# Demihexeract

Demihexeract | |
---|---|

Rank | 6 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Hax |

Coxeter diagram | x3o3o *b3o3o3o () |

Elements | |

Peta | 32 hexatera, 12 demipenteracts |

Tera | 192 pentachora, 60 hexadecachora |

Cells | 160+480 tetrahedra |

Faces | 640 triangles |

Edges | 240 |

Vertices | 32 |

Vertex figure | Rectified hexateron, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dipetal angles | Hin–pen–hix: |

Hin–hex–hin: 90° | |

Height | |

Central density | 1 |

Number of pieces | 44 |

Level of complexity | 4 |

Related polytopes | |

Army | Hax |

Regiment | Hax |

Dual | Semistellated hexacontatetrapeton |

Conjugate | None |

Abstract properties | |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | D_{6}, order 23040 |

Convex | Yes |

Nature | Tame |

The **demihexeract**, or **hax**, also called the **hemihexeract** or **6-demicube**, is a convex uniform polypeton. It has 12 demipenteracts and 32 hexatera as facets, with 6 of each at a vertex forming a rectified hexateron as the vertex figure. It is the 6-dimensional demihypercube and is formed by alternating the hexeract.

It is a segmentopeton, as a demipenteractic antiprism. It is also a tetrahedral duoantiprism and digonal trioantiprism.

The demihexeract contains the vertices of a tetrahedral duoprism, in fact being the convex hull of 2 tetrahedral duoprisms, one of which has bases in dual orientation compared to the other.

## Vertex coordinates[edit | edit source]

The vertices of a demihexeract of edge length 1, centered at the origin, are given by all even sign changes of:

## Representations[edit | edit source]

A demihexeract has the following Coxeter diagrams:

- x3o3o *b3o3o3o (full symmetry)
- s4o3o3o3o3o () (as alternated hexeract)
- xo3oo3ox *b3oo3oo&#x (D
_{5}axial, demipenteract antiprism) - xoo3ooo3oxo3ooo3oox&#xt (A
_{5}axial, hexateron-first) - oooo3oxoo3oooo3ooxo3oooo&#x (A
_{5}axial, vertex-first) - oxo xox3ooo3ooo *c3oxo&#xt (D
_{4}×A_{1}axial, hexadecachoron-first) - xo3oo3ox *b3oo xo ox&#zx (D
_{4}×A_{1}×A_{1}symmetry) - oxoo3ooxo xoxo3oooo3oxox&#xt (A
_{3}×A_{2}axial, tetrahedron-first) - xo3oo3ox xo3oo3ox&#zx (A
_{3}×A_{3}symmetry, hull of two tetrahedral duoprisms)

## Gallery[edit | edit source]

## Related polytopes[edit | edit source]

The regiment of the demihexeract contains a total of 145 members, 66 fissaries, and 18 compounds. of these, 15 of the main members and one fissary has full D6 symmetry, all the others were only found in 2019 and have various subsymmetries.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Primary Polypeta" (#9).

- Klitzing, Richard. "hax".

- Wikipedia Contributors. "6-demicube".