# Elongated pentagonal cupola

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Elongated pentagonal cupola | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Epcu |

Coxeter diagram | oxx5xxx&#xt |

Elements | |

Faces | 5 triangles, 5+5+5 squares, 1 pentagon, 1 decagon |

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

Vertices | 5+10+10 |

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

10 trapezoids, edge lengths 1, √2, √2, √2 | |

10 isosceles triangles, edge lengths √(5+√5)/2, √2, √2 | |

Measures (edge length 1) | |

Volume | |

Dihedral angles | 3–4 cupolaic: |

4–5: | |

4–4 prismatic: 144° | |

3–4 join: | |

4–4 join: | |

4–10: 90° | |

Central density | 1 |

Number of external pieces | 22 |

Level of complexity | 18 |

Related polytopes | |

Army | Epcu |

Regiment | Epcu |

Dual | Decakis order-10 truncated semibisected pentagonal trapezohedron |

Conjugate | Elongated retrograde pentagrammic cupola |

Abstract & topological properties | |

Flag count | 180 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Convex | Yes |

Nature | Tame |

The **elongated pentagonal cupola** is one of the 92 Johnson solids (J_{20}). It consists of 5 triangles, 5+5+5 squares, 1 pentagon, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal cupola.

If a second cupola is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobicupola. If the second cupola is rotated 36º instead, the result is the elongated pentagonal gyrobicupola.

## Vertex coordinates[edit | edit source]

An elongated pentagonal cupola of edge length 1 has the following vertices:

## External links[edit | edit source]

- Klitzing, Richard. "epcu".
- Quickfur. "The Elongated Pentagonal Cupola".

- Wikipedia contributors. "Elongated pentagonal cupola".
- McCooey, David. "Elongated Pentagonal cupola"