Regular dodecahedron an edge length of one unit has the properties: The "regular dodecahedron " has two special orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A and H Coxeter planes. In perspective projection, viewed above a pentagonal face, the regular dodecahedron can be seen as a linear-edged Schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions. The regular dodecahedron can also be represented as a spherical tiling. The following Cartesian coordinates define the 20 vertices
Bilinski dodecahedron of lengths of the two diagonals is the golden ratio; that is, the faces of this shape are golden rhombi. In contrast, for the standard rhombic dodecahedron the corresponding ratio is the square root of 2. This shape is a zonohedron. Like the rhombic dodecahedron, it can tile three-dimensional space by translation, making it a parallelohedron. The Bilinski dodecahedron and rhombic dodecahedron have the same topology: their vertices, edges, and faces correspond one-for-one, in an adjacency-preserving way. However, their geometry is different. In a 1962 paper, H. S. M. Coxeter claimed that the Bilinski dodecahedron could be obtained by an