# Cuboctahedron

The **cuboctahedron**, or **co**, is a quasiregular polyhedron and one of the 13 Archimedean solids. It consists of 8 equilateral triangles and 6 squares, with two of each joining at a vertex. It also has 4 hexagonal pseudofaces. It can be derived as a rectified cube or octahedron, or by expanding the faces of the tetrahedron outward.

Cuboctahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Co |

Coxeter diagram | o4x3o () |

Stewart notation | B_{4} |

Elements | |

Faces | 8 triangles, 6 squares |

Edges | 24 |

Vertices | 12 |

Vertex figure | Rectangle, edge lengths 1 and √2 |

Measures (edge length 1) | |

Circumradius | 1 |

Volume | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 2 |

Related polytopes | |

Army | Co |

Regiment | Co |

Dual | Rhombic dodecahedron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 96 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Convex | Yes |

Nature | Tame |

The cuboctahedron has the property that its circumradius equals its edge length. This relates to the fact that the cuboctahedron is the vertex figure of the Euclidean tetrahedral-octahedral honeycomb. Other notable polytopes that satisfy this property are the hexagon, the tesseract, and the icositetrachoron.

## Vertex coordinatesEdit

A cuboctahedron of side length 1 has vertex coordinates given by all permutations of

## RepresentationsEdit

A cuboctahedron has the following Coxeter diagrams:

- o4x3o (full symmetry)
- x3o3x (A3 subsymmetry, small rhombitetratetrahedron)
- s4x3o (A3 symmetry, triangle-alternated truncated cube)
- xxo3oxx&#xt (A2 axial, triangular gyrobicupola)
- xox4oqo&#xt (BC2 axial, square-first)
- oxuxo oqoqo&#xt (A1×A1 axial, vertex-first)
- qo xo4oq&#zx (BC2×A1 symmetry, rectified square prism)
- x(uo)x x(ou)x&#xt (square-first under rectangle subsymmetry)
- qqo qoq oqq&#zx (A1×A1×A1 symmetry, rectified cuboid)

## VariationsEdit

A cuboctahedron can also be constructed in A3 symmetry, as the cantellated tetrahedron. This figure is named the small rhombitetratetrahedron, also commonly known as simply the rhombitetratetrahedron. In this form, the 8 triangles split into 2 sets of 4, and the squares alternately join to the two kinds of triangles. It can be represented as x3o3x.

## Related polyhedraEdit

The cuboctahedron is the colonel of a three-member regiment that also includes the octahemioctahedron and the cubohemioctahedron.

A cuboctahedron can be cut in half along an equatorial hexagonal section to produce 2 triangular cupolas. Since the two cupolas are in opposite orientations, this means the cuboctahedron can be called the **triangular gyrobicupola**. If one cupola is rotated 60° and then rejoined, so that triangles join to triangles and squares join to squares, the result is the triangular orthobicupola. If a hexagonal prism is inserted between the halves of a cuboctahedron, the result is an elongated triangular gyrobicupola.

The antirhombicosicosahedron is a uniform polyhedron compound composed of 5 cuboctahedra.

The square faces of the cuboctahedron can be subdivided into triangles to form a polyhedron which is abstractly equivalent to the icosahedron.

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Cube | cube | {4,3} | x4o3o | |

Truncated cube | tic | t{4,3} | x4x3o | |

Cuboctahedron | co | r{4,3} | o4x3o | |

Truncated octahedron | toe | t{3,4} | o4x3x | |

Octahedron | oct | {3,4} | o4o3x | |

Small rhombicuboctahedron | sirco | rr{4,3} | x4o3x | |

Great rhombicuboctahedron | girco | tr{4,3} | x4x3x | |

Snub cube | snic | sr{4,3} | s4s3s |

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Tetrahedron | tet | {3,3} | x3o3o | |

Truncated tetrahedron | tut | t{3,3} | x3x3o | |

Tetratetrahedron = Octahedron | oct | r{3,3} | o3x3o | |

Truncated tetrahedron | tut | t{3,3} | o3x3x | |

Tetrahedron | tet | {3,3} | o3o3x | |

Small rhombitetratetrahedron = Cuboctahedron | co | rr{3,3} | x3o3x | |

Great rhombitetratetrahedron = Truncated octahedron | toe | tr{3,3} | x3x3x | |

Snub tetrahedron = Icosahedron | ike | sr{3,3} | s3s3s |

## External linksEdit

- Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#21).

- Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#1 under co).

- Klitzing, Richard. "co".

- Quickfur. "The Cuboctahedron".

- Wikipedia Contributors. "Cuboctahedron".
- McCooey, David. "Cuboctahedron"

- Hi.gher.Space Wiki Contributors. "Stauromesohedron".