6. Fullerenes on other surfaces Different regular homoatomic C plane nets are discussed by
T.Balaban [18]. They could be derived
in the same way as the general fullerenes: by introducing digons in the
vertices of 4regular graphs or by an edgecoloring of a 3regular graph,
resulting in 4regular one. For example, we could start from the regular
tessellation {4,4}, Archimedean tiling (3,6,3,6) or 2uniform tiling (3,4^{2},6;3,6,3,6)
that all are 4regular [19], and
introduce digons in their vertices, or from regular tiling {6,3} that is
3regular and color its edges (Fig.
18). From 3regular tilings we could also derive "perfect" plane nets
in the same way as before.
For different surfaces, the necessary condition for general
fullerenes follows from Euler theorem ve+f=22g,
where g is the genus of the surface. For the torus g=1, so
accepting 5/6 restriction we conclude that for 3regular graphs n_{5}=
0. In this case, the only possibility is the regular tessellation {6,3},
consisting of b^{2}+bc+c^{2} hexagons
(where b,c are natural numbers) [5].
This tessellation we could obtain identifying opposite sides of the rectangle
(Fig. 19).
From such finite {6,3} we could simply derive the corresponding
"perfect" hexagonal fullerene on torus. The proposed approach could be
extended also to the double, triple, etc. torus with g=2,3,… Similar
transformations of C nets from one surface to the other (e.g. from plane
to cylinder, and then to torus) maybe could explain the formation of certain
fullerenes and their growing process [20].
Accepting that the faces could be also heptagons or octagons,
from the relationship 2e=3v and Euler formula, follows that
n_{5}n_{7}2n_{8}= 12(1g).
For a sphere without octagons, n_{5}n_{7}=12,
and for a torus without octagons n_{5}=n_{7}
[21]. To obtain such general fullerenes
with a higher degree of symmetry, we could start from different vertextransitive
structures (e.g. uniform polyhedra, stellated regular and semiregular
polyhedra or infinite polyhedra) [21].
For example, different uniform 4valent polyhedra of the type (3,q,3,q)
(q=7,8,9,10,12,18) could be used for the derivation of the corresponding
"perfect" fullerenes with qgonal holes on a double torus (g=2)
(Fig. 20, q=8 [22]).
For this, we use the regular vertexbifurcation of triangular faces, transforming
all of them into hexagons. In the same way, the uniform tessellations of
the type (4,q,4,q) (q=5,6,8,12) or (5,10,5,10) of
a double torus may result in different finite general fullerenes. The interesting
classes of infinite general fullerenes with noneuclidean plane symmetry
groups could be derived from the tessellations of hyperbolic plane H^{2}.
For example, from the uniform tessellation (3,7,3,7) we derive the infinite
perfect 6/7 fullerene in H^{2} with heptagonal holes (Fig.
21)[21]).
