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
Regular icosahedron Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length = = "ϕ" − 1. The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in