# Quasirhombicuboctahedron

Quasirhombicuboctahedron Rank3
TypeUniform
SpaceSpherical
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$\frac{\sqrt{5-2\sqrt2}}{2} \approx 0.73681$ Volume$2\frac{5\sqrt2-6}{3} \approx 0.71404$ Dihedral angles4–4: 45°
4–3: $\arccos\left(\frac{\sqrt6}{3}\right) \approx 35.26439^\circ$ Central density5
Number of pieces488
Level of complexity73
Related polytopes
ArmyTic
RegimentGocco
DualGreat deltoidal icositetrahedron
ConjugateSmall rhombicuboctahedron
Convex coreChamfered cube
Abstract properties
Flag count192
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryB3, order 48
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 BC2 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.

o4/3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4/3,3} x4/3o3o (       )
Quasitruncated hexahedron quith t{4/3,3} x4/3x3o (       )
Cuboctahedron co r{3,4/3} o4/3x3o (       )
Truncated octahedron toe t{3,4/3} o4/3x3x (       )
Octahedron oct {3,4/3} o4/3o3x (       )
Quasirhombicuboctahedron querco rr{3,4/3} x4/3o3x (       )
Quasitruncated cuboctahedron quitco tr{3,4/3} x4/3x3x (       )
(degenerate, oct+6(4)) o4/3o3ß (       )
Icosahedron ike s{3,4/3} o4/3s3s (       )