# Tetrakis hexahedron

Tetrakis hexahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Tekah |

Coxeter diagram | o4m3m () |

Conway notation | kC |

Elements | |

Faces | 24 isosceles triangles |

Edges | 12+24 |

Vertices | 6+8 |

Vertex figure | 6 squares, 8 hexagons |

Measures (edge length 1) | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 24 |

Level of complexity | 3 |

Related polytopes | |

Army | Tekah |

Regiment | Tekah |

Dual | Truncated octahedron |

Conjugate | None |

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 **tetrakis hexahedron** is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-4 and 8 order-6 vertices. It is the dual of the uniform truncated octahedron.

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 less than times the edge length of the cube (including if the two have the same edge length) gives a fully symmetric variant of this polyhedron. The lower limit is times that of the cube, where the octahedron's vertices will coincide with the cube's face centers.

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

- ,
- .

## Variations[edit | edit source]

In addition to its fully-symmetric variants, the tetrakis hexahedron has variants that remain isotopic under tetrahedral symmetry. This variant can be called the disdyakis hexahedron and uses scalene triangles for faces.

## External links[edit | edit source]

- Klitzing, Richard. "tekah".
- Wikipedia contributors. "Tetrakis hexahedron".
- McCooey, David. "Tetrakis Hexahedron"
- Quickfur. "The Tetrakis Hexahedron".