# Prismatorhombated tesseract

Prismatorhombated tesseract | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Prit |

Coxeter diagram | |

Elements | |

Cells | 24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, 8 small rhombicuboctahedra |

Faces | 64 triangles, 48+96+96 squares, 64 hexagons |

Edges | 96+192+192 |

Vertices | 192 |

Vertex figure | Isosceles trapezoidal pyramid, base edge lengths 1, √2, √2, √2; lateral edge lengths √2, √2, √3, √3 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tut–6–hip: 150° |

Hip–4–cube: | |

Sirco–4–cube: 135° | |

Sirco–4–hip: | |

Sirco–3–tut: 120° | |

Central density | 1 |

Number of pieces | 80 |

Level of complexity | 16 |

Related polytopes | |

Army | Prit |

Regiment | Prit |

Dual | Deltopyramidal hecatonenneacontadichoron |

Conjugate | Prismatoquasirhombated tesseract |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | B_{4}, order 384 |

Convex | Yes |

Nature | Tame |

The **prismatorhombated tesseract**, or **prit**, also commonly called the **runcitruncated 16-cell**, is a convex uniform polychoron that consists of 24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, and 8 small rhombicuboctahedra. 1 cube, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexadecachoron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a prismatorhombated tesseract of edge length 1 are given by all permutations of:

## Representations[edit | edit source]

A prismatorhombated tesseract has the following Coxeter diagrams:

- x4o3x3x (full symmetry)
- xxxxxx4ooxxoo3xuxxux&#xt (BC
_{3}axial, small rhombicuboctahedron-first) - qo3xx3oq *b3xx&#zx (D
_{4}symmetry) - Xwx xxx4oox3xux&#zx (BC
_{3}×A_{1}symmetry)

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

The prismatorhombated tesseract has a semi-uniform variant of the form x4o3y3z that maintains its full symmetry. This variant uses 8 small rhombicuboctahedra of form x4o3y, 16 truncated tetrahedra of form z3y3o, 24 square prisms of form z x4o, and 32 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4o3b3c), its circumradius is given by .

It has coordinates given by all permutations of:

## Related polychora[edit | edit source]

The prismatorhombated tesseract is the colonel of a 3-member regiment that also includes the small cubihexadecadisoctachoron and the small rhombiprismic disoctachoron.

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in small rhombicuboctahedron-first orientation.

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

Tesseract | tes | x4o3o3o | |

Truncated tesseract | tat | x4x3o3o | |

Rectified tesseract | rit | o4x3o3o | |

Tesseractihexadecachoron | tah | o4x3x3o | |

Rectified hexadecachoron = Icositetrachoron | ico | o4o3x3o | |

Truncated hexadecachoron | thex | o4o3x3x | |

Hexadecachoron | hex | o4o3o3x | |

Small rhombated tesseract | srit | x4o3x3o | |

Great rhombated tesseract | grit | x4x3x3o | |

Small rhombated hexadecachoron = Rectified icositetrachoron | rico | o4x3o3x | |

Great rhombated hexadecachoron = Truncated icositetrachoron | tico | o4x3x3x | |

Small disprismatotesseractihexadecachoron | sidpith | x4o3o3x | |

Prismatorhombated hexadecachoron | proh | x4x3o3x | |

Prismatorhombated tesseract | prit | x4o3x3x | |

Great disprismatotesseractihexadecachoron | gidpith | x4x3x3x |

## External links[edit | edit source]

- Bowers, Jonathan. "Category 10: Prismatorhombates" (#355).

- Bowers, Jonathan. "Tessic Isogonals".

- Klitzing, Richard. "prit".

- Quickfur. "The Runcitruncated 16-cell".

- Wikipedia Contributors. "Runcitruncated 16-cell".