|Bowers style acronym||Sirco|
|Coxeter diagram||x4o3x ()|
|Faces||8 triangles, 6+12 squares|
|Vertex figure||Isosceles trapezoid, edge lengths 1, √, √, √ |
|Measures (edge length 1)|
|Number of pieces||26|
|Level of complexity||4|
|Symmetry||B3, order 48|
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.
Vertex coordinates[edit | edit source]
A small rhombicuboctahedron of edge length 1 has vertex coordinates given by all permutations of:
Representations[edit | edit source]
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[edit | edit source]
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 and its volume is given by .
It has coordinates given by all permutations of:
Variations[edit | edit source]
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[edit | edit source]
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.
|Name||OBSA||Schläfli symbol||CD diagram||Picture|
[edit | edit source]
- Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#36).
- Klitzing, Richard. "sirco".
- Quickfur. "The Rhombicuboctahedron".
- Hi.gher.Space Wiki Contributors. "Stauroperihedron".