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 "at" uni-graz.at, May 10,
2016
