# Tesseractihexadecachoron

Tesseractihexadecachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Tah |

Coxeter diagram | o4x3x3o () |

Elements | |

Cells | 16 truncated tetrahedra, 8 truncated octahedra |

Faces | 32 triangles, 24 squares, 64 hexagons |

Edges | 96+96 |

Vertices | 96 |

Vertex figure | Digonal disphenoid, edge lengths 1 (base 1), √2 (base 2) and √3 (sides) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Toe–6–tut: 120° |

Tut–3–tut: 120° | |

Toe–4–toe: 90° | |

Central density | 1 |

Number of pieces | 24 |

Level of complexity | 6 |

Related polytopes | |

Army | Tah |

Regiment | Tah |

Dual | Disphenoidal enneacontahexachoron |

Conjugate | None |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | B_{4}, order 384 |

Convex | Yes |

Nature | Tame |

The **tesseractihexadecachoron**, or **tah**, also commonly called the **bitruncated tesseract**, is a convex uniform polychoron that consists of 16 truncated tetrahedra and 8 truncated octahedra. 2 truncated tetrahedra and 2 truncated octahedra join at each vertex. It is the medial stage of the truncation series between a tesseract and its dual hexadecachoron. As such is could also be called a **bitruncated 16-cell**.

## Gallery[edit | edit source]

Alternate view, centered on truncated octahedron

Card with vertex figure, cell counts, and cross-sections

## Vertex coordinates[edit | edit source]

The vertices of a tesseractihexadecachoron of edge length 1 are all permutations of:

Alternatively it can be given under D4 symmetry as all even sign changes of:

## Representations[edit | edit source]

The tesseractihexadecachoron has the following Coxeter diagrams:

- o4x3x3o (full symmetry)
- x3x3x *b3o (D4 symmetry, great rhombated demitesseract)
- s4o3x3x (as runcicantic tesseract)
- s4x3x3o (also as snub)
- ooqoo4xuxux3xooox&#xt (BC3 axial, truncated octahedron-first)
- xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
- Qqo ooq4xux3xoo&#zx (BC3×A1 symmetry)

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

The tesseractihexadecachoron has a semi-uniform variant of the form o4x3y3o that maintains its full symmetry. This variant uses 16 semi-uniform truncated tetrahedra of form x3y3o and 8 semi-uniform truncated octahedra of form o4x3y as cells, with 2 edge lengths.

With edges of length a (of square faces) and b (of triangular faces), its circumradius is given by and its hypervolume is given by .

It has coordinates given by all permutations of:

There is also a semi-uniform variant with D4 symmetry known as the great rhombated demitesseract.

## Related polychora[edit | edit source]

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 7: Bitruncates" (#299).

- Bowers, Jonathan. "Tessic Isogonals".

- Klitzing, Richard. "tah".

- Quickfur. "The Bitruncated Tesseract".

- Wikipedia Contributors. "Bitruncated tesseract".