# Triangular duoantiprism

Triangular duoantiprism | |
---|---|

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Notation | |

Bowers style acronym | Triddap |

Coxeter diagram | s6o2s6o () |

Elements | |

Cells | 18 tetragonal disphenoids, 12 triangular antiprisms |

Faces | 72 isosceles triangles, 12 triangles |

Edges | 36+36 |

Vertices | 18 |

Vertex figure | Gyrobifastigium |

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

Edge lengths | Lacing edges (36): |

Edges of triangles (36): 1 | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Triddap |

Regiment | Triddap |

Dual | Triangular duoantitegum |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | (G_{2}≀S_{2})/2, order 144 |

Convex | Yes |

Nature | Tame |

The **triangular duoantiprism** or **triddap**, also known as the **triangular-triangular duoantiprism**, the **3 duoantiprism** or the **3-3 duoantiprism**, is a convex isogonal polychoron that consists of 12 triangular antiprisms and 18 tetragonal disphenoids. 4 triangular antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the hexagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal triangular dihedral swirlchora.

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

## Vertex coordinates[edit | edit source]

The vertices of a triangular duoantiprism based on triangles of edge length 1, centered at the origin, are given by:

These coordinates show that a triangular duoantiprism can be obtained as the convex hull of two inversely oriented triangular duoprisms.

## Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Triangular antiprism (12): Hexagonal duotegum
- Tetragonal disphenoid (18): Triangular duoantiprism
- Triangle (12): Hexagonal duotegum
- Isosceles triangle (72): Hexagonal ditetragoltriate
- Edge (36): Hexagonal duoprism
- Edge (36): Triangular double gyroantiprismoid

## Variations[edit | edit source]

The triangular duoantiprism has isogonal variants where the two base triangles are of different edge lengths., and the tetragonal disphenoids become digonal disphenoids while the antiprisms split into 2 sets of 6.

If the ratio of the sides of the two triangular bases is , the resulting polychoron has 6 regular octahedra, 6 octahedral variants formed from removing 2 vertices from a cube, and 18 disphenoids formed from removing two adjacent vertices from an octahedron. In fact this polychoron is nothing but a diminishing of the icositetrachoron created by removing 6 vertices that form an equatorial hexagon.

## External links[edit | edit source]

- Klitzing, Richard. "triddap".