# Cuboctahedron

Cuboctahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymCo
Coxeter diagramo4x3o ()
Stewart notationB4
Elements
Faces8 triangles, 6 squares
Edges24
Vertices12
Vertex figureRectangle, edge lengths 1 and 2
Measures (edge length 1)
Volume${\displaystyle \frac{5\sqrt2}{3} ≈ 2.35702}$
Dihedral angle${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ}$
Central density1
Number of pieces14
Level of complexity2
Related polytopes
ArmyCo
RegimentCo
DualRhombic dodecahedron
ConjugateNone
Abstract properties
Flag count96
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

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.

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 coordinates

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

• ${\displaystyle \left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0\right).}$

## Representations

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)

## Variations

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 polyhedra

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.

o4o3o truncations
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
o3o3o truncations
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