# Joined hecatonicosachoron

The **joined hecatonicosachoron**, also known as the **pentagonal-tegmatic heptacosiicosachoron** or **pibhaki**, is a convex isochoric polychoron with 720 pentagonal tegums as cells. It can be obtained as the dual of the rectified hexacosichoron.

Joined hecatonicosachoron | |
---|---|

Rank | 4 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Pibhaki |

Coxeter diagram | o5o3m3o () |

Elements | |

Cells | 720 pentagonal tegums |

Faces | 3600 isosceles triangles |

Edges | 1200+2400 |

Vertices | 120+600 |

Vertex figure | 600 cubes, 120 dodecahedra |

Measures (edge length 1) | |

Dichoral angle | |

Central density | 1 |

Related polytopes | |

Dual | Rectified hexacosichoron |

Abstract & topological properties | |

Flag count | 43200 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{4}, order 14400 |

Convex | Yes |

Nature | Tame |

It can also be obtained as the convex hull of a hecatonicosachoron and a hexacosichoron, where the edges of the hexacosichoron are times the length of those of the hecatonicosachoron.

The ratio between the longest and shortest edges is 1: ≈ 1:1.44721. Each face is an isosceles triangle that uses one short and two long edges.

## Isogonal derivatives edit

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Pentagonal tegum (720): Rectified hexacosichoron
- Isosceles triangle (3600): Semi-uniform small rhombated hexacosichoron
- Edge (1200): Rectified hecatonicosachoron
- Edge (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
- Vertex (120): Hexacosichoron
- Vertex (600): Hecatonicosachoron

## External links edit

- Klitzing, Richard. "pibhaki".