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 also has 6 octagonal pseudofaces. 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. Rectifying the cuboctahedron gives a semi-uniform variant of the rhombicuboctahedron.

6 of the squares in this figure have full B2 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:
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).$$

Representations
A small rhombicuboctahedron has the following Coxeter diagrams:


 * x4o3x (full symmetry)
 * x4s3s (BC3+ symmetry, is an edge-snub 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)

Semi-uniform variant
The small rhombicuboctahedron has a semi-uniform variant of the form x4o3y that maintains its full symmetry. This variant has 6 squares of side length x, 8 triangles of size length y, and 12 rectangles as faces.

With edges of length a (of squares) and b (of triangles), its circumradius is given by $$\frac{\sqrt{3a^2+2b^2+2ab\sqrt2}}{2}$$ and its volume is given by $$a^3+3ab^2+(9a^2b+b^3)\frac{\sqrt2}{3}$$.

It has coordinates given by all permutations of:


 * $$\left(±\frac{a+b\sqrt2}{2},\,±\frac{a}{2},\,±\frac{a}{2}\right).$$

Variations
Besides the semi-uniform variation, another variation, the pyritosnub cube, can be obtained as an alternated faceting of the great rhombicuboctahedron with pyritohedral symmetry. This faceting has 6 rectangles, 8 triangles, and 12 trapezoids as faces.

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.

The rhombisnub rhombicosicosahedron is a uniform polyhedron compound composed of 5 small rhombicuboctahedra.

The Überoctoplex is abstractly identical and also isogonal, but is not convex and only has pyritohedral symmetry.