# Pentagonal rotunda

Pentagonal rotunda | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Pero |

Coxeter diagram | ofx5xox&#xt |

Stewart notation | R_{5} |

Elements | |

Faces | |

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

Vertices | 5+5+10 |

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

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

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–5: |

3–10: | |

5–10: | |

Central density | 1 |

Number of external pieces | 17 |

Level of complexity | 14 |

Related polytopes | |

Army | Pero |

Regiment | Pero |

Dual | Semibisected pentagonal rhombitrapezohedron |

Conjugate | Pentagrammic rotunda |

Abstract & topological properties | |

Flag count | 140 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Flag orbits | 14 |

Convex | Yes |

Nature | Tame |

The **pentagonal rotunda** (OBSA: **pero**) is one of the 92 Johnson solids (J_{6}). It consists of 5+5 triangles, 1+5 pentagons, and 1 decagon. It is a rotunda based on a pentagon, and the only rotunda that results in a Johnson solid.

It can be constructed by cutting an icosidodecahedron in half along one of its decagonal circles of edges. This produces two pentagonal rotundas with the bases in opposite orientation, so the icosidodecahedron can be thought of as the pentagonal gyrobirotunda.

## Vertex coordinates[edit | edit source]

Coordinates for a pentagonal rotunda of edge length 1 has vertices are given by the following:

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

These coordinates create a pentagonal rotunda with a decagonal base on the xy plane.

An alternate set of coordinates can be obtained as a subset of the vertices of the icosidodecahedron:

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

## Related polyhedra[edit | edit source]

Two pentagonal rotundas can be attached at their decagonal bases in the same orientation to form a pentagonal orthobirotunda. If the second rotunda is rotated by 36° the result is the pentagonal gyrobirotunda, better known as the icosidodecahedron. If a pentagonal cupola is attached, the result is either a pentagonal orthocupolarotunda (if the base pentagons are in the same orientation) or a pentagonal gyrocupolarotunda (if the base pentagons are rotated 36°).

A decagonal prism can be attached to the pentagonal rotunda's decagonal base to form the elongated pentagonal rotunda. If a decagonal antiprism is attached instead, the result is the gyroelongated pentagonal rotunda.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#4 under id).

- Klitzing, Richard. "pero".
- Quickfur. "The Pentagonal Rotunda".

- Weisstein, Eric W. "Pentagonal Rotunda" ("Johnson solid") at MathWorld.

- Wikipedia contributors. "Pentagonal rotunda".
- McCooey, David. "Pentagonal Rotunda"