Some cycle indicesTopIntroduction

Introduction

In 1985 the first C60 clusters were produced and it was assumed [21] that its 60 carbon atoms are placed at the vertices of a truncated icosahedron (which is commonly called a foot ball or soccer ball.) The truncated icosahedron is one of the 14 Archimedean solids already known to the ancient Greeks. Five years later [20][24] the original hypotheses on the structure of C60 could be confirmed. A nice introduction into the theory of the fullerenes providing historical and chemical background information can be found in [10].

From the mathematical point of view in the present paper we are interested in the combinatorics of the symmetry group of the truncated icosahedron and we will discuss some examples in how many ways the soccer ball can be coloured in essentially different ways. From the chemical point of view we will determine the number of all possible placements of double bonds for the C60 fullerene. Furthermore investigating the C60H60-molecule (i.e. a hypothetic molecule where all the thirty double bonds of the C60 are hydrated), we will compute the numbers of all C60HkCl60-k molecules. Finally the implementation of these cycle index methods in the computer algebra system SYMMETRICA [23] will be discussed and some further cycle indices will be listed.


harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

Some cycle indicesTopIntroduction