Quasirhombicuboctahedron

Quasirhombicuboctahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Querco
Coxeter diagram	x4/3o3x ()
Elements
Faces	8 triangles, 6+12 squares
Edges	24+24
Vertices	24
Vertex figure	Crossed 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 angles	4–4: 45°
	4–3: $\arccos\left(\frac{\sqrt6}{3}\right) \approx 35.26439^\circ$
Central density	5
Number of external pieces	488
Level of complexity	73
Related polytopes
Army	Tic, edge length $\sqrt2-1$
Regiment	Gocco
Dual	Great deltoidal icositetrahedron
Conjugate	Small rhombicuboctahedron
Convex core	Chamfered cube
Abstract & topological properties
Flag count	192
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

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 ()

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#46).

• Klitzing, Richard. "querco".

• Wikipedia Contributors. "Nonconvex great rhombicuboctahedron".
• McCooey, David. "Uniform Great Rhombicuboctahedron"