# Pentagonal antitegum

Pentagonal antitegum | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Pat |

Coxeter diagram | p2p10o () |

Conway notation | dA5 |

Elements | |

Faces | 10 kites |

Edges | 10+10 |

Vertices | 2+10 |

Vertex figure | 2 pentagons, 10 triangles |

Measures (edge lengths 1, ) | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 4 |

Related polytopes | |

Army | Pat |

Regiment | Pat |

Dual | Pentagonal antiprism |

Conjugate | Pentagrammic retroantitegum |

Abstract & topological properties | |

Flag count | 80 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | (I_{2}(10)×A_{1})/2, order 20 |

Convex | Yes |

Nature | Tame |

The **pentagonal antitegum**, also known as the **pentagonal trapezohedron**, is an antitegum based on the pentagon, constructed as the dual of a pentagonal antiprism. It has 10 kites as faces, with 2 order–5 and 10 order–3 vertices.

Each face of this polyhedron is a kite with its longer edges times the length of its shorter edges. These kites have 3 angles measuring 108° and 1 measuring 36°.

The pentagonal antitegum can be constructed from the regular dodecahedron by augmenting tall pyramids on two opposite faces.

In tabletop games, it is a popular design for a 10-sided die or "d10." A standard set of dice used in role-playing games comprises the five Platonic solids and two d10s, one marked with the numbers 0-9 and the other with the multiples of 10 from 00-90.

The pentagonal antitegum (with theoretical edge length 1) is the vertex figure of the triangular-gyroprismatic hexacosichoron, which cannot be made uniform.

## External links[edit | edit source]

- Klitzing, Richard. "p2p5p".
- Wikipedia contributors. "Pentagonal trapezohedron".
- McCooey, David. "Pentagonal Trapezohedron"