# Compound of five cubes

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Compound of five cubes | |
---|---|

Rank | 3 |

Type | Weakly regular compound |

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 |

Number of external pieces | 360 |

Level of complexity | 18 |

Related polytopes | |

Army | Doe, edge length |

Regiment | Sidtid |

Dual | Compound of five octahedra |

Conjugate | Compound of five cubes |

Convex core | Rhombic triacontahedron |

Abstract & topological properties | |

Flag count | 240 |

Schläfli type | {4,3} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 2 |

Convex | No |

Nature | Tame |

History | |

Discovered by | Edmond Hess |

First discovered | 1876 |

The **rhombihedron**, **rhom**, or **compound of five cubes** is a weakly-regular 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 called 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 pentagyroprism, which is seven-dimensional.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

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

- ,

along with all even permutations of:

- .

## External links[edit | edit source]

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

- Klitzing, Richard. "rhom".
- Wikipedia contributors. "Compound of five cubes".