Polytope 240-Based Rod Structures Made of Tetrahedral Atoms: Comparison with Structures Obtained Using a Modular Design Method

Some rod structures consisting of tetrahedral atoms and differing in symmetry are considered. The structures are obtained as a stereographic projection of polytope 240 vertices into three-dimensional Euclidean space when the polytope rolls along a straight line belonging to this space. The axes of the rod structures are the projections of great circles of a three-dimensional sphere containing the polytope 240 vertices. It has been shown that the rod structures obtained earlier by modular design are topologically equivalent to those considered in this paper, but they have a changed symmetry. Three possible conformations following from the results of this work are presented for a C_nH_2n hydrocarbon chain with close packing of atoms, conservation of the tetrahedral coordination of carbon atoms, and noncrystallographic symmetry.