Small rhombicuboctahedron

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Small rhombicuboctahedron
Bowers style acronymSirco
Coxeter diagramx4o3x ()
Conway notationeC
Stewart notationE4
Vertex figureIsosceles trapezoid, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Dihedral angles4–3:
 4–4: 135°
Central density1
Number of external pieces26
Level of complexity4
Related polytopes
DualDeltoidal icositetrahedron
Abstract & topological properties
Flag count192
Euler characteristic2
SymmetryB3, order 48
Flag orbits4

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 K2 symmetry with respect to the whole polyhedron.

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:

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 side 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:

The variant where the squares have edge lengths times those of the triangles is the direct rectification of the cuboctahedron.

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]

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 B3/2 symmetry.

External links[edit | edit source]