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
Truncated trapezohedron Truncated trapezohedron An "n"-gonal truncated trapezohedron is a polyhedron formed by a "n"-gonal trapezohedron with "n"-gonal pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism. The vertices exist as 4 "n"-gons in four parallel planes, with alternating orientation in the middle creating the pentagons. The regular dodecahedron is the most common polyhedron in this class, being a platonic solid, with 12 congruent pentagonal faces. A "truncated trapezohedron" has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces.