# Truncated hexacosichoron

Truncated hexacosichoron | |
---|---|

Rank | 4 |

Type | Uniform |

Notation | |

Bowers style acronym | Tex |

Coxeter diagram | o5o3x3x () |

Elements | |

Cells | 600 truncated tetrahedra, 120 icosahedra |

Faces | 2400 triangles, 1200 hexagons |

Edges | 720+3600 |

Vertices | 1440 |

Vertex figure | Pentagonal pyramid, edge lengths 1 (base) and √3 (sides) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tut–6–tut: |

Ike–3–tut: | |

Central density | 1 |

Number of external pieces | 729 |

Level of complexity | 4 |

Related polytopes | |

Army | Tex |

Regiment | Tex |

Dual | Dodecakis hecatonicosachoron |

Conjugate | Truncated grand hexacosichoron |

Abstract & topological properties | |

Flag count | 57600 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{4}, order 14400 |

Flag orbits | 4 |

Convex | Yes |

Nature | Tame |

The **truncated hexacosichoron**, or **tex**, also commonly called the **truncated 600-cell**, is a convex uniform polychoron that consists of 120 regular icosahedra and 600 truncated tetrahedra. 1 icosahedron and five truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexacosichoron.

It is also isogonal under H4/5 symmetry, with the icosahedra having the symmetry of snub tetrahedra, and 480 of the truncated tetrahedra having trigonal symmetry only.

## Cross-sections[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a truncated hexacosichoron of edge length 1 are given by all even permutations of:

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

## Semi-uniform variant[edit | edit source]

The truncated hexacosichoron has a semi-uniform variant of the form o5o3y3x that maintains its full symmetry. This variant uses 120 icosahedra of size y and 600 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of icosahedra), its circumradius is given by .

## Related polychora[edit | edit source]

The truncated hexacosichoron is the colonel of a two-member regiment that also includes the truncated faceted hexacosichoron.

Uniform polychoron compounds composed of truncated hexacosichora include:

- Truncated chirotetrahedral dishexacosichoron (2)
- Truncated snub pentishecatonicosachoron (5)
- Truncated snub decahecatonicosachoron (10)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 2: Truncates" (#23).

- Klitzing, Richard. "tex".
- Quickfur. "The Truncated 600-cell".

- Wikipedia contributors. "Truncated 600-cell".