# Tesseractihemioctachoron

Tesseractihemioctachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Tho |

Coxeter diagram | /2 |

Elements | |

Cells | 8 tetrahedra, 4 octahedra |

Faces | 32 triangles |

Edges | 24 |

Vertices | 8 |

Vertex figure | Tetrahemihexahedron, edge length 1 |

Edge figure | tet 3 oct 3 tet 3 oct 3 |

Measures (edge length 1) | |

Circumradius | |

Dichoral angle | 60° |

Number of external pieces | 40 |

Level of complexity | 5 |

Related polytopes | |

Army | Hex |

Regiment | Hex |

Company | Hex |

Dual | Tesseractihemioctacron |

Conjugate | None |

Abstract & topological properties | |

b_{0} | 1 |

b_{1} | 0 |

b_{2} | 3 |

b_{3} | 0 |

Euler characteristic | 4 |

Orientable | No |

Properties | |

Symmetry | D_{4}, order 192 |

Convex | No |

Nature | Tame |

The **tesseractihemioctachoron**, or **tho**, is a uniform hemipolychoron. It has 8 regular tetrahedra and 4 central octahedra as cells, with 4 tetrahedra and 3 octahedra at each vertex in the form of a tetrahemihexahedron. It is the 4D demicross.

It is a faceting of the hexadecachoron, analogous to how the tetrahemihexahedron is a faceting of the octahedron. It shares the hexadecachoron's vertices, edges, and faces. The 8 tetrahedra are half of those of the hexadecachoron, while the octahedra are its original vertex figures.

Notably, as well as being isogonal, the tesseractihemioctachoron is also edge- and face-transitive. It is one of only four non-regular uniform polychora to be vertex, edge, and face transitive.

## Cross-sections[edit | edit source]

## Vertex coordinates[edit | edit source]

Its vertices are the same as those of its regiment colonel, the hexadecachoron.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Regular Polychora" (#17).

- Klitzing, Richard. "tho".