   The Leapfrog principle

## The Leapfrog principle

In  a method is described how to construct a fullerene C3n from a fullerene Cn having the same or even a bigger symmetry group as Cn. This method is called the Leapfrog principle. If we are starting with a Cn cluster with icosahedral symmetry all the new clusters will be of the same symmetry, since this is the biggest symmetry group in 3-dimensional space. In the first step you have to put an extra vertex into the centre of each face of Cn. Then connect these new vertices with all the vertices surrounding the corresponding face. Then the dual polyhedron is again a fullerene having 3n vertices 12 pentagonal and (3n/2)-10 hexagonal faces. Knowing the 3-dimensional cycle index of S(Cn) acting on the sets of vertices, edges and faces it is very easy to compute the cycle index for the induced action of S( Cn) on the set of vertices of C3n. We just have to identify the vertices of Cn with the n new hexagonal faces of C3n. This can be done by identifying the two families of indeterminates describing the action on the sets of vertices and faces of Cn.

For computing the cycle indices for the action on the sets of vertices and edges of C3n we have to proceed in the following way: Let p be an element of S(Cn) given as a permutation of the vertices of Cn and pf the induced permutation of the faces of Cn. Then [^p] , a permutation representation of p acting on the faces of C3n, can be defined as

[^p] (i) :=p(i) if i£n
[^p] :=pf(i-n)+n if i>n.

The permutation representation of p acting on the set of edges of C3n is the induced operation of [^p] on the union of the set of all edges of Cn and the set of all pairs (i,k) where i is an edge of the vertex of the face k-n. In the same way the permutation representation of p acting on the set of vertices of C3n is the induced operation of [^p] acting on the set of all triples (i,j,k), where {i,j} is an edge of the face k-n in Cn. From these permutation representations the cycle indices for the action on the sets of vertices or edges can be computed. For instance the C60 can be constructed from the C20 by the Leapfrog principle.

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