Cuboctahedron

Cuboctahedron
Dimensionality	3
Schläfli Symbol	$t_{1}\{4,3\}$
Symmetry	Full octahedral symmetry (Oh)
Coxeter–Dynkin diagram
Bowers acronym	co
Subfacets
Vertices	12 points
Edges	24 line segments
Faces	8 triangles 6 squares
Cells	1 cuboctahedron
Hypervolumes
Vertex Count	$12$
Edge Length	$24l$
Surface Area	$\left(6+2{\sqrt {3}}\right)l^{2}$
Surcell Volume	${\frac {5}{3}}{\sqrt {2}}l^{3}$

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 (O_h).

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 = $12$
• edge length = $24l$
• surface area = $\left(6+2{\sqrt {3}}\right)l^{2}$
• surcell volume = ${\frac {5}{3}}{\sqrt {2}}l^{3}$

Subfacets

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

Radii

• Vertex radius: $l$
• 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

• Dual: Rhombic dodecahedron
• Vertex figure: Rectangle, edges 1 and ${\sqrt {2}}$
• 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"

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

$t_{1}\{2,3\}$	$t_{1}\{3,3\}$	$t_{1}\{4,3\}$	$t_{1}\{5,3\}$	$t_{1}\{6,3\}$	$t_{1}\{7,3\}$	$t_{1}\{8,3\}$	...	$t_{1}\{\infty ,3\}$	$t_{1}\{{\frac {i\pi }{\lambda }},3\}$
Trigonal dihedron	Octahedron	Cuboctahedron	Icosidodecahedron	Trihexagonal tiling	Triheptagonal tiling	Trioctagonal tiling	...	Triapeirogonal tiling	Tripseudogonal tiling

$t_{0,2}\{2,3\}$	$t_{0,2}\{3,3\}$	$t_{0,2}\{4,3\}$	$t_{0,2}\{5,3\}$	$t_{0,2}\{6,3\}$	$t_{0,2}\{7,3\}$	$t_{0,2}\{8,3\}$	...	$t_{0,2}\{\infty ,3\}$	$t_{0,2}\{{\frac {\pi i}{\lambda }},3\}$
Triangular prism	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Rhombitrihexagonal tiling	Rhombitriheptagonal tiling	Rhombitrioctagonal tiling	...	Rhombitriapeirogonal tiling	Rhombitripseudogonal tiling