# Rhombohedron

Rhombohedron | |
---|---|

Rank | 3 |

Space | Spherical |

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 |

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.

## Special cases[edit | edit source]

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

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 | edit source]

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

## External links[edit | edit source]

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

## References[edit | edit source]

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