# Great rhombicuboctahedron

Great rhombicuboctahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Girco |

Coxeter diagram | x4x3x () |

Conway notation | bC |

Stewart notation | K_{4} |

Elements | |

Faces | 12 squares, 8 hexagons, 6 octagons |

Edges | 24+24+24 |

Vertices | 48 |

Vertex figure | Scalene triangle, edge lengths √2, √3, √2+√2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 6–4: |

8–4: 135° | |

8–6: | |

Central density | 1 |

Number of external pieces | 26 |

Level of complexity | 6 |

Related polytopes | |

Army | Girco |

Regiment | Girco |

Dual | Disdyakis dodecahedron |

Conjugate | Quasitruncated cuboctahedron |

Abstract & topological properties | |

Flag count | 288 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Flag orbits | 6 |

Convex | Yes |

Nature | Tame |

The **great rhombicuboctahedron** or **girco**, also commonly known as the **truncated cuboctahedron**, is one of the 13 Archimedean solids. It consists of 12 squares, 8 hexagons, and 6 octagons, with one of each type of face meeting per vertex. It can be obtained by cantitruncation of the cube or octahedron, or equivalently by truncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

This is one of three Wythoffian non-prismatic polyhedra with a Coxeter diagram having all ringed nodes, the other two being the great rhombitetratetrahedron and the great rhombicosidodecahedron.

It can be alternated into the snub cube after equalizing edge lengths.

## Naming[edit | edit source]

**Rhombi** refers to the twelve square faces on the axis of the rhombic dodecahedron, **cub**(e) refers to the six faces on the axis of the cube, and **octahedron** for the eight hexagons on the axis of the octahedron.

Alternate names include:

**Truncated cuboctahedron**(a translation of Kepler's Latin name), because it can be derived by truncating the cuboctahedron. However, it is not a true truncation of the rhombicuboctahedron, as a true truncation would result in rectangles rather than squares. Kepler used a different word for this sense of "truncated", but it was lost in translation.**Rhombitruncated cuboctahedron**(similar to the above, but also refers to the planes of the truncated faces).**Great rhombcuboctahedron**(no "i") - alternate spelling.

## Vertex coordinates[edit | edit source]

A great rhombicuboctahedron of edge length 1 has vertex coordinates given by all permutations of:

## Representations[edit | edit source]

A great rhombicuboctahedron has the following Coxeter diagrams:

- x4x3x (full symmetry)
- xxwwxx4xuxxux&#xt (B
_{2}axial, octagon-first) - wx3xx3xw&#zx (A
_{3}symmetry, as hull of two inverse great rhombitetratetrahedra) - Xwx xxw4xux&#zx (B
_{2}×A_{1}symmetry) - xxuUxwwx3xwwxUuxx&#xt (A
_{2}axial, hexagon-first)

## Semi-uniform variant[edit | edit source]

The great rhombicuboctahedron has a semi-uniform variant of the form x4y3z that maintains its full symmetry. This variant has 6 ditetragons, 8 ditrigons, and 12 rectangles as faces.

With edges of length a (ditetragon-rectangle), b (ditetragon-ditrigon), and c (ditrigon-rectangle), its circumradius is given by and its volume is given by .

It has coordinates given by all permutations of:

- .

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#57).

- Klitzing, Richard. "Girco".
- Quickfur. "The Great Rhombicuboctahedron".

- Wikipedia contributors. "Truncated cuboctahedron".
- McCooey, David. "Truncated Cuboctahedron"
- Hi.gher.Space Wiki Contributors. "Stauropantohedron".