# Elongated pentagonal rotunda

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Elongated pentagonal rotunda | |
---|---|

Rank | 3 |

Type | CRF |

Space | Spherical |

Notation | |

Bowers style acronym | Epro |

Coxeter diagram | ofxx5xoxx&#xt |

Elements | |

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

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

Vertices | 5+5+10+10 |

Vertex figures | 5 rectangles, edge lengths 1 and (1+√5)/2 |

10 irregular tetragons, edge lengths 1, (1+√5)/2, √2, √2 | |

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

Measures (edge length 1) | |

Volume | |

Dihedral angles | 3–4: |

5–4: | |

4–4: 144° | |

3–5: | |

4–10: 90° | |

Central density | 1 |

Related polytopes | |

Army | Epro |

Regiment | Epro |

Dual | Decakis order-10 truncated semibisected pentagonal rhombitrapezohedron |

Conjugate | Elongated pentagrammic rotunda |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Convex | Yes |

Nature | Tame |

The **elongated pentagonal rotunda** is one of the 92 Johnson solids (J_{21}). It consists of 5+5 triangles, 5+5 squares, 1+5 pentagons, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal rotunda.

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

## Vertex coordinates[edit | edit source]

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

## Related polyhedra[edit | edit source]

Three quasi-convex Stewart toroids are made by tunnelling the elongated pentagonal rotunda. Two of these are distinct elongations of the tunnelled pentagonal rotunda.

## External links[edit | edit source]

- Klitzing, Richard. "epro".

- Quickfur. "The Elongated Pentagonal Rotunda".

- Wikipedia Contributors. "Elongated pentagonal rotunda".
- McCooey, David. "Elongated Pentagonal Rotunda"