Faces of Platonic solids in all dimensions

A polytope is called Platonic if it has faces of one kind in every dimension, 0 (vertices), 1 (edges), 2 (faces), …. The three-dimensional Platonic solids/polytopes have been known since antiquity: tetrahedron, octahedron, cube, dodecahedron and icosahedron. Platonic solids in any dimension n < ∞ were identified in the 20th century. In this paper a simple recursive method of decoration of the Coxeter–Dynkin diagram is used to describe the faces of all types and dimensions.

Icosahedral symmetry breaking: C₆₀ to C₇₈, C₉₆ and to related nanotubes

The structure of a particular type of hollow cage fullerenes (C_60+N18) is described in detail, and their existence explained from a symmetry-breaking mechanism starting from the perfect icosahedral symmetry of C₆₀ to specific subgroups A₂. This mechanism expands previous results describing the existence of other groups of fullerenes (C_60+N10) based on the breaking of the icosahedral symmetry of C₆₀ to the subgroup H₂. The mechanism is extended to describe the cases that generate carbon nanotubes, as well as stereoisomers of the C₇₈ molecule.