# Great disprismatotesseractihexadecachoron

Great disprismatotesseracti-hexadecachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Gidpith |

Coxeter diagram | |

Elements | |

Cells | 32 hexagonal prisms, 24 octagonal prisms, 16 truncated octahedra, 8 great rhombicuboctahedra |

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

Edges | 192+192+192+192 |

Vertices | 384 |

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

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Toe–6–hip: 150° |

Op–4–hip: | |

Toe–4–op: 135° | |

Girco–8–op: 135° | |

Girco–4–hip: | |

Girco–6–toe: 120° | |

Central density | 1 |

Number of pieces | 80 |

Level of complexity | 24 |

Related polytopes | |

Army | Gidpith |

Regiment | Gidpith |

Dual | Tetrahedral triacosioctacontatetrachoron |

Conjugate | Great quasidisprismatotesseracti-hexadecachoron |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | B_{4}, order 384 |

Convex | Yes |

Nature | Tame |

The **great disprismatotesseractihexadecachoron**, or **gidpith**, also commonly called the **omnitruncated tesseract** or **omnitruncated hexadecachoron**, is a convex uniform polychoron that consists of 32 hexagonal prisms, 24 octagonal prisms, 16 truncated octahedra, and 8 great rhombicuboctahedra. 1 of each type of cell join at each vertex. It is the omnitruncate of the BC_{4} family of uniform polychora.

The great disprismatotesseractihexadecachoron can be vertex-inscribed into a prismatorhombated icositetrachoron.

This polychoron can be alternated into a snub tesseract, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritosnub tesseract, which is also nonuniform. If the octagons are alternated alternately, then the resulting polychoron is a truncated icositetrachoron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

Vertex coordinates for a great disprismatotesseractihexadecachoron of edge length 1 are given by all permutations of:

## Representations[edit | edit source]

A great disprismatotesseractihexadecachoron has the following Coxeter diagrams:

- x4x3x3x (full symmetry)
- xxxwwxxx4xxuxxuxx3xuxxxxux&#xt (BC3 axial, great rhombicuboctahedron-first)
- wx3xx3xw *b3xx&#zx (D
_{4}symmetry)

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

The great disprismatotesseractihexadecachoron has a semi-uniform variant of the form a4b3c3d that maintains its full symmetry. This variant uses 8 great rhombicuboctahedra of form a4b3c, 16 great rhombitetratetrahedra of form b3c3d, 24 ditetragonal prisms of form d a4b, and 32 ditrigonal prisms of form a c3d as cells, with 4 edge lengths.

With edges of length a, b, c, and d (such that it forms a4b3c3d), its circumradius is given by .

It has coordinates given by all permutations of:

## Related polychora[edit | edit source]

A great disprismatotesseractihexadecachoron can be augmented by joining truncated octahedron atop great rhombicuboctahedron segmentochora to its great rhombicuboctahedral cells. If all eight great rhombicuboctahedra are augmented, the result is the prismatorhombated icositetrachoron, as the octagonal prisms merge with square cupolas coming from the augments.

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 9: Omnitruncates" (#331).

- Bowers, Jonathan. "Tessic Isogonals".

- Klitzing, Richard. "gidpith".

- Quickfur. "The Omnitruncated Tesseract".

- Wikipedia Contributors. "Omnitruncated tesseract".