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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
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.

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
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