# Hecatonicosihexapentacosiheptacontahexaexon

Hecatonicosihexapentacosiheptacontahexaexon | |
---|---|

Rank | 7 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Naq |

Coxeter diagram | o3o3o3o *c3o3o3x () |

Elements | |

Exa | 576 heptapeta, 126 hexacontatetrapeta |

Peta | 2016+4032 hexatera |

Tera | 12096 pentachora |

Cells | 10080 tetrahedra |

Faces | 4032 triangles |

Edges | 756 |

Vertices | 56 |

Vertex figure | Icosiheptaheptacontadipeton, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diexal angles | Gee–hix–hop: |

Gee–hix–gee: 120° | |

Central density | 1 |

Number of pieces | 702 |

Level of complexity | 3 |

Related polytopes | |

Army | Naq |

Regiment | Naq |

Dual | Pentacontahexaexon |

Conjugate | None |

Abstract properties | |

Euler characteristic | 2 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | E_{7}, order 2903040 |

Convex | Yes |

Nature | Tame |

The **hecatonicosihexapentacosiheptacontahexaexon**, or **naq**, also called the **3 _{21} polytope** or

**biambohexadecaexon**, is a convex uniform polyexon. It has 126 hexacontatetrapeta and 576 heptapeta as facets, with 27 hexacontatetrapeta and 72 heptapeta at a vertex forming an icosiheptaheptacontadipeton as the vertex figure.

The hecatonicosihexapentacosiheptacontahexaexon contains the vertices of a hexacontatetrapetic prism and a triangular-hexateric duoprism. It is also the convex hull of two oppositely oriented rectified octaexa.

## Vertex coordinates[edit | edit source]

The vertices of a hecatonicosihexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by:

- and all permutations of first 6 coordinates
- and all even sign changes of the first 6 coordinates.

Alternate coordinates can be given in 8 dimensions, as all permutations of:

- .

These give a naq of edge length , and come from the fact that naq is the convex hull of two opposite rectified octaexa.

## Representations[edit | edit source]

A hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:

- o3o3o3o *c3o3o3x (full symmetry)
- oxo3ooo3ooo *b3ooo3ooo3xox&#xt (D6 axial, hexacontatetrapeton-first)
- oxoo3oooo3oooo3oooo3ooxo *c3oooo&#xt (E6 axial, vertex-first)
- xoxox oxooo3ooooo3oooxo *c3ooooo3ooxoo&#xt (D5×A1 axial, edge-first)
- xooo3ooxo3oooo3oooo3oxoo3ooox&#xt (A6 axial, heptapeton-first)
- oo3xo3oo3oo3oo3ox3oo&#zx (A7 symmetry, hull of 2 rectified octaexa)
- ox xo3oo3oo *c3oo3oo3ox&#zx (D6×A1 symmetry)
- xxoo xoxo xoox oxoo3oooo3ooxo *e3ooox&#zx (D4×A1×A1×A1 symmetry)

## External links[edit | edit source]

- Klitzing, Richard. "naq".

- Wikipedia Contributors. "3_21 polytope".