# Rhombihedron

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Rhombihedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Rhom |

Elements | |

Components | 5 cubes |

Faces | 30 squares |

Edges | 60 |

Vertices | 20 |

Vertex figure | Golden hexagram, edge length √2 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | 5 |

Dihedral angle | 90° |

Central density | 5 |

Related polytopes | |

Army | Doe |

Regiment | Sidtid |

Dual | Small icosicosahedron |

Conjugate | Rhombihedron |

Convex core | Rhombic triacontahedron |

Abstract & topological properties | |

Schläfli type | {4,3} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | No |

Nature | Tame |

The **rhombihedron**, **rhom**, or **compound of five cubes** is a uniform polyhedron compound. It consists of 30 squares. The vertices coincide in pairs, leading to 20 vertices where 6 squares join.

It has the same edges as the small ditrigonary icosidodecahedron.

This compound is sometimes considered to be regular, but it is not flag-transitive, despite the fact it is vertex, edge, and face-transitive. It is however regular if you consider conjugacies along with its other symmetries.

Its quotient prismatic equivalent is the cubic pentachoroorthowedge, which is seven-dimensional.

## Gallery

## Vertex coordinates

The vertices of a rhombihedron of edge length 1 are given by:

along with all even permutations of:

## External links

- Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#4).

- Klitzing, Richard. "rhom".

- Wikipedia Contributors. "Compound of five cubes".