# Hecatonicosoctaexon

(Redirected from 7-orthoplex)

Jump to navigation
Jump to search
Hecatonicosoctaexon | |
---|---|

Rank | 7 |

Type | Regular |

Space | Spherical |

Bowers style acronym | Zee |

Info | |

Coxeter diagram | o4o3o3o3o3o3x |

Schläfli symbol | {3,3,3,3,3,4} |

Symmetry | BC7, order 645120 |

Army | Zee |

Regiment | Zee |

Elements | |

Vertex figure | Hexacontatetrapeton, edge length 1 |

Exa | 128 heptapeta |

Peta | 448 hexatera |

Tera | 672 pentachora |

Cells | 560 tetrahedra |

Faces | 280 triangles |

Edges | 84 |

Vertices | 14 |

Measures (edge length 1) | |

Circumradius | √2/2 ≈ 0.70711 |

Inradius | √14/14 ≈ 0.26726 |

Hypervolume | √2/630 ≈ 0.0022448 |

Diexal angle | acos(–5/7) ≈ 135.58470° |

Height | √14/7 ≈ 0.53452 |

Central density | 1 |

Euler characteristic | 2 |

Related polytopes | |

Dual | Hepteract |

Conjugate | Hecatonicosoctaexon |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

The **hecatonicosoctaexon**, or **zee**, also called the **heptacross** or **7-orthoplex**, is one of the 3 regular polyexa. It has 128 regular heptapeta as facets, joining 4 to a pentachoron peak and 64 to a vertex in a hexacontatetrapetal arrangement. It is the 7-dimensional orthoplex. It is also a convex segmentoexon, as a heptapetal antiprism.

## Vertex coordinates[edit | edit source]

The vertices of a regular hecatonicosoctaexon of edge length 1, centered at the origin, are given by all permutations of:

- (±√2/2, 0, 0, 0, 0, 0, 0).

## Representations[edit | edit source]

A hecatonicosoctaexon has the following Coxeter diagrams:

- o4o3o3o3o3o3x (full symmetry)
- o3o3o *b3o3o3o3x (D7 symmetry)
- xo3oo3oo3oo3oo3ox&#x (A6 axial, heptapetal antiprism)
- ooo4ooo3ooo3ooo3ooo3oxo&#xt (BC6 axial, hexacontatetrapetal bipyramid)
- qo oo4oo3oo3oo3oo3ox&#zx (BC6×A1 symmetry)
- xox ooo4ooo3ooo3ooo3oxo&#xt (BC5×A1 axial, dege-first)
- xox ooo3ooo3ooo *b3ooo3oxo&#xt (D5×A1 axial, edge-first with half symmetry)
- xoox oxoo3oooo3oooo3ooxo&#xr (A4×A1 symmetry)
- xoo3oox ooo4ooo3ooo3oox&#xt (BC4×A2 axial, triangle-first)
- xoo3oox oxo3ooo3ooo *d3ooo&#xt (D4×2 symmetry, triangle-first with half symmetry)
- oxo4oooo xoo3ooo3ooo3oox&#xt (A4×BC2 axial, pentachoron-first)
- oxo oxo xoo3ooo3ooo3oox&#xt (A4×A1×A1 symmetry, pentachoron-first with half symmetry)
- xo4oo oo4oo3oo3oo3ox&#zx (BC5×BC2 symmetry, square-triacontaditeral duotegum)
- xoo3ooo3oox ooo4ooo3oxo&#xt (BC3×A3 axial, tetrahedron-first)
- xoo3ooo3oox ooo3oxo3ooo&#xt (A3×A3 axial, tetrahedron-first with half symmetry)
- oo4oo3xo oo4oo3oo3ox&#zx (BC4×BC3 symmetry, octahedral-hexadecachoric duotegum)
- oo3xo3oo ox3oo3oo *e3oo&#zx (D4×A3 symmetry, tetratetrahedral-demitesseractic duotegum)
- oxoo3oooo3oooo3oooo3ooxo&#xr (A5 symmetry)
- oqo xoo3ooo3ooo3ooo3oox&#xt (A5×A1 axial, hexateron-first)
- xooo3ooxo oxoo3oooo3ooox&#xr (A3×A2 symmetry)
- ooxoo4ooooo oxooo3ooooo3oooxor (A3×BC2 symmetry)
- oxooo3oooxo ooxoo3ooooo4ooooor (BC3×A2 symmetry)
- o(xoo)o o(xoo)o o(oxo)o o(oxo)o o(oox)o o(oox)o&#xt (square triotegum bipyramid)

## External links[edit | edit source]

- Klitzing, Richard. "Zee".

- Wikipedia Contributors. "7-orthoplex".