Structure of the truncated icosahedron (e.g. fullerene or \(\text{C}_{60}\), viral coatings) and a \(60\)-element conjugacy class in \(\text{PSl}(2,11)\)
type: Item
instance of: scholarly article
title: Structure of the truncated icosahedron (e.g. fullerene or \(\text{C}_{60}\), viral coatings) and a \(60\)-element conjugacy class in \(\text{PSl}(2,11)\)
zbMATH Open document ID: 0830.20073
DOI: 10.1007/BF01614076
published in: Selecta Mathematica. New Series
publication date: 1995-06-11T00:00:00Z
review text: The truncated icosahedron is a polyhedron \(P\) with 60 vertices \(V\), 90 edges \(E\), 12 pentagons and 20 hexagons as faces. The icosahedral group \(A\) of the proper symmetry rotations of the icosahedron \(\pi\), which is isomorphic to \(A_5\), is also a symmetry group for \(P\). The action of \(A\) on the graph \(\Delta\) of the vertices \(V\) and edges \(E\) makes \(A\) a symmetry group of \(\Delta\). Furthermore, the vertices \(V\) form a principal homogeneous space for \(A\) and the structure of \(\Delta\) (which determines \(P\)) is given by the pairs of vertices yielding \(E\). The idea of the author is to investigate by group-theoretical methods the structure of \(\Delta\) by making the elements of \(V\) group elements. Considered is the group \(\text{PSl} (2,11)\) of order 660 which operates transitively on a set of 11 elements (taken to be the finite field \(F_{11})\) and has the icosahedral group as an isotropy subgroup \(A\). Once such a subgroup \(A\) of \(\text{PSl} (2,11)\) is chosen, there is a corresponding conjugacy class \(M\) of elements of order 11 in \(\text{PSl} (2,11)\) which is \(A\)-invariant (under conjugation) and has 60 elements. One can, accordingly, make \(M\) a model of \(V\). The vertices are now group elements of order 11. The edge structure \(E\) can then be expressed in terms of order 2 elements \(x^{-1} y\), with \(x, y \in V\). In this way, one gets a group theoretical model of \(\Delta\), and thus one of \(P\). In the paper this model is extensively investigated. In particular, the 12 pentagons correspond to the 12 sets of maximally commutative subsets of \(M\). There are many more relations revealing the rich mathematical structure associated with the geometry of a truncated icosahedron.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 52B15
Mathematics Subject Classification ID: 82D25
zbMATH DE Number: 754158
zbMATH Keywords: fullerene
zbMATH Keywords: truncated icosahedron
zbMATH Keywords: icosahedral group
zbMATH Keywords: symmetry rotations
zbMATH Keywords: action
zbMATH Keywords: homogeneous space
zbMATH Keywords: isotropy subgroup
zbMATH Keywords: conjugacy class
zbMATH Keywords: commutative subsets
author: Bertram Kostant
reviewed by: A. Janner
MaRDI profile type: Publication
cites work: Q3684278
cites work: Q4869233
cites work: Q4039784
description: scientific article; zbMATH DE number 754158