# Great prismatodecachoron

Great prismatodecachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Gippid |

Coxeter diagram | x3x3x3x () |

Elements | |

Cells | 20 hexagonal prisms, 10 truncated octahedra |

Faces | 30+60 squares, 20+40 hexagons |

Edges | 120+120 |

Vertices | 120 |

Vertex figure | Phyllic disphenoid, edge lengths √2 (3) and √3 (3) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Hip–4–hip: |

Toe–6–hip: | |

Toe–4–hip: | |

Toe–6–toe: | |

Central density | 1 |

Number of external pieces | 30 |

Level of complexity | 12 |

Related polytopes | |

Army | Gippid |

Regiment | Gippid |

Dual | Disphenoidal hecatonicosachoron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 2880 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}×2, order 240 |

Convex | Yes |

Nature | Tame |

The **great prismatodecachoron**, or **gippid**, also commonly called the **omnitruncated 5-cell** or **omnitruncated pentachoron**, is a convex uniform polychoron that consists of 20 hexagonal prisms and 10 truncated octahedra. 2 hexagonal prisms and 2 truncated octahedra join at each vertex. It is the omnitruncate of the A_{4} family of uniform polychora.

This polychoron can be alternated into a snub decachoron, although it cannot be made uniform.

Like the omnitruncated simplex of any dimension, this polychoron can tile 4D space. The resulting tetracomb is called the omnitruncated cyclopentachoric tetracomb.

It is the 5th-order permutohedron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a great prismatodecachoron of edge length 1 are given by the following points:

Much simpler coordinates can be given in five dimensions, as all permutations of:

## Representations[edit | edit source]

A great prismatodecachoron has the following Coxeter diagrams:

- x3x3x3x (full symmetry)
- xxxux3xxuxx3xuxxx&#xt (A
_{3}axial, truncated octahedron-first)

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

The great prismatodecachoron has a semi-uniform variant of the form x3y3y3x that maintains its full symmetry. This variant uses 10 great rhombitetratetrahedra of form x3y3y and 20 ditrigonal prisms of form x x3y as cells, with 2 edge lengths.

With edges of length a and b (so that it is represented by a3b3b3a), its circumradius is given by .

If it has only single pentachoric symmetry, the variant is called a great disprismatopentapentachoron.

## Related polychora[edit | edit source]

The antifrustary prismatohexacosichoron is a uniform polychoron compound composed of 60 great prismatodecachora.

Name | OBSA | CD diagram | Picture |
---|---|---|---|

Pentachoron | pen | ||

Truncated pentachoron | tip | ||

Rectified pentachoron | rap | ||

Decachoron | deca | ||

Rectified pentachoron | rap | ||

Truncated pentachoron | tip | ||

Pentachoron | pen | ||

Small rhombated pentachoron | srip | ||

Great rhombated pentachoron | grip | ||

Small rhombated pentachoron | srip | ||

Great rhombated pentachoron | grip | ||

Small prismatodecachoron | spid | ||

Prismatorhombated pentachoron | prip | ||

Prismatorhombated pentachoron | prip | ||

Great prismatodecachoron | gippid |

## External links[edit | edit source]

- Bowers, Jonathan. "Category 9: Omnitruncates" (#330).

- Bowers, Jonathan. "Pennic and Decaic Isogonals".

- Klitzing, Richard. "gippid".

- Quickfur. "The Omnitruncated 5-cell".

- Wikipedia Contributors. "Omnitruncated 5-cell".