Cuboctahedron

A **cuboctahedron** is a uniform three-dimensional polyhedron that can be constructed by rectifying a cube. It can also be created by rectifying an octahedron, or by cantellating a tetrahedron (when considered with this symmetry, it can be called a **rhombitetratetrahedron**).

As a rectified octahedron, it has the same symmetry group as the octahedron, namely octahedral symmetry (Oh).

The dual of the cuboctahedron is called a rhombic dodecahedron.

The Bowers acronym for the cuboctahedron is **co**.

## Symbols

Dynkin based symbols of the cuboctahedron include:

* o4x3o (full symmetry)
* x3o3x (rhombitetratetrahedron)
* xox4oqo&#xt (square symmetry)
* xxo3oxx&#xt (triangular gyrobicupola)
* oxuxo oqoqo&#xt (vertex first)
* qo xo4oq&#zx (square prism symmetry)
* qqo qoq oqq&#zx (block symmetry)

## Structure and Sections

On each vertex of the cuboctahedron, two triangles and two squares join.

### Hypervolumes

* vertex count =

12

{\displaystyle 12}
* edge length =

24
l

{\displaystyle 24 l}
* surface area =

(

6
2

3

)

l

2

{\displaystyle \left(6+2{\sqrt {3}}\right)l^{2}}
* surcell volume =

5
3

2

l

3

{\displaystyle {\frac {5}{3}}{\sqrt {2}}l^{3}}

### Subfacets

* 12 points (0D)
* 24 line segments (1D)
* 8 triangles (2D)
* 6 squares (2D)
* 1 cuboctahedron (3D)

### Radii

* Vertex radius:

l

{\displaystyle l}
* Edge radius:

3

2

l

{\displaystyle \frac{\sqrt{3}}{2}l}
* Triangle radius:

6

3

l

{\displaystyle {\frac {\sqrt {6}}{3}}l}
* Square radius:

2

2

l

{\displaystyle \frac{\sqrt{2}}{2}l}

### Angles

* Dihedral angle:

a
r
c
c
o
s
(

3

3

)

{\displaystyle arccos({\frac {\sqrt {3}}{3}})}

### Vertex coordinates

The vertex coordinates of a cuboctahedron with length 2 are all permuations of (±√2,±√2,0).

### Related shapes

* Dual: Rhombic dodecahedron
* Vertex figure: Rectangle, edges 1 and

2

{\displaystyle \sqrt{2}}
* Caps: Triangular cupola (triangle first)
* Gyrations: Triangular orthobicupola
* Regiment members: 3 (other members: Octahemioctahedron, Cubohemioctahedron)

## See Also

* Polytope Wiki. "Cuboctahedron"
* Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars"

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