# Retrograde pentagrammic cupola

Retrograde pentagrammic cupola | |
---|---|

Rank | 3 |

Type | Segmentotope |

Notation | |

Bowers style acronym | Rastacu |

Coxeter diagram | ox5/3xx&#x |

Elements | |

Faces | 5 triangles, 5 squares, 1 pentagram, 1 decagram |

Edges | 5+5+5+10 |

Vertices | 5+10 |

Vertex figures | 5 crossed isosceles trapezoids, edge lengths 1, √2, (1-√5)/2, √2 |

10 scalene triangles, edge lengths 1, √2, √(5-√5)/2 | |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–10/3: |

3–4: | |

4–5/2: | |

4–10/3: | |

Height | |

Related polytopes | |

Army | Non-CRF Pecu |

Regiment | Rastacu |

Dual | Semibisected pentagrammic trapezohedron |

Conjugate | Pentagonal cupola |

Convex hull | Pentagonal cupola |

Abstract & topological properties | |

Flag count | 100 |

Orientable | Yes |

Properties | |

Symmetry | H_{2}×I, order 10 |

Convex | No |

Nature | Tame |

The **retrograde pentagrammic cupola** or **rastacu**, also called the **crossed pentagrammic cupola** or sometimes just the **pentagrammic cupola**, is a cupola based on the pentagram, seen as a 5/3-gon rather than 5/2. It consists of 5 triangles, 5 squares, 1 pentagram, and 1 decagram.

It can be obtained as a segment of the quasirhombicosidodecahedron, just as its conjugate, the convex pentagonal cupola, can be obtained from the small rhombicosidodecahedron.

It also appears in some scaliform polychora, namely the retroprismatorhombisnub hecatonicosachoron and retroprismatorhombiretrosnub hecatonicosachoron.

## Vertex coordinates[edit | edit source]

A retrograde pentagrammic cupola of edge length 1 has vertices given by the following coordinates:

## Related polyhedra[edit | edit source]

The retrograde pentagrammic cupola is the pentagram-first cap of the quasirhombicosidodecahedron.

## External links[edit | edit source]

- Klitzing, Richard. "rastacu".
- Webb, Robert. "Crossed Pentagrammic Cupola”.