# Tetrahedral tegum

Tetrahedral tegum | |
---|---|

Rank | 4 |

Type | CRF |

Space | Spherical |

Notation | |

Bowers style acronym | Tete |

Coxeter diagram | oxo3ooo3ooo&#xt |

Elements | |

Cells | 8 tetrahedra |

Faces | 4+12 triangles |

Edges | 6+8 |

Vertices | 2+4 |

Vertex figures | 2 tetrahedra, edge length 1 |

4 skewed triangular tegums, edge length 1 | |

Measures (edge length 1) | |

Inradius | |

Hypervolume | |

Dichoral angles | Tet–3–tet equatorial: |

Tet–3–tet pyramidal: | |

Height | |

Central density | 1 |

Related polytopes | |

Army | Tete |

Regiment | Tete |

Dual | Sem-uniform Tetrahedral prism |

Conjugate | None |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A3×A1, order 48 |

Convex | Yes |

Nature | Tame |

The **tetrahedral tegum** or **tete**, also called the **tetrahedral bipyramid**, is a CRF polychoron with 8 identical regular tetrahedra as cells. As such it is also a Blind polytope. As the name suggests, it is a tegum based on the tetrahedron, formed by attaching two regular pentachora at a common cell.

## Vertex coordinates[edit | edit source]

The vertices of a tetrahedral tegum of edge length 1 are given by:

## Representations[edit | edit source]

A tetrahedral tegum has the following Coxeter diagrams:

- oxo3ooo3ooo&#xt
- yo ox3oo3oo&#xt (y = , as full tegum)
- oyo oox3ooo&#xt (as triangular pyramidal tegum)

## Variations[edit | edit source]

The tetrahedral tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 8 non-CRF triangular pyramids as cells.

One notable variation can be obtained as the dual of the uniform tetrahedral prism, which can be represented by m2m3o3o. In this variation the height between the top and bottom vertices of the tegum is exactly times the length of the edges of the base tetrahedron, and all the dichoral angles are .

## External links[edit | edit source]

- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

- Klitzing, Richard. "tete".