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
Hexagonal bipyramid Hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and because its faces cannot be equilateral triangles. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal