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
Great dodecahedron pentagram, its two-dimensional analogue, via the extension of the ("n-1")-D pentagonal polytope faces of the core "n"D polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes. It shares the same edge arrangement as the convex regular icosahedron; the compound with both is the small complex icosidodecahedron. If only the visible surface is considered, it has the same topology as a triakis icosahedron with concave pyramids rather than convex ones. The excavated dodecahedron can be seen as the same process applied to a regular dodecahedron. A truncation process applied to the great dodecahedron