Small rhombicuboctahedron

The small rhombicuboctahedron, also commonly known as simply the rhombicuboctahedron, or sirco is one of the 13 Archimedean solids. It consists of 8 triangles and 6+12 squares, with one triangle and three squares meeting at each vertex. It can be obtained by cantellation of the cube or octahedron, or equivalently by pushing either polyhedron's faces outward and filling the gaps with the corresponding polygons.

6 of the squares in this figure have full BC2 symmetry, while 12 of them have only A1×A1 symmetry with respect to the whole polyhedron.

Vertex coordinates
A small rhombicuboctahedron of edge length 1 has vertex coordinates given by all permutations of:
 * (±(1+$\sqrt{5+2√2}$)/2, ±1/2, ±1/2).

Representations
A small rhombicuboctahedron has the following Coxeter diagrams:


 * x4o3x (full symmetry)
 * x4s3s (BC2+ symmetry, is a bialternatosnub octahedron/edge-alternated great rhombicuboctahedron)
 * xxxx4oxxo&#xt (BC2 axial, main square-first)
 * xxwoqo3oqowxx&#xt (A2 axial, triangle-first)
 * qo3xx3oq&#zx (A3 subsymmetry, hull of two opposite truncated tetrahedra)


 * wx xx4ox&#zx (BC2×A1 symmetry)
 * wxx xwx xxw&#zx (A1×A1×A1 symmetry)
 * xowqwox xwxwxwx&#xt (A1×A1 axial)

Related polyhedra
The small rhombicuboctahedron is the colonel of a three-member regiment that also includes the small cubicuboctahedron and the small rhombihexahedron.

It is possible to diminish the small rhombicuboctahedron by removing square cupolas. In fact, it is the result of attaching two square cupolas to an octagonal prism's bases, and can be called an elongated square orthobicupola. If one is removed the result is the elongated square cupola. If one cupola is rotated by 45º, the result is the elongated square gyrobicupola, or pseudo-rhombicuboctahedron. If the central prism is removed and the two cupolas are connected at their octagonal face, the result is a square orthobicupola.