Regular dodecahedron whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges. A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions. In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes. The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1 : "ϕ", or ("ϕ" − 1) : 1. The ratio of a regular dodecahedron's volume to the volume of a cube embedded inside
Small stellated dodecahedron the form of doubly-wound pentagons overlapping the first 12 pentagons. The latter faces are formed by truncating the original pentagrams. When an {n/d}-gon is truncated, it becomes a {2n/d}-gon. For example, a truncated pentagon {5/1} becomes a decagon {10/1}, so truncating a pentagram {5/2} becomes a doubly-wound pentagon {10/2} (the common factor between 10 and 2 mean we visit each vertex twice to complete the polygon). Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of