# Triangular duotegum

Triangular duotegum | |
---|---|

Rank | 4 |

Type | Noble |

Space | Spherical |

Notation | |

Bowers style acronym | Triddit |

Coxeter diagram | m3o2m3o |

Elements | |

Cells | 9 tetragonal disphenoids |

Faces | 18 isosceles triangles |

Edges | 6+9 |

Vertices | 6 |

Vertex figure | Triangular tegum |

Measures (based on triangles of edge length 1) | |

Edge lengths | Lacing (9): |

Base (6): 1 | |

Circumradius | |

Inradius | |

Hypervolume | |

Central density | 1 |

Related polytopes | |

Army | Triddit |

Regiment | Triddit |

Dual | Triangular duoprism |

Conjugate | None |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{2}≀S_{2}, order 72 |

Convex | Yes |

Nature | Tame |

The **triangular duotegum** or **triddit**, also known as the **triangular-triangular duotegum**, the **3 duotegum**, or the **3-3 duotegum**, is a convex noble duotegum that consists of 9 tetragonal disphenoids and 6 vertices, with 6 cells joining at a vertex. It is the simplest possible duotegum, and is also the 6-2 step prism. It is the first in an infinite family of isogonal triangular hosohedral swirlchora and also the first in an infinite family of isochoric triangular dihedral swirlchora.

It shares the same vertex and edge configuration with the 5-dimensional hexateron. In fact, it is the simplest polytope that is not a simplex, but every pair of vertices is joined by an edge. Every n-2 step prism also has this property.

The ratio between the longest and shortest edges is 1: ≈ 1:1.22474.

It is notable as the Birkhoff polytope *B*_{3}.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a triangular duotegum based on 2 unit-edge triangles, centered at the origin, are given by:

Due to this polytope being a Birkhoff polytope, the vertices of a triangular duotegum can also be positioned in 9D by taking all 3 × 3 permutation matrices and unraveling them in reading order:

- (1, 0, 0, 0, 1, 0, 0, 0, 1)
- (1, 0, 0, 0, 0, 1, 0, 1, 0)
- (0, 0, 1, 1, 0, 0, 0, 1, 0)
- (0, 1, 0, 1, 0, 0, 0, 0, 1)
- (0, 1, 0, 0, 0, 1, 1, 0, 0)
- (0, 0, 1, 0, 1, 0, 1, 0, 0)

The longer edge length in this case is , and the shorter one 2. It is centered on .

## External links[edit | edit source]

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

- Klitzing, Richard. "triddit".

- Hi.gher.Space Wiki Contributors. "Duotrianglone".