# Prismatorhombated hexadecachoron

Prismatorhombated hexadecachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Proh |

Coxeter diagram | |

Elements | |

Cells | 32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, 8 truncated cubes |

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

Edges | 96+192+192 |

Vertices | 192 |

Vertex figure | Skewed rectangular pyramid, base edge lengths 1, √2, 1, √2; lateral edge lengths √2, √2, √2+√2, √2+√2 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Co–3–trip: 150° |

Op–4–trip: | |

Co–4–op: 135° | |

Tic–8–op: 135° | |

Tic–3–co: 120° | |

Central density | 1 |

Number of pieces | 80 |

Level of complexity | 16 |

Related polytopes | |

Army | Proh |

Regiment | Proh |

Dual | Rhombipyramidal hecatonenneacontadichoron |

Conjugate | Quasiprismatorhombated hexadecachoron |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | B_{4}, order 384 |

Convex | Yes |

Nature | Tame |

The **prismatorhombated hexadecachoron**, or **proh**, also commonly called the **runcitruncated tesseract**, is a convex uniform polychoron that consists of 32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, and 8 truncated cubes. 1 triangular prism, 2 octagonal prisms, 1 cuboctahedron, and 1 truncated cube join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the tesseract.

The prismatorhombated hexadecachoron can be edge-inscribed into a small rhombated icositetrachoron. Blending 3 prismatorhombated hexadecachora results in the small rhombic disicositetrachoron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

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

## Representations[edit | edit source]

A prismatorhombated hexadecachoron has the following Coxeter diagrams:

- x4x3o3x (full symmetry)
- xxwwxx4xxooxx3oxxxxo&#xt (BC3 axial, truncated cube-first)
- wx3oo3xw *b3xx&#zx (D4 symmetry)
- Xwx xxw4xxo3oxx&#zx (BC3×A1 symmetry)

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

The prismatorhombated hexadecachoron has a semi-uniform variant of the form x4y3o3z that maintains its full symmetry. This variant uses 8 truncated cubes of form x4y3o, 16 rhombitetratetrahedra of form y3o3z, 32 triangular prisms of form x z3o, and 24 ditetragonal prisms of form z x4y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4b3o3c), its circumradius is given by .

It has coordinates given by all permutations of:

## Related polychora[edit | edit source]

The prismatorhombated hexadecachoron is the colonel of a 3-member regiment that also includes the small prismatohexadecadisoctachoron and small rhombiprismic tesseractihexadecachoron.

The segmentochoron truncated cube atop great rhombicuboctahedron can be obtained as a cap of the prismatorhombated hexadecachoron.

The truncated cubes of the prismatorhombated hexadecachoron can be augmented by cuboctahedron atop truncated cube segmentochora. If all 8 truncated cubes are augmented, the result is the small rhombated icositetrachoron, as square cupolas from the segmentochora merge with octagonal prisms to form small rhombicuboctahedra.

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" (#358).

- Bowers, Jonathan. "Tessic Isogonals".

- Klitzing, Richard. "proh".

- Quickfur. "The Runcitruncated Tesseract".

- Wikipedia Contributors. "Runcitruncated tesseract".