Regular dodecahedron dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry I, Coxeter group [5,3], order 120, with an abstract group structure of "A" × "Z". When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...), which ratio is approximately , or in exact terms: or . A regular dodecahedron has 12 faces and 20 vertices,
Diminished rhombic dodecahedron a cuboctahedron. It retains 5 square faces, and 4 equilateral triangle faces of the cuboctahedron. The 13 Cartesian coordinate can be positioned as: Augmenting two opposite squares will create a dihedral rhombic dodecahedron, doubling the symmetry to "D" symmetry, order 16. Augmenting pyramids on all six square faces, with merged faces will produce a regular octahedron, restoring full octahedral symmetry, order 48. Diminished rhombic dodecahedron In geometry, a diminished rhombic dodecahedron is a rhombic dodecahedron with one or more vertices removed. This article describes diminishing one 4-valence vertex. This diminishment creates one new square face while 4 rhombic faces are