# Small rhombicuboctahedron

Small rhombicuboctahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSirco
Coxeter diagramx4o3x ()
Elements
Faces8 triangles, 6+12 squares
Edges24+24
Vertices24
Vertex figureIsosceles trapezoid, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897}$
Volume${\displaystyle 2\frac{6+5\sqrt2}{3} ≈ 8.71404}$
Dihedral angles4–3: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ}$
4–4: 135°
Central density1
Number of pieces26
Level of complexity4
Related polytopes
ArmySirco
RegimentSirco
DualDeltoidal icositetrahedron
ConjugateQuasirhombicuboctahedron
Abstract properties
Flag count192
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

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 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:

• ${\displaystyle \left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).}$

## Representations

A small rhombicuboctahedron has the following Coxeter diagrams:

## 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 ${\displaystyle \frac{\sqrt{3a^2+2b^2+2ab\sqrt2}}{2}}$ and its volume is given by ${\displaystyle a^3+3ab^2+(9a^2b+b^3)\frac{\sqrt2}{3}}$.

It has coordinates given by all permutations of:

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

## 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.

o4o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4,3} x4o3o
Truncated cube tic t{4,3} x4x3o
Cuboctahedron co r{4,3} o4x3o
Truncated octahedron toe t{3,4} o4x3x
Octahedron oct {3,4} o4o3x
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Great rhombicuboctahedron girco tr{4,3} x4x3x
Snub cube snic sr{4,3} s4s3s