# Dodecahedron

Dodecahedron Rank3
TypeRegular
SpaceSpherical
Bowers style acronymDoe
Coxeter diagramx5o3o (     )
Schläfli symbol{5,3}
Elements
Vertex figureTriangle, edge length (1+5)/2 Faces12 pentagons
Edges30
Vertices20
Measures (edge length 1)
Circumradius$\frac{\sqrt3+\sqrt{15}}{4} ≈ 1.40126$ Edge radius$\frac{3+\sqrt5}{4} ≈ 1.30902$ Inradius$\sqrt{\frac{25+11\sqrt5}{40}} ≈ 1.11352$ Volume$\frac{15+7\sqrt5}{4} ≈ 7.66312$ Dihedral angle$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$ Central density1
Euler characteristic2
Flag count120
Number of pieces12
Level of complexity1
Related polytopes
ArmyDoe
RegimentDoe
DualIcosahedron
Petrie dualPetrial dodecahedron
ConjugateGreat stellated dodecahedron
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

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+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

Dodecahedra in vertex figures
Name Picture Schläfli symbol Edge length
Small stellated hecatonicosachoron {5/2,5,3} $\frac{\sqrt{5}-1}{2}$ Icosahedral honeycomb {3,5,3}

## 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:

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: