Dodecahedron

The dodecahedron, or doe, is one of the five Platonic solids. It has 12 pentagons as faces, joining 3 to a vertex.

It is the only Platonic solid that does not appear as the vertex figure in one of the convex regular polychora. It does, however, appear as a cell of the hecatonicosachoron.

Vertex coordinates
The vertices of a dodecahedron of edge length 1, centered at the origin, are given by:


 * $$(±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4}),$$

along with all even permutations of:


 * $$(±\frac{3+\sqrt{5}}{4},\,±\frac{1}{2},\,0).$$

The first set of vertices corresponds to a cube of edge length (1+$\sqrt{5}$)/2 which can be inscribed into the dodecahedron's vertices.

Representations
A regular dodecahedron has the following Coxeter diagrams:


 * x5o3o (full symmetry)
 * x4oo5oo4x&#xt (H2 axial, face-first)
 * ofxfoo3oofxfo&#xt (A2 axial, vertex-first)
 * xfoFofx ofFxFxo&#xt (A1×A1 axial, edge-first)
 * oxfF xFfo Fofx&#zx (A1×A1×A1 symmetry)

In vertex figures
The dodecahedron appears as a vertex figure in one regular polychoron, that being the small stellated hecatonicosachoron. This vertex figure has an edge length of ($\sqrt{5}$–1)/2.

Related polyhedra
Several Johnson solids can be formed by augmenting the faces of the dodecahedron with pentagonal pyramids:


 * Augmented dodecahedron - One face is augmented
 * Parabiaugmented dodecahedron - Two opposite faces are augmented
 * Metabiaugmented dodecahedron - Two non-adjacent, non-opposite faces are augmented
 * Triaugmented dodecahedron - Three mutually non-adjacent faces are augmented

The dodecahedron has three stellations, namely the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron.