Cuboctahedron  

(Click here for rotating model) 

Type  Archimedean solid 
Elements  F = 14, E = 24, V = 12 (χ = 2) 
Faces by sides  8{3}+6{4} 
Schläfli symbol  t_{1}{4,3} t_{0,2}{3,3} 
Wythoff symbol  2  3 4 3 3  2 
CoxeterDynkin  
Symmetry  O_{h} (*432) T_{d} (*332) 
References  U_{07}, C_{19}, W_{11} 
Properties  Semiregular convex quasiregular 
Colored faces 
3.4.3.4 (Vertex figure) 
Rhombic dodecahedron (dual polyhedron) 
Net 
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertextransitive and edgetransitive.
Its dual polyhedron is the rhombic dodecahedron.
Contents 
The area A and the volume V of the cuboctahedron of edge length a are:
The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin are:
A cuboctahedron can be obtained by taking an appropriate cross section of a fourdimensional 16cell.
A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.
The cuboctahedron is a rectified cube and also a rectified octahedron.
It is also a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3  2.
A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertexuniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group.
The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola.
Both triangular bicupolae are important in sphere packing. The distance from the solid's centre to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a facecentered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal closepacked lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's centre.
Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform polychora.
The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.
Cube 
Truncated cube 
cuboctahedron 
Truncated octahedron 
Octahedron 
The cuboctahedron shares its edges and vertex arrangement with two nonconvex uniform polyhedra: the cubohemioctahedron (having the square faces in common) and the octahemioctahedron (having the triangular faces in common).
Cuboctahedron 
Cubohemioctahedron 
Octahemioctahedron 
The cuboctahedron can be decomposed into a regular octahedron and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cellfirst parallel projection of the 24cell into 3 dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into 6 square faces, a regular octahedron, and 8 irregular octahedra. These elements correspond with the images of 6 of the octahedral cells in the 24cell, the nearest and farthest cells from the 4D viewpoint, and the remaining 8 pairs of cells, respectively.

