# Augmented truncated dodecahedron

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Augmented truncated dodecahedron | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Autid |

Elements | |

Faces | |

Edges | 5+5+5+5+5+5+5+5+5+10+10+10+10+10+10 |

Vertices | 5+5+5+5+5+10+10+10+10 |

Vertex figures | 5 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2 |

10 irregular tetragons, edge length 1, √2, 1, √(5+√5)/2 | |

50 isosceles triangles, edge lengths 1, √2+√2, √2+√2 | |

Measures (edge length 1) | |

Volume | |

Dihedral angles | 3–4 join: |

3–4 cupolaic: | |

3–10 join: | |

4–5: | |

3–10 tid: | |

10–10: | |

Central density | 1 |

Number of external pieces | 42 |

Level of complexity | 42 |

Related polytopes | |

Army | Autid |

Regiment | Autid |

Dual | Rhombirhombistellated triakis icosahedron |

Conjugate | Augmented quasitruncated great stellated dodecahedron |

Abstract & topological properties | |

Flag count | 420 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Flag orbits | 42 |

Convex | Yes |

Nature | Tame |

The **augmented truncated dodecahedron** (OBSA: **autid**) is one of the 92 Johnson solids (J_{68}). It consists of 5+5+5+5+5 triangles, 5 squares, 1 pentagon, and 1+5+5 decagons. It can be constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron.

## Vertex coordinates[edit | edit source]

An augmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of:

- ,
- ,
- ,

plus the following additional vertices:

- ,
- ,
- .

## External links[edit | edit source]

- Klitzing, Richard. "autid".
- Quickfur. "The Augmented Truncated Dodecahedron".

- Wikipedia contributors. "Augmented truncated dodecahedron".
- McCooey, David. "Augmented Truncated Dodecahedron"