# Compound of ten tetrahedra

Compound of ten tetrahedra | |
---|---|

Rank | 3 |

Type | Regular compound |

Notation | |

Bowers style acronym | E |

Elements | |

Components | 10 tetrahedra |

Faces | 40 triangles as 20 golden hexagrams |

Edges | 60 |

Vertices | 20 |

Vertex figure | Golden hexagram, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 10 |

Number of external pieces | 180 |

Level of complexity | 10 |

Related polytopes | |

Army | Doe, edge length |

Regiment | E |

Dual | Compound of ten tetrahedra |

Conjugate | Compound of ten tetrahedra |

Convex core | Icosahedron |

Abstract & topological properties | |

Flag count | 240 |

Schläfli type | {3,3} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 2 |

Convex | No |

Nature | Tame |

The **icosicosahedron**, **e**, or **compound of ten tetrahedra** is a weakly-regular polyhedron compound. It consists of 40 triangles which form 20 coplanar pairs, combining into golden hexagrams. The vertices also coincide in pairs, leading to 20 vertices where 6 triangles join. It can be seen as a compound of the two chiral forms of the chiricosahedron. It can also be seen as a rhombihedron, the compound of five cubes, with each cube replaced by a stella octangula.

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 tetrahedral decagyroprism, which is twelve-dimensional.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of an icosicosahedron of edge length 1 are given by:

- ,

plus all even permutations of:

- .

## Related polyhedra[edit | edit source]

It has connections to all weakly regular polyhedra and polyhedron compounds. It can be decomposed into 10 tetrahedra, 5 stella octangulas, or 2 chiricosahedra. It and each chiricosahedron has a dodecahedron convex hull and an icosahedron convex core while each stella octangula has a cube convex hull and an octahedron convex core, which form a rhombihedron and small icosicosahedron respectively.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#3).

- Klitzing, Richard. "e".
- Wikipedia contributors. "Compound of ten tetrahedra".