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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

Subfacets

Radii

  • Vertex radius: $ l $
  • Edge radius: $ \frac{\sqrt{3}}{2}l $
  • Triangle radius: $ \frac{\sqrt{6}}{3}l $
  • Square radius: $ \frac{\sqrt{2}}{2}l $

Angles

  • Dihedral angle: $ 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

See Also

Regular
$ t_0 \{4,3\} $
Rectified
$ t_1 \{4,3\} $
Birectified
$ t_2 \{4,3\} $
Truncated
$ t_{0,1} \{4,3\} $
Bitruncated
$ t_{1,2} \{4,3\} $
Cantellated
$ t_{0,2} \{4,3\} $
Cantitruncated
$ t_{0,1,2} \{4,3\} $
Cube Cuboctahedron Octahedron Truncated cube Truncated octahedron Rhombicuboctahedron Great rhombicuboctahedron
Regular
$ t_0 \{3,3\} $
Rectified
$ t_1 \{3,3\} $
Birectified
$ t_2 \{3,3\} $
Truncated
$ t_{0,1} \{3,3\} $
Bitruncated
$ t_{1,2} \{3,3\} $
Cantellated
$ t_{0,2} \{3,3\} $
Cantitruncated
$ t_{0,1,2} \{3,3\} $
Tetrahedron Octahedron Tetrahedron Truncated tetrahedron Truncated tetrahedron Cuboctahedron Truncated octahedron
$ t_1 \{2,4\} $ $ t_1 \{3,4\} $ $ t_1 \{4,4\} $ $ t_1 \{5,4\} $ $ t_1 \{6,4\} $ ... $ t_1 \{\aleph_0,3\} $
Square dihedron Cuboctahedron Square tiling Tetrapentagonal tiling Tetrahexagonal tiling ... Tetrapeirogonal tiling
$ t_1 \{2,3\} $ $ t_1 \{3,3\} $ $ t_1 \{4,3\} $ $ t_1 \{5,3\} $ $ t_1 \{6,3\} $ $ t_1 \{7,3\} $ $ t_1 \{8,3\} $ ... $ t_1 \{\aleph_0,3\} $
Trigonal dihedron Octahedron Cuboctahedron Icosidodecahedron Trihexagonal tiling Triheptagonal tiling Trioctagonal tiling ... Triapeirogonal tiling
$ t_{0,2} \{2,3\} $ $ t_{0,2} \{3,3\} $ $ t_{0,2} \{4,3\} $ $ t_{0,2} \{5,3\} $ $ t_{0,2} \{6,3\} $ $ t_{0,2} \{7,3\} $ $ t_{0,2} \{8,3\} $ ... $ t_{0,2} \{\aleph_0,3\} $
Triangular prism Cuboctahedron Rhombicuboctahedron Rhombicosidodecahedron Rhombitrihexagonal tiling Rhombitriheptagonal tiling Rhombitrioctagonal tiling ... Rhombitriapeirogonal tiling