3.3.5  Fullerenes on other surfaces

Different 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 4-valent graphs or by an edge-coloring of a 3-valent graph, resulting in a 4-valent graph. For example, we can start from the square regular tessellation {4,4}, Archimedean tiling (3,6,3,6) or 2-uniform tiling (3,42,6;3,6,3,6) that are all 4-valent (Grünbaum and Shephard, 1986), and introduce digons in their vertices, or from the regular tiling {6,3} that is 3-valent and color its edges. In the same way, from arbitrary 3-valent tiling we can derive perfect plane nets. 

For different surfaces, the necessary condition for general fullerenes follows from the Euler theorem v-e+f = 2-2g, where g is the genus of the surface. For the torus g = 1, so accepting the 5/6 restriction we conclude that for 3-valent graphs n5 = 0. In this case, the only possibility is the regular tessellation {6,3}, consisting of b2+bc+c2 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 n5-n7-2n8 = 12(1-g). For a sphere without octagons, n5-n7 = 12, and for a torus without octagons n5 = n7 (Mackay and Terrones, 1993). To obtain such general fullerenes with a higher degree of symmetry, we can start from different vertex-transitive structures (e.g., uniform polyhedra, stellated regular and semi-regular polyhedra or infinite polyhedra) (Mackay and Terrones, 1993). For example, different uniform 4-valent 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 q-gonal holes on a double torus (g = 2, q = 8) (Bilinski, 1985). For that, we use the regular vertex-bifurcation 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 non-euclidean plane symmetry groups can be derived from the tessellations of the hyperbolic plane H2. For example, from the uniform tessellation (3,7,3,7) we derive the infinite perfect 6/7 fullerene in H2 with heptagonal holes (Mackay and Terrones, 1993). 

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