C70, C80, C90and carbon nanotubes by breaking of the icosahedral symmetry of C60

Abstract

The icosahedral symmetry groupH3of order 120 and its dihedral subgroupH2of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for theH3orbit of the fullerene C60to the subgroupH2yields a union of eight orbits ofH2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three ornadditional decagonal orbits ofH2, one builds the polytopes C70, C80, C90and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number ofH2decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60are described.