Cuboctahedron

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.

Cuboctahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Co
Coxeter diagram	o4x3o ()
Conway notation	aC
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	${\frac {5{\sqrt {2}}}{3}}\approx 2.35702$
Dihedral angle	$\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
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
Flag orbits	2
Convex	Yes
Nature	Tame

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

• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)$.

Representations

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

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

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.

External links

• Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#21).

• Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#1 under co).

• Klitzing, Richard. "co".
• Quickfur. "The Cuboctahedron".

• Wikipedia contributors. "Cuboctahedron".
• McCooey, David. "Cuboctahedron"
• Hi.gher.Space Wiki Contributors. "Stauromesohedron".