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
Pentakis dodecahedron buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom. The "pentakis dodecahedron" is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a "pentakis dodecahedron". The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge: Pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of