C₇₀, C₈₀, C₉₀ and carbon nanotubes by breaking of the icosahedral symmetry of C₆₀

The icosahedral symmetry group H₃ of order 120 and its dihedral subgroup H₂ of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H₃ orbit of the fullerene C₆₀ to the subgroup H₂ yields a union of eight orbits of H₂: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H₂, one builds the polytopes C₇₀, C₈₀, C₉₀ and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H₂ decagons 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 C₆₀ are described.