# Truncated octahedral prism

Truncated octahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Tope |

Coxeter diagram | x o4x3x () |

Elements | |

Cells | 6 cubes, 8 hexagonal prisms, 2 truncated octahedra |

Faces | 12+12+24 squares, 16 hexagons |

Edges | 24+24+48 |

Vertices | 48 |

Vertex figure | Sphenoid, edge lengths √2, √3, √3 (base), √2 (legs) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Cube–4–hip: |

Hip–4–hip: | |

Toe–4–cube: 90° | |

Toe–6–hip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 16 |

Level of complexity | 12 |

Related polytopes | |

Army | Tope |

Regiment | Tope |

Dual | Tetrakis hexahedral tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 1152 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×A_{1}, order 96 |

Convex | Yes |

Nature | Tame |

The **truncated octahedral prism** or **tope** is a prismatic uniform polychoron that consists of 2 truncated octahedra, 6 cubes, and 8 hexagonal prisms. Each vertex joins 1 truncated octahedron, 1 cube, and 2 hexagonal prisms. As the name suggests, it is a prism based on the truncated octahedron. As such it is also a convex segmentochoron (designated K-4.89 on Richard Klitzing's list).

This polychoron can be alternated into a pyritohedral icosahedral antiprism, although it cannot be made uniform.

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, toe atop toe

## Vertex coordinates[edit | edit source]

The vertices of a truncated octahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

## Representations[edit | edit source]

A truncated octahedral prism has the following Coxeter diagrams:

- x o4x3x (full symmetry)
- x x3x3x () (bases have A
_{3}symmetry) - s2s4x3x () (bases have A
_{3}symmetry, as snub) - oo4xx3xx&#x (bases considered separately)
- xx3xx3xx&#x (bases separately under A
_{3}) - xxxxx xuxux4ooqoo&#xt (BC
_{2}×A_{1}axial, cube-first) - xxxx xuxx3xxux&#xt (A
_{2}×A_{1}axial, hexagonal prism-first)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#900).

- Klitzing, Richard. "Tope".

- Wikipedia Contributors. "Truncated octahedral prism".