# Octahedral prism

Octahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Ope |

Coxeter diagram | x o4o3x () |

Bracket notation | [<III>I] |

Elements | |

Cells | 8 triangular prisms, 2 octahedra |

Faces | 16 triangles, 12 squares |

Edges | 6+24 |

Vertices | 12 |

Vertex figure | Square pyramid, edge lengths 1 (base), √2 (legs) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Trip–4–trip: |

Oct–3–trip: 90° | |

Heights | Oct atop oct: 1 |

Trip atop gyro trip: | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 4 |

Related polytopes | |

Army | Ope |

Regiment | Ope |

Dual | Cubic tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 384 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×A_{1}, order 96 |

Convex | Yes |

Nature | Tame |

The **octahedral prism** or **ope** is a prismatic uniform polychoron that consists of 2 octahedra and 8 triangular prisms. Each vertex joins 1 octahedron and 4 triangular prisms. It is a prism based on the octahedron. As such it is also a convex segmentochoron (designated K-4.11 on Richard Klitzing's list).

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, oct atop oct

Segmentochoron display, trip atop gyro trip

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of an octahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

## Representations[edit | edit source]

An octahedral prism has the following Coxeter diagrams:

- x o4o3x (full symmetry)
- x o3x3o () (base has A
_{3}symmetry, tetratetrahedral prism) - x2s2s3s () (triangular antiprismatic prism)
- x2s2s6o () (base has G
_{2}×A_{1}+ symmetry) - oo4oo3xx&#x (bases considered separately)
- oo3xx3oo&#x (bases considered in tet symmetry)
- xx xo3ox&#x (A
_{2}×A_{1}axial. trip atop gyro trip) - xxx oxo4ooo&#xt (BC
_{2}×A_{1}symmetry, as square bipyramidal prism) - xxx oxo oxo&#xt (A
_{1}×A_{1}×A_{1}symmetry, as rectangular bipyramidal prism) - xxx xox oqo&#xt (A
_{1}×A_{1}×A_{1}axial, bases are edge-first) - xxx qoo oqo ooq&#zx (A
_{1}×A_{1}×A_{1}×A_{1}symmetry, as rhombic bipyramidal prism) - xx qo ox4oo&#zx (BC2×A1×A1 symmetry)

## Related polychora[edit | edit source]

An octahedral prism can be cut into 2 square pyramidal prisms joining at a common cubic cell. If one half is rotated the result is instead a dyadic gyrotegmipucofastegium, which is also a segmentochoron.

The regiment of the octahedral prism also includes the tetrahemihexahedral prism.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#890).

- Klitzing, Richard. "Ope".

- Wikipedia Contributors. "Octahedral prism".
- Hi.gher.Space Wiki Contributors. "Octahedral prism".