# Compound of five octahedra

Compound of five octahedra | |
---|---|

Rank | 3 |

Type | Weakly regular compound |

Notation | |

Bowers style acronym | Se |

Elements | |

Components | 5 octahedra |

Faces | 40 triangles as 20 golden hexagrams |

Edges | 60 |

Vertices | 30 |

Vertex figure | Square, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 5 |

Number of external pieces | 120 |

Level of complexity | 6 |

Related polytopes | |

Army | Id, edge length |

Regiment | Se |

Dual | Compound of five cubes |

Conjugate | Compound of five octahedra |

Convex core | Icosahedron |

Abstract & topological properties | |

Flag count | 240 |

Schläfli type | {3,4} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | No |

Nature | Tame |

The **small icosicosahedron**, **se**, or **compound of five octahedra** is a weakly-regular polyhedron compound. It consists of 40 triangles which form 20 coplanar pairs, combining into golden hexagrams. 4 triangles join at each vertex.

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.

It can be derived as a rectified chiricosahedron. It is also related to the icosicosahedron. If each stella octangula in the icosicosahedron is replaced with the intersection of the two tetrahedra (an octahedron), the result is a small icosicosahedron.

Its quotient prismatic equivalent is the octahedral pentagyroprism, which is seven-dimensional.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a small icosicosahedron of edge length 1 are given by all permutations of:

- ,

Plus all even permutations of:

- .

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#5).

- Klitzing, Richard. "se".
- Wikipedia contributors. "Compound of five octahedra".