zbl 0820.20002

Kostant, Bertram

Proc. Natl. Acad. Sci. USA 91, No. 24, 11714-11717 (1994).

A truncated icosahedron is the polyhedron \(P\), obtained by cutting off each of the 12 vertices of an icosahedron, when each cut is made sufficiently close to the vertex to be removed. The faces of the resulting \(P\) are either pentagons \((n = 12)\) or hexagons \((n = 20)\). The proper symmetry group of a truncated icosahedron \(P\) is the icosahedral group PSl(2,5). However, knowning the symmetry group is not enough to specify the graph structure (e.g., the carbon bonds for fullerene, \(C_{60}\)) 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. The author shows that \(M\) exhibits a model for \(P\) where the graph structure is expressed 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.

Reviewer: Chen Chengdong (Shanghai)

MSC:

20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
20H15	Other geometric groups, including crystallographic groups
52B15	Symmetry properties of polytopes
51M25	Length, area and volume in real or complex geometry
51F15	Reflection groups, reflection geometries
82D25	Statistical mechanical studies of crystals

Keywords:

truncated icosahedron; polyhedron; icosahedral group; symmetry group; graph structure; carbon bonds; conjugacy class; maximal commuting subsets