# Triakis tetrahedron

Triakis tetrahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Tikit |

Coxeter diagram | m3m3o () |

Conway notation | kT |

Elements | |

Faces | 12 isosceles triangles |

Edges | 6+12 |

Vertices | 4+4 |

Vertex figure | 4 hexagons, 4 triangles |

Measures (edge length 1) | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 3 |

Related polytopes | |

Army | Tikit |

Regiment | Tikit |

Dual | Truncated tetrahedron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 72 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{3}, order 24 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

The **triakis tetrahedron** is one of the 13 Catalan solids. It has 12 isosceles triangles as faces, with 4 order-6 and 4 order-3 vertices. It is the dual of the uniform truncated tetrahedron.

It can also be obtained as the convex hull of two dually-oriented tetrahedra, where one has edges exactly times the length of those of the other. If the ratio of the edge lengths of the two tetrahedra is varied to be anything between 1:1 (producing the cube) and 1:3 (in which case the vertices of the small tetrahedron are the face centers of the larger), a fully symmetric variant of the triakis tetrahedron is produced.

Each face of this polyhedron is an isosceles triangle with base side length times those of the side edges. These triangles have apex angle and base angles .

## Vertex coordinates[edit | edit source]

A triakis tetrahedron with dual edge length 1 has vertex coordinates given by all even sign changes of:

- ,

as well as all odd sign changes of:

- .

## External links[edit | edit source]

- Klitzing, Richard. "tikit".
- Wikipedia contributors. "Triakis tetrahedron".
- McCooey, David. "Triakis Tetrahedron"

- Quickfur. "The Triakis Tetrahedron".