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 the vertex figure of the nonconvex small stellated hecatonicosachoron and the hyperbolic icosahedral honeycomb. It also appears as a cell of the hecatonicosachoron.

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


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

along with all even permutations of:


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

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)

Variations
The dodecahedron has a number of variations that retain its face-transitivity:


 * Pyritohedron - has 12 mirror-symmetric pentagonal faces
 * Tetartoid - has 12 generally irregular pentagonal faces, chiral tetrahedral symmetry

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 regular stellations, namely the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron. It also has an uncounted number of stellations with pyritohedral or chiral-tetrahedral symmetry.

The dodecahedron can be constructed by augmenting a cube with 6 specifically-proportioned wedges, such that adjacent triangular and trapezoidal faces of the wedges combine into regular pentagons.

Stellations
A dodecahedron has three stellations:


 * The small stellated dodecahedron
 * The great dodecahedron
 * The great stellated dodecahedron