Quasirhombicosidodecahedron

The quasirhombicosidodecahedron, also commonly known as simply the nonconvex great rhombicosidodecahedron, or qrid is a uniform polyhedron. It consists of 20 triangles, 30 squares, and 12 pentagrams, with one triangle, two squares, and one pentagram meeting at each vertex. It can be obtained by quasicantellation of the great stellated dodecahedron or great icosahedron, or equivalently by pushing either polyhedron's faces inward and filling the gaps with appropriate faces.

Quasirhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Qrid
Coxeter diagram	x5/3o3x ()
Elements
Faces	20 triangles, 30 squares, 12 pentagrams
Edges	60+60
Vertices	60
Vertex figure	Crossed isosceles trapezoid, edge lengths 1, √2, (√5–1)/2, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{11-4\sqrt5}}{2} \approx 0.71689$
Volume	$\frac{29\sqrt5-60}{3} \approx 1.61532$
Dihedral angles	4–3: $\arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) \approx 69.09484^\circ$
	5/2–4: $\arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) \approx 58.28253^\circ$
Central density	13
Number of external pieces	980
Level of complexity	59
Related polytopes
Army	Semi-uniform Ti, edge lengths $\sqrt5-2$ (pentagons), $\frac{3-\sqrt5}{2}$ (between ditrigons)
Regiment	Gaddid
Dual	Great deltoidal hexecontahedron
Conjugate	Small rhombicosidodecahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	480
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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

It is a faceting of the great dodecicosidodecahedron, using the original's 12 pentagrams and 20 triangles along with 30 additional squares.

Vertex coordinates

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

Related polyhedra

o5/3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Great stellated dodecahedron	gissid	{5/3,3}	x5/3o3o ()
Quasitruncated great stellated dodecahedron	quit gissid	t{5/3,3}	x5/3x3o ()
Great icosidodecahedron	gid	r{3,5/3}	o5/3x3o ()
Truncated great icosahedron	tiggy	t{3,5/3}	o5/3x3x ()
Great icosahedron	gike	{3,5/3}	o5/3o3x ()
Quasirhombicosidodecahedron	qrid	rr{3,5/3}	x5/3o3x ()
Great quasitruncated icosidodecahedron	gaquatid	tr{3,5/3}	x5/3x3x ()
Great inverted snub icosidodecahedron	gisid	sr{3,5/3}	s5/3s3s ()

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#55).

• Klitzing, Richard. "qrid".

• Wikipedia Contributors. "Nonconvex great rhombicosidodecahedron".
• McCooey, David. "Uniform Great Rhombicosidodecahedron"