# Great rhombitetratetrahedron

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Great rhombitetratetrahedron | |
---|---|

Rank | 3 |

Type | Semi-uniform |

Notation | |

Bowers style acronym | Gratet |

Coxeter diagram | x3y3z |

Elements | |

Faces | 6 rectangles, 4+4 ditrigons |

Edges | 12+12+12 |

Vertices | 24 |

Vertex figure | Scalene triangle |

Measures (edge length 1) | |

Dihedral angles | 6–4: |

6–6: | |

Central density | 1 |

Related polytopes | |

Army | Gratet |

Regiment | Gratet |

Dual | Disdyakis hexahedron |

Conjugate | Great rhombitettratetrahedron |

Abstract & topological properties | |

Flag count | 144 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{3}, order 24 |

Flag orbits | 6 |

Convex | Yes |

Nature | Tame |

The **great rhombitetratetrahedron**, or **gratet**, is a convex semi-uniform polyhedron. It is the result of relaxing the truncated octahedron so that it only needs to possess tetrahedral symmetry rather than cubic symmetry. It has 2 sets of 4 ditrigons and 6 rectangles as faces. It generally has 3 types of edge lengths, connecting each pair of face types.

It can be alternated into a snub tetrahedron.

## Vertex coordinates[edit | edit source]

A great rhombitetratetrahedron with edges of length a , b , and c , where a and c are the rectangle edges, and b edges are between the two types of ditrigons, has vertex coordinates given by all permutation and even sign changes of:

- .

## Measures[edit | edit source]

Letting a , b , and c be as before, we have the following measures:

- Circumradius =
- Volume =