The regular dodecahedron is a Platonic solid bounded by 12 regular pentagons. It has 20 vertices and 30 edges.
In order to be able to identify the dodecahedron in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the viewpoints that are commonly encountered:
Vertex-first parallel projection.
Edge-first parallel projection. Four of the dodecahedron's faces project onto 4 of the edges of the hexagonal envelope.
Face-first parallel projection.
Parallel projection parallel to top and bottom faces. The top and bottom faces project to the top and bottom edges of the projection envelope.
The canonical Cartesian coordinates for the dodecahedron are:
- (±1, ±1, ±1)
- (0, ±φ−1, ±φ)
- (±φ−1, ±φ, 0)
- (±φ, 0, ±φ−1)
where φ=(1+√5)/2 is the Golden Ratio.
These coordinates give a dodecahedron with edge length 2/φ, or approximately 1.236.