# Compound of five tetrahedra

Compound of five tetrahedra | |
---|---|

Rank | 3 |

Type | Weakly regular compound |

Notation | |

Bowers style acronym | Ki |

Elements | |

Components | 5 tetrahedra |

Faces | 20 triangles |

Edges | 30 |

Vertices | 20 |

Vertex figure | Equilateral triangle, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 5 |

Number of external pieces | 60 |

Level of complexity | 10 |

Related polytopes | |

Army | Doe, edge length |

Regiment | Ki |

Dual | Compound of five tetrahedra |

Conjugate | Compound of five tetrahedra |

Convex core | Icosahedron |

Abstract & topological properties | |

Flag count | 120 |

Schläfli type | {3,3} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}+, order 60 |

Flag orbits | 2 |

Convex | No |

Nature | Tame |

The **chiricosahedron**, **ki**, or **compound of five tetrahedra** is a weakly-regular polyhedron compound. It consists of 20 triangles, 3 joining at each vertex. As the name suggests, it is chiral, and has faces in planes parallel to those of the convex icosahedron.

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 tetrahedral pentagyroprism, which is seven-dimensional.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a chiricosahedron of edge length 1 are given by:

- ,

plus all even permutations of:

- .

## Related polyhedra[edit | edit source]

The icosicosahedron is a compound of the two opposite chiral forms of the chiricosahedron.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#2).

- Klitzing, Richard. "ki".
- Wikipedia contributors. "Compound of five tetrahedra".