Quasirhombicosidodecahedron

Quasirhombicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymQrid
Coxeter diagramx5/3o3x ()
Elements
Faces20 triangles, 30 squares, 12 pentagrams
Edges60+60
Vertices60
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 2, (5–1)/2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11-4\sqrt5}}{2} \approx 0.71689}$
Volume${\displaystyle \frac{29\sqrt5-60}{3} \approx 1.61532}$
Dihedral angles4–3: ${\displaystyle \arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) \approx 69.09484^\circ}$
5/2–4: ${\displaystyle \arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) \approx 58.28253^\circ}$
Central density13
Number of pieces980
Level of complexity59
Related polytopes
ArmySemi-uniform Ti
DualGreat deltoidal hexecontahedron
ConjugateSmall rhombicosidodecahedron
Convex coreRhombic triacontahedron
Abstract properties
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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.

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 ()