View source for Cuboctahedron

{{Infobox polytope
|name = Cuboctahedron
|image = Cuboctahedron.png
|3d = Cuboctahedron.stl
|off = Cuboctahedron.off
|dim = 3
|type=[[Uniform]]
|obsa = {{BSAhelp|C|cube|o|octahedron}}
|coxeter = {{ACD|o4x3o|both=yes}}
|stewart = {{Stew|B4}}
|symmetry = [[Cubic symmetry|B<sub>3</sub>]], order 48
|conway = aC
|army = '''Co'''
|reg = '''Co'''
|faces = 8 [[triangle]]s, 6 [[square]]s
|edges = 24
|vertices = 12
|verf = [[Rectangle]], edge lengths 1 and {{radic|2}} <br> {{small|Cuboctahedron vertfig.png}}
|circum = 1
|volume = <math>\frac{5\sqrt2}{3} \approx 2.35702</math>
|dih = <math>\arccos\left(-\frac{\sqrt3}{3}\right) \approx 125.26439^\circ</math>
|flags = 96
|pieces = 14
|loc = 2
|orbits = 2
|dual = [[Rhombic dodecahedron]]
|conjugate = None
|conv = Yes
|orientable = Yes
|nat = Tame
}}
The '''cuboctahedron''', or '''co''', is a [[quasiregular polyhedron]] and one of the 13 [[Archimedean solids]]. It consists of 8 equilateral [[triangle]]s and 6 [[square]]s, with two of each joining at a vertex. It also has 4 [[hexagon]]al pseudofaces. It can be derived as a [[rectification|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 
* <math>\left(\pm\frac{\sqrt2}{2},\,\pm\frac{\sqrt2}{2},\,0\right)</math>.

==Representations==

A cuboctahedron has the following [[Coxeter diagram]]s:

*{{ACD|o4x3o|both=yes}} (full symmetry)
*{{ACD|x3o3x|both=yes}} ([[A3]] subsymmetry, [[small rhombitetratetrahedron]])
*{{ACD|s4x3o|both=yes}} (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 [[cantellation|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 [[hexagon]]al section to produce 2 [[triangular cupola]]s. 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]].

==External links==

*{{Bowers|quasireg|Polyhedron Category 3: Quasiregulars|21}}
*{{Bowers|octco|Batch 1: Oct and Co Facetings|1 under co}}
*{{Incmats|co}}
*{{Eusebeia|cuboctahedron|The Cuboctahedron}}
*{{Wikipedia|Cuboctahedron}}
*{{Dmccooey|Cuboctahedron|Cuboctahedron}}
*{{hgs_wiki|Stauromesohedron}}

{{o4o3o truncations}}
{{o3o3o truncations}}

[[Category:Convex quasiregular polyhedra]]
[[Category:Co regiment]]
[[Category:B3 symmetry]]
[[Category:Polyhedra with 14 faces]]
[[Category:4-3-4-3 acrohedra]]
[[Category:Facetings of the cuboctahedron]]