# Rhombohedron

A **rhombohedron** or **rhombic hexahedron** is a polytope with 6 rhombic faces. The rhombohedron is a special case of the parallelepiped where all edges are the same length. It is a zonohedron and can tile Euclidean space in a rhombohedral honeycomb.

Rhombohedron | |
---|---|

Rank | 3 |

Elements | |

Faces | 2 + 2 + 2 rhombi |

Edges | 2 + 2 + 2 + 2 + 2 + 2 edges |

Vertices | 2 + 2 + 2 + 2 vertices |

Measures (edge length 1) | |

Central density | 1 |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | S_{2}, order 2 |

Convex | Yes |

Nature | Tame |

## Special cases edit

### Golden rhombohedra^{[1]}^{[2]}^{[3]}
edit

Two of the five golden isozonohedra are rhombohedra. They are called the **acute golden rhombohedron** and the **obtuse golden rhombohedron**. They are constructed as rhombohedra with golden rhombi as faces. The acute golden rhombohedron is constructed with an angle of at the apex and the obtuse golden rhombohedron is constructed with an angle of .

The golden rhombohedra with side length 1 have volumes:

- Acute:
- Obtuse:

Polyhedra such as the rhombic hexecontahedron and rhombic triacontahedron can be dissected into golden rhombohedra and thus their volumes can be derived from the volumes of the golden rhombohedra.

### Other cases edit

- Triangular antitegum: isotopic.
- Cube: all faces are squares.

## External links edit

- Wikipedia contributors. "Rhombohedron".
- Weisstein, Eric W. "Rhombohedron" at MathWorld.

## References edit

- ↑ Wikipedia contributors. "Golden rhombohedra".
- ↑ Weisstein, Eric W. "Acute Golden Rhombohedron" at MathWorld.
- ↑ Weisstein, Eric W. "Obtuse Golden Rhombohedron" at MathWorld.