# Cuboctahedral prism

Cuboctahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Cope |

Coxeter diagram | x o4x3o () |

Elements | |

Cells | 8 triangular prisms, 6 cubes, 2 cuboctahedra |

Faces | 16 triangles, 12+24 squares |

Edges | 12+48 |

Vertices | 24 |

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

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Cube–4–trip: |

Co–4–cube: 90° | |

co–3–trip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 16 |

Level of complexity | 8 |

Related polytopes | |

Army | Cope |

Regiment | Cope |

Dual | Rhombic dodecahedral tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 768 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×A_{1}, order 96 |

Convex | Yes |

Nature | Tame |

The **cuboctahedral prism** or **cope** is a prismatic uniform polychoron that consists of 2 cuboctahedra, 6 cubes and 8 triangular prisms. Each vertex joins 1 cuboctahedron, 2 cubes, and 2 triangular prisms. As the name suggests, it is a prism based on the cuboctahedron. As such it is also a convex segmentochoron (designated K-4.43 on Richard Klitzing's list).

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, co atop co

## Vertex coordinates[edit | edit source]

The vertices of a cuboctahedral prism of edge length 1 are given by all permutations and sign changes of the first three coordinates of:

## Representations[edit | edit source]

A cuboctahedral prism has the following Coxeter diagrams:

- x o4x3o (full symmetry)
- x x3o3x () (rhombitetratetrahedral prism)
- s2s4x3o () (bases as rhombitetratetrahedra)
- oo4xx3oo&#x (bases considered separately)
- xx3oo3xx&#x (rhombitetratetrahedral bases considered separately)
- xxx xox4oqo&#xt (BC
_{2}×A_{1}axial, cube-first) - xxx xxo3oxx&#xt (A
_{2}×A_{1}axial, triangular prism-first) - xxx qqo qoq oqq&#zx (A
_{1}×A_{1}×A_{1}×A_{1}symmetry)

## Related polychora[edit | edit source]

A cuboctahedral prism can be cut in half to produce two triangular cupolic prisms with the base triangular prisms in rotated orientations.

The regiment of the cuboctahedral prism also includes the octahemioctahedral prism and the cubohemioctahedral prism.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#908).

- Klitzing, Richard. "Cope".

- Wikipedia Contributors. "Cuboctahedral prism".