# Quasirhombicuboctahedron

(Redirected from Querco)
Quasirhombicuboctahedron
Rank3
TypeUniform
Notation
Bowers style acronymQuerco
Coxeter diagramx4/3o3x ()
Elements
Faces8 triangles, 6+12 squares
Edges24+24
Vertices24
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5-2{\sqrt {2}}}}{2}}\approx 0.73681}$
Volume${\displaystyle 2{\frac {5{\sqrt {2}}-6}{3}}\approx 0.71404}$
Dihedral angles4–4: 45°
4–3: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{3}}\right)\approx 35.26439^{\circ }}$
Central density5
Number of external pieces488
Level of complexity73
Related polytopes
ArmyTic, edge length ${\displaystyle {\sqrt {2}}-1}$
RegimentGocco
DualGreat deltoidal icositetrahedron
ConjugateSmall rhombicuboctahedron
Convex coreChamfered cube
Abstract & topological properties
Flag count192
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 48
Flag orbits4
ConvexNo
NatureTame

The quasirhombicuboctahedron, also commonly known as the nonconvex great rhombicuboctahedron, or querco is a uniform polyhedron. It consists of 8 triangles and 6+12 squares, with one triangle and three squares meeting at each vertex. It also has 6 octagrammic pseudofaces. It can be obtained by quasicantellation of the cube or octahedron, or equivalently by pushing either polyhedron's faces inward and filling the gaps with squares.

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.

It is also sometimes called a great rhombicuboctahedron, but is not to be confused with the convex polyhedron with the same name.

It is a faceting of the great cubicuboctahedron, using the original's squares and triangles, while also introducing 12 additional squares.

## Related polyhedra

The rhombisnub quasirhombicosicosahedron is a uniform polyhedron compound composed of 5 quasirhombicuboctahedra.

The quasirhombicuboctahedron can be constructed as an octagrammic prism augmented with retrograde square cupolas facing inwards on the octagrammic faces.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the great cubicuboctahedron.