# Great rhombicosidodecahedron

Great rhombicosidodecahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Grid |

Coxeter diagram | x5x3x () |

Elements | |

Faces | 30 squares, 20 hexagons, 12 decagons |

Edges | 60+60+60 |

Vertices | 120 |

Vertex figure | Scalene triangle, edge lengths √2, √3, √(5+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 6–4: |

10–4: | |

10–6: | |

Central density | 1 |

Number of external pieces | 62 |

Level of complexity | 6 |

Related polytopes | |

Army | Grid |

Regiment | Grid |

Dual | Disdyakis triacontahedron |

Conjugate | Great quasitruncated icosidodecahedron |

Abstract & topological properties | |

Flag count | 720 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **great rhombicosidodecahedron** or **grid**, also commonly known as the **truncated icosidodecahedron**, is the most complex of the 13 Archimedean solids. It consists of 12 decagons, 20 hexagons, and 30 squares, with one of each type of face meeting per vertex. It can be obtained by cantitruncation of the dodecahedron or icosahedron, or equivalently by truncating the vertices of an icosidodecahedron and then adjusting the edge lengths to be all equal.

This is one of three Wythoffian non-prismatic polyhedra whose Coxeter diagram nodes are all ringed, the other two being the great rhombitetratetrahedron and the great rhombicuboctahedron.

It can be alternated into the snub dodecahedron after edge lengths are equalized.

## Vertex coordinates[edit | edit source]

A great rhombicosidodecahedron of edge length 1 has vertex coordinates given by all permutations of

along with all even permutations of:

## Representations[edit | edit source]

A great rhombicosidodecahedron has the following Coxeter diagrams:

- x5x3x (full symmetry)
- xuxxuAxFVFxx5xxFVFxAuxxux&#xt (H2 axial, decagon-first)

## Semi-uniform variant[edit | edit source]

The great rhombicosidodecahedron has a semi-uniform variant of the form x5y3z that maintains its full symmetry. This variant has 12 dipentagons, 20 ditrigons, and 30 rectangles as faces.

With edges of length a (dipentagon-rectangle), b (dipentagon-ditrigon), and c (ditrigon-rectangle), its circumradius is given by and its volume is given by .

## Related polyhedra[edit | edit source]

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Dodecahedron | doe | {5,3} | x5o3o | |

Truncated dodecahedron | tid | t{5,3} | x5x3o | |

Icosidodecahedron | id | r{5,3} | o5x3o | |

Truncated icosahedron | ti | t{3,5} | o5x3x | |

Icosahedron | ike | {3,5} | o5o3x | |

Small rhombicosidodecahedron | srid | rr{5,3} | x5o3x | |

Great rhombicosidodecahedron | grid | tr{5,3} | x5x3x | |

Snub dodecahedron | snid | sr{5,3} | s5s3s |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#58).

- Klitzing, Richard. "grid".

- Quickfur. "The Great Rhombicosidodecahedron".

- Wikipedia Contributors. "Truncated icosidodecahedron".
- McCooey, David. "Truncated Icosidodecahedron"

- Hi.gher.Space Wiki Contributors. "Rhodopantohedron".