# Great rhombated icositetrachoron

Great rhombated icositetrachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Notation | |

Bowers style acronym | Grico |

Coxeter diagram | x3x4x3o () |

Elements | |

Cells | 96 triangular prisms, 24 truncated cubes, 24 great rhombicuboctahedra |

Faces | 192 triangles, 288 squares, 96 hexagons, 144 octagons |

Edges | 288+288+576 |

Vertices | 576 |

Vertex figure | Sphenoid edge lengths 1 (1), √2 (2), √3 (1), and √2+√2 (2) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tic–3–trip: 150° |

Girco–4–trip: | |

Girco–8–tic: 135° | |

Girco–6–girco: 120° | |

Central density | 1 |

Number of external pieces | 144 |

Level of complexity | 12 |

Related polytopes | |

Army | Grico |

Regiment | Grico |

Dual | Sphenoidal pentacosiheptacontahexachoron |

Conjugate | Great quasirhombated icositetrachoron |

Abstract & topological properties | |

Flag count | 13824 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | F_{4}, order 1152 |

Convex | Yes |

Nature | Tame |

The **great rhombated icositetrachoron**, or **grico**, also commonly called the **cantitruncated 24-cell**, is a convex uniform polychoron that consists of 96 triangular prisms, 24 truncated cubes, and 24 great rhombicuboctahedra. 1 triangular prism, 1 truncated cube, and 2 great rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the icositetrachoron.

## Cross-sections[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a great rhombated icositetrachoron of edge length 1 are given by all permutations of:

- ,
- .

The cantitruncation of the dual icositetrachoron has coordinates given by all permutations of:

- ,
- ,
- .

## Representations[edit | edit source]

A great rhombated icositetrachoron has the following Coxeter diagrams:

- x3x4x3o () (full symmetry)
- xo4xw3xx3wx&#zx (B
_{4}symmetry) - xux4wxx3ooo3xwX (B
_{4}symmetry, dual ico)

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

The great rhombated icositetrachoron has a semi-uniform variant of the form x3y4z3o that maintains its full symmetry. This variant uses 24 great rhombicuboctahedra of form z4y3x, 24 truncated cubes of form y4z3o, and 96 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a3b4c3o), its circumradius is given by .

## Related polychora[edit | edit source]

Uniform polychoron compounds composed of great rhombated icositetrachora include:

- Great rhombated stellated tetracontoctachoron (2)
- Great rhombated dodecahedronary hexacosichoron (25)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 8: Great Rhombates" (#309).

- Klitzing, Richard. "grico".
- Quickfur. "The Cantitruncated 24-cell".

- Wikipedia contributors. "Cantitruncated 24-cell".