# Great rhombicuboctahedral prism

Great rhombicuboctahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Gircope |

Coxeter diagram | x x4x3x () |

Elements | |

Cells | 12 cubes, 8 hexagonal prisms, 6 octagonal prisms, 2 great rhombicuboctahedra |

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

Edges | 48+48+48+48 |

Vertices | 96 |

Vertex figure | Irregular tetrahedron, edge lengths √2, √3, √2+√2 (base), √2 (legs) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Cube–4–hip: |

Cube–4–op: 135° | |

Hip–4–op: | |

Girco–8–op: 90° | |

Girco–6–hip: 90° | |

Girco–4–cube: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 28 |

Level of complexity | 24 |

Related polytopes | |

Army | Gircope |

Regiment | Gircope |

Dual | Disdyakis dodecahedral tegum |

Conjugate | Quasitruncated cuboctahedral prism |

Abstract & topological properties | |

Flag count | 2304 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×A_{1}, order 96 |

Convex | Yes |

Nature | Tame |

The **great rhombicuboctahedral prism**, or **gircope**, is a prismatic uniform polychoron that consists of 2 great rhombicuboctahedra, 6 octagonal prisms, 8 hexagonal prisms, and 12 cubes. Each vertex joins one of each type of cell. As the name suggests, it is a prism based on the great rhombicuboctahedron. As such it is also a convex segmentochoron (designated K-4.125 on Richard Klitzing's list).

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in rhombicuboctahedron-first orientation.

This polychoron can be alternated into a snub cubic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritosnub alterprism, which is also nonuniform.

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, girco atop girco

## Vertex coordinates[edit | edit source]

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

## Representations[edit | edit source]

The great rhombicuboctahedral prism has the following Coxeter diagrams:

- x x4x3x (full symmetry)
- xx4xx3xx&#x (bases considered separately)
- xxxxxx xuxxux4xxwwxx&#xt (BC
_{2}×A_{1}symmetry, octagonal prism-first)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#944).

- Klitzing, Richard. "Gircope".

- Wikipedia Contributors. "Truncated cuboctahedral prism".