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 π be an element of S(Cn) given as
a permutation of the vertices of Cn and πf
the induced permutation of the faces of Cn. Then
π̂, a permutation representation of π acting on the
faces of C3n, can be defined as
| π̂ :=
πf(i-n)+n if i>n... |
The permutation representation of π acting on the set of
edges of C3n is the induced operation of π̂ 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 π acting on
the set of vertices of C3n is the induced operation of
π̂ 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 "at" uni-graz.at, May 10,
2016
