Final stellation of the icosahedron 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2. The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio When regarded as a three-dimensional solid object with edge lengths
Chamfered dodecahedron faces, 110 hexagons and 12 pentagons. It is constructed by a chamfer operation to the truncated icosahedron, adding new hexagons in place of original edges. It can also be constructed as a zip (= dk = dual of kis of) operation from the "chamfered dodecahedron". In other words, raising pentagonal and hexagonal pyramids on a chamfered dodecahedron (kis operation) will yield the (2,2) geodesic polyhedron. Taking the dual of that yields the (2,2) Goldberg polyhedron, which is the chamfered truncated icosahedron, and is also Fullerene C. Chamfered dodecahedron The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges,