# Triangular-rectified pentachoric duoprism

The **triangular-rectified pentachoric duoprism** or **trarap** is a convex uniform duoprism that consists of 3 rectified pentachoric prisms, 5 triangular-octahedral duoprisms, and 5 triangular-tetrahedral duoprisms. Each vertex joins 2 rectified pentachoric prisms, 2 triangular-tetrahedral duoprisms, and 3 triangular-octahedral duoprisms. It is a duoprism based on a triangle and a rectified pentachoron, and is thus also a convex segmentopeton, as a rectified pentachoron atop rectified pentachoric prism.

Triangular-rectified pentachoric duoprism | |
---|---|

Rank | 6 |

Type | Uniform |

Notation | |

Bowers style acronym | Trarap |

Coxeter diagram | x3o o3x3o3o () |

Elements | |

Peta | 5 triangular-tetrahedral duoprisms, 5 triangular-octahedral duoprisms, 3 rectified pentachoric prisms |

Tera | 15 tetrahedral prisms, 10+20 triangular duoprisms, 15 octahedral prisms, 3 rectified pentachora |

Cells | 15 tetrahedra, 30+30+60 triangular prisms, 15 octahedra |

Faces | 10+30+60 triangles, 90 squares |

Edges | 30+90 |

Vertices | 30 |

Vertex figure | Triangular prismatic scalene, edge lengths 1 (base tetrahedron and top edge) and √2 (sides) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dipetal angles | Tratet–triddip–troct: |

Rappip–ope–troct: 90° | |

Rappip–tepe–tratet: 90° | |

Troct–triddip–troct: | |

Rappip–rap–rappip: 60° | |

Heights | Rap atop rappip: |

Tratet atop troctt: | |

Central density | 1 |

Number of external pieces | 13 |

Level of complexity | 45 |

Related polytopes | |

Army | Trarap |

Regiment | Trarap |

Dual | Triangular-joined pentachoric duotegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 32400 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}×A_{2}, order 720 |

Convex | Yes |

Nature | Tame |

## Vertex coordinates edit

The vertices of a triangular-rectified pentachoric duoprism of edge length 1 are given by:

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

## Representations edit

A triangular-rectified pentachoric duoprism has the following Coxeter diagrams:

- x3o o3x3o3o ( ) (full symmetry)
- xx3oo xo3ox3oo&#x (A
_{3}×A_{2}symmetry, triangular-tetrahedral duoprism atop triangular-octahedral duoprism) - ox oo3xx3oo3oo&#x (A
_{4}×A_{1}symmetry, rectified pentachoron atop rectified pentachoric prism)

## External links edit

- Klitzing, Richard. "trarap".