3.3.5 Fullerenes on other surfacesDifferent regular homoatomic carbon plane nets are discussed by T. Balaban (1989). They can be derived in the same way as the general fullerenes: by introducing digons in the vertices of 4valent graphs or by an edgecoloring of a 3valent graph, resulting in a 4valent graph. For example, we can start from the square regular tessellation {4,4}, Archimedean tiling (3,6,3,6) or 2uniform tiling (3,4^{2},6;3,6,3,6) that are all 4valent (Grünbaum and Shephard, 1986), and introduce digons in their vertices, or from the regular tiling {6,3} that is 3valent and color its edges. In the same way, from arbitrary 3valent tiling we can derive perfect plane nets. For different surfaces, the necessary condition for general fullerenes follows from the Euler theorem ve+f = 22g, where g is the genus of the surface. For the torus g = 1, so accepting the 5/6 restriction we conclude that for 3valent graphs n_{5} = 0. In this case, the only possibility is the regular tessellation {6,3}, consisting of b^{2}+bc+c^{2} hexagons (bÎN, c ÎN) (Coxeter and Moser, 1980). We can obtain this tessellation by identifying opposite sides of the rectangle. From finite {6,3} tessellation, we can easily derive the corresponding perfect hexagonal fullerene on torus. The proposed approach can be extended also to the double, triple, etc. torus with g = 2,3,... Similar transformations of carbon nets from one surface to the other (e.g., from a plane to a cylinder, and then to torus) can perhaps explain the formation of certain fullerenes and process of their growing (Kroto, 1989). Allowing heptagons or octagons for faces, from the relationship 2e = 3v and the Euler formula, it 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} (Mackay and Terrones, 1993). To obtain such general fullerenes with a higher degree of symmetry, we can start from different vertextransitive structures (e.g., uniform polyhedra, stellated regular and semiregular polyhedra or infinite polyhedra) (Mackay and Terrones, 1993). For example, different uniform 4valent polyhedra of the type (3,q,3,q) (q = 7,8,9,10,12,18) can be used for the derivation of the corresponding perfect fullerenes with qgonal holes on a double torus (g = 2, q = 8) (Bilinski, 1985). For that, 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. Interesting classes of infinite general fullerenes with noneuclidean plane symmetry groups can be derived from the tessellations of the 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 (Mackay and Terrones, 1993).
