The Leapfrog principle
In [12][13]
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
