# Dodecahedron

Dodecahedron | |
---|---|

Rank | 3 |

Type | Regular |

Notation | |

Bowers style acronym | Doe |

Coxeter diagram | x5o3o () |

Schläfli symbol | {5,3} |

Conway notation | D |

Stewart notation | D_{5} |

Elements | |

Faces | 12 pentagons |

Edges | 30 |

Vertices | 20 |

Vertex figure | Triangle, edge length (1+√5)/2 |

Petrie polygons | 6 skew decagons |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 1 |

Related polytopes | |

Army | Doe |

Regiment | Doe |

Dual | Icosahedron |

Petrie dual | Petrial dodecahedron |

κ ^{?} | Petrial great stellated dodecahedron |

Conjugate | Great stellated dodecahedron |

Abstract & topological properties | |

Flag count | 120 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Skeleton | Dodecahedral graph |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 1 |

Convex | Yes |

Net count | 43380^{[1]} |

Nature | Tame |

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 as well as the nonconvex grand hecatonicosachoron.

## Vertex coordinates[edit | edit source]

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

- ,

along with all even permutations of:

- .

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[edit | edit source]

A regular dodecahedron has the following Coxeter diagrams:

- x5o3o () (full symmetry)
- xfoo5oofx&#xt (H
_{2}axial, face-first) - ofxfoo3oofxfo&#xt (A
_{2}axial, vertex-first) - xfoFofx ofFxFxo&#xt (K
_{2}axial, edge-first) - oxfF xFfo Fofx&#zx (K
_{3}symmetry)

## Related polytopes[edit | edit source]

### Alternative realizations[edit | edit source]

The dodecahedron and the great stellated dodecahedron are conjugates. Thus they are realizations of the same underlying abstract regular polytope {5,3}. These are the only faithful symmetric realizations of this polytope in 3-dimensional Euclidean space, however there are many more skew faithful symmetric realizations. In total there are 28 faithful symmetric realizations, of which 3 are pure.

### Johnson solids[edit | edit source]

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 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[edit | edit source]

A dodecahedron has three regular stellations:

It also has an uncounted number of stellations with pyritohedral or chiral-tetrahedral symmetry.

### Variations[edit | edit source]

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

### In vertex figures[edit | edit source]

Name | Picture | Schläfli symbol | Edge length |
---|---|---|---|

Small stellated hecatonicosachoron | {5/2,5,3} | ||

Icosahedral honeycomb | {3,5,3} |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#4).

- Klitzing, Richard. "Doe".
- Quickfur. "The Dodecahedron".

- Nan Ma. "Dodecahedron {5, 3}".

- McCooey, David. "Dodecahedron"
- Hi.gher.Space Wiki Contributors. "Dodecahedron".

- Wikipedia contributors. "Dodecahedron".
- Hartley, Michael. "{5,3}*120".
- Wedd, N. The dodecahedron

## References[edit | edit source]

- ↑ Edkins, Jo (2007). "Dodecahedron".
*Solid shapes and their nets*. Archived from the original on 2019-12-26.