# Triakis octahedron

Triakis octahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Tikko |

Coxeter diagram | m4m3o () |

Conway notation | kO |

Elements | |

Faces | 24 isosceles triangles |

Edges | 12+24 |

Vertices | 6+8 |

Vertex figure | 6 octagons, 8 triangles |

Measures (edge length 1) | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 24 |

Level of complexity | 3 |

Related polytopes | |

Army | Tikko |

Regiment | Tikko |

Dual | Truncated cube |

Conjugate | Great triakis octahedron |

Abstract & topological properties | |

Flag count | 144 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

The **triakis octahedron** is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-8 and 8 order-3 vertices. It is the dual of the uniform truncated cube.

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are times the length of those of the cube. Using an octahedron that is any number more than times the edge length of the cube gives a fully symmetric variant of this polyhedron. The upper limit is , where the cube's vertices coincide with the face centers of the octahedron.

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 octahedron with dual edge length 1 has vertex coordinates given by all permutations of:

- ,
- .

## External links[edit | edit source]

- Klitzing, Richard. "tikko".
- Wikipedia contributors. "Triakis octahedron".
- McCooey, David. "Triakis Octahedron"
- Quickfur. "The Triakis Octahedron".