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
Rhombic dodecahedron Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites (deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by ("a","b"), where "b" is determined from "a" for planar faces. (1,1) is the rhombic solution. As ("a") approaches , ("b") approaches infinity. When projected onto a sphere (see right), it