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
Dodecahedron pyritohedron, has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}. The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex. The convex regular dodecahedron also has three stellations, all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron {5/2, 5}, the great dodecahedron {5, 5/2},