# Truncated dodecahedron

Truncated dodecahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Tid |

Coxeter diagram | x5x3o () |

Conway notation | tD |

Stewart notation | T_{5} |

Elements | |

Faces | 20 triangles, 12 decagons |

Edges | 30+60 |

Vertices | 60 |

Vertex figure | Isosceles triangle, edge lengths 1, √(5+√5)/2, √(5+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 10–3: |

10–10: | |

Central density | 1 |

Number of external pieces | 32 |

Level of complexity | 3 |

Related polytopes | |

Army | Tid |

Regiment | Tid |

Dual | Triakis icosahedron |

Conjugate | Quasitruncated great stellated dodecahedron |

Abstract & topological properties | |

Flag count | 360 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

The **truncated dodecahedron**, or **tid**, is one of the 13 Archimedean solids. It consists of 20 triangles and 12 decagons. Each vertex joins one triangle and two decagons. As the name suggests, it can be obtained by truncation of the dodecahedron.

## Vertex coordinates[edit | edit source]

A truncated dodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

- ,
- ,
- .

## Representations[edit | edit source]

A truncated dodecahedron has the following Coxeter diagrams:

- x5x3o () (full symmetry)
- xooxFVFx5xFVFxoox&#xt (H
_{2}axial, decagon-first) - ooxFBVFxFVFx3xFVFxFVBFxoo&#xt (A
_{2}axial, triangle-first)

## Semi-uniform variant[edit | edit source]

The truncated dodecahedron has a semi-uniform variant of the form x5y3o that maintains its full symmetry. This variant has 20 triangles of size y and 12 dipentagons as faces.

With edges of length a (between two dipentagons) and b (between a dipentagon and a triangle), its circumradius is given by and its volume is given by .

It has coordinates given by all even permutations of:

- ,
- ,
- .

where .

## Related polyhedra[edit | edit source]

The truncated dodecahedron can be augmented by attaching pentagonal cupolae onto its decagonal faces, with the squares of the pentagonal cupola adjacent to the triangles of the truncated dodecahedron. This leads to several Johnson solids:

- Augmented truncated dodecahedron – One decagon is augmented.
- Parabiaugmented truncated dodecahedron – Two opposite decagons are augmented.
- Metabiaugmented truncated dodecahedron – Two non-adjacent, non-opposite decagons are augmented.
- Triaugmented truncated dodecahedron – Three mutually non-adjacent decagons are augmented.

If the cupolae are gyrated so that the triangular faces of both solids are adjacent, these faces turn out coplanar, so they don't create any new Johnson solids.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#13).

- Klitzing, Richard. "Tid".
- Quickfur. "The Truncated Dodecahedron".

- Wikipedia contributors. "Truncated dodecahedron".
- McCooey, David. "Truncated Dodecahedron"
- Hi.gher.Space Wiki Contributors. "Dodecahedral truncate".