Cuboctahedron

Cuboctahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Co
Coxeter diagram	o4x3o ()
Stewart notation	B4
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	B3, order 48
Convex	Yes
Nature	Tame

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[edit | edit source]

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

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

Representations[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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.

o4o3o truncations
Name	OBSA	Schläfli symbol	CD diagram
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
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