# Pentagonal cupolic prism

The **pentagonal cupolic prism**, or **pecupe**, is a CRF segmentochoron (designated K-4.117 on Richard Klitzing's list). It consiss of 2 pentagonal cupolas, 5 triangular prisms, 5 cubes, 1 pentagonal prism, and 1 decagonal prism.

Pentagonal cupolic prism | |
---|---|

Rank | 4 |

Type | Segmentotope |

Space | Spherical |

Notation | |

Bowers style acronym | Pecupe |

Coxeter diagram | xx ox5xx&#x |

Elements | |

Cells | 5 triangular prisms, 5 cubes, 1 pentagonal prism, 2 pentagonal cupolas, 1 decagonal prism |

Faces | 10 triangles, 5+5+5+10+10 squares, 2 pentagons, 2 decagons |

Edges | 5+10+10+10+10+20 |

Vertices | 10+20 |

Vertex figures | 10 isosceles trapezoidal pyramids, base edge lengths 1, √2, (1+√5)/2, √2, side edge length √2 |

20 irregular tetrahedra, edge lengths 1 (1), √2 (4), and √(5+√5)/2 (1) | |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Trip–4–cube: |

Cube–4–pip: | |

Pecu–3–trip: 90° | |

Pecu–4–cube: 90° | |

Pecu–5–pip: 90° | |

Pecu–10–dip: 90° | |

Trip–4–dip: | |

Cube–4–dip: | |

Heights | Pecu atop pecu: 1 |

Pip atop dip: | |

Central density | 1 |

Related polytopes | |

Army | Pecupe |

Regiment | Pecupe |

Dual | Semibisected pentagonal trapezohedral tegum |

Conjugate | Retrograde pentagrammic cupolic prism |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{2}×A_{1}×I, order 20 |

Convex | Yes |

Nature | Tame |

As the name suggests, it is a prism based on the pentagonal cupola. As such, it is a segmentochoron between two pentagonal cupolas. It can also be viewed as a segmentochoron between a decagonal prism and a pentagonal prism.

It can be obtained as a segment of the small rhombicosidodecahedral prism.

## Vertex coordinatesEdit

Coordinates of the vertices of a pentagonal cupolic prism of edge length 1 centered at the origin are given by:

## External linksEdit

- Klitzing, Richard. "pecupe".