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
Bilinski dodecahedron Bilinski dodecahedron In geometry, the Bilinski dodecahedron is a 12-sided convex polyhedron with congruent rhombic faces. It has the same topology but different geometry from the face transitive rhombic dodecahedron, another 12-sided polyhedron with congruent rhombic faces. This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus. It is named after Stanko Bilinski, who rediscovered it in 1960. Bilinski himself called it the rhombic dodecahedron of the second kind. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces. In the rhombi of the Bilinski dodecahedron, the ratio