# 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 | |
---|---|

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 | |

Volume | |

Dihedral angles | 4–3: |

5/2–4: | |

Central density | 13 |

Number of external pieces | 980 |

Level of complexity | 59 |

Related polytopes | |

Army | Semi-uniform Ti, edge lengths (pentagons), (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 | H_{3}, 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 coordinatesEdit

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

## Related polyhedraEdit

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 linksEdit

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

- Klitzing, Richard. "qrid".

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