## The Leapfrog principle

In [12][13]
a method is described how to
construct a fullerene C_{3n} from a fullerene C_{n} having
the same or even a bigger symmetry group as C_{n}.
This method is called the *Leapfrog principle*.
If we are starting with a C_{n} 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 C_{n}.
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(C*_{n}) acting
on the sets of vertices, edges and faces it is very easy
to compute the cycle index for the induced action of *S(
C*_{n}) on the set of vertices of C_{3n}.
We just have to identify the vertices of C_{n} with the *n* new
hexagonal faces of C_{3n}.
This can be done by identifying the two families of indeterminates
describing the action on the sets of vertices and faces of C_{n}.
For computing the cycle indices for the action on the sets of
vertices and edges of C_{3n} we have to proceed
in the following way:
Let *p* be an element of *S(C*_{n}) given as a permutation
of the vertices of C_{n} and *p*_{f} the induced permutation
of the faces of C_{n}.
Then *[^p] *,
a permutation representation of *p* acting on the faces of
C_{3n}, can be defined as

* [^p] (i) :=p(i) if i£n*

*[^p] :=p*_{f}(i-n)+n if i>n.

The permutation representation of *p* acting on the set
of edges of C_{3n} is the induced operation of *[^p] * on
the union
of the set of all edges of C_{n} 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 C_{3n} 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 C_{n}.
From these permutation representations the cycle indices for the
action on the sets of vertices or edges can be computed.
For instance the C_{60} can be constructed from
the C_{20} by the Leapfrog principle.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001