On geometric tubular polyhedra of types 201 and 21.

In this paper we classify the two-orbit polyhedra in classes 201 and 21 that have a line preserved by their symmetry groups. Other classes of two-orbit polyhedra that have been previously classified include the chiral (or type 2) polyhedra, and the polyhedra in classes 20 and 22. Since all of those polyhedra are affinely irreducible, none of their symmetry groups preserve a line as we see in the examples studied here. The skeleta of the polyhedra described in this paper bear some similarities to carbon nanotubes; however there the vertices of their skeleta all have degree three, whereas the polyhedra studied here all have degree four.