# 7-orthoplex

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7-orthoplex | |
---|---|

Rank | 7 |

Type | Regular |

Notation | |

Bowers style acronym | Zee |

Coxeter diagram | o4o3o3o3o3o3x () |

Schläfli symbol | {3,3,3,3,3,4} |

Bracket notation | <IIIIIII> |

Elements | |

Exa | 128 6-simplices |

Peta | 448 5-simplices |

Tera | 672 pentachora |

Cells | 560 tetrahedra |

Faces | 280 triangles |

Edges | 84 |

Vertices | 14 |

Vertex figure | 6-orthoplex, edge length 1 |

Petrie polygons | 23040 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Hypervolume | |

Diexal angle | |

Height | |

Central density | 1 |

Number of external pieces | 128 |

Level of complexity | 1 |

Related polytopes | |

Army | Zee |

Regiment | Zee |

Dual | 7-cube |

Conjugate | None |

Abstract & topological properties | |

Flag count | 645120 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | B_{7}, order 645120 |

Flag orbits | 1 |

Convex | Yes |

Net count | 33064966 |

Nature | Tame |

The **7-orthoplex** (also called the **7-cross-polytope**, **heptacross**, **hecatonicosoctaexon**, or **zee**) is a regular 7-polytope. It has 128 regular 6-simplices 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 7-segmentotope, as a 6-simplicial 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:

- .

## Representations[edit | edit source]

A hecatonicosoctaexon has the following Coxeter diagrams:

- o4o3o3o3o3o3x () (full symmetry)
- o3o3o3o3o3x *b3o () (D
_{7}symmetry) - xo3oo3oo3oo3oo3ox&#x (A
_{6}axial, heptapetal antiprism) - ooo4ooo3ooo3ooo3ooo3oxo&#xt (B
_{6}axial, hexacontatetrapetal bipyramid) - qo oo4oo3oo3oo3oo3ox&#zx (B
_{6}×A_{1}symmetry) - xox ooo4ooo3ooo3ooo3oxo&#xt (B
_{5}×A_{1}axial, dege-first) - xox ooo3ooo3ooo *b3ooo3oxo&#xt (D
_{5}×A_{1}axial, edge-first with half symmetry) - xoox oxoo3oooo3oooo3ooxo&#xr (A
_{4}×A_{1}symmetry) - xoo3oox ooo4ooo3ooo3oox&#xt (B
_{4}×A_{2}axial, triangle-first) - xoo3oox oxo3ooo3ooo *d3ooo&#xt (D
_{4}×2 symmetry, triangle-first with half symmetry) - oxo4oooo xoo3ooo3ooo3oox&#xt (A
_{4}×B_{2}axial, pentachoron-first) - oxo oxo xoo3ooo3ooo3oox&#xt (A
_{4}×A_{1}×A_{1}symmetry, pentachoron-first with half symmetry) - xo4oo oo4oo3oo3oo3ox&#zx (B
_{5}×B_{2}symmetry, square-triacontaditeral duotegum) - xoo3ooo3oox ooo4ooo3oxo&#xt (B
_{3}×A_{3}axial, tetrahedron-first) - xoo3ooo3oox ooo3oxo3ooo&#xt (A
_{3}×A_{3}axial, tetrahedron-first with half symmetry) - oo4oo3xo oo4oo3oo3ox&#zx (B
_{4}×B_{3}symmetry, octahedral-hexadecachoric duotegum) - oo3xo3oo ox3oo3oo *e3oo&#zx (D
_{4}×A_{3}symmetry, tetratetrahedral-demitesseractic duotegum) - oxoo3oooo3oooo3oooo3ooxo&#xr (A
_{5}symmetry) - oqo xoo3ooo3ooo3ooo3oox&#xt (A
_{5}×A_{1}axial, hexateron-first) - xooo3ooxo oxoo3oooo3ooox&#xr (A
_{3}×A_{2}symmetry) - ooxoo4ooooo oxooo3ooooo3oooxor (A
_{3}×B_{2}symmetry) - oxooo3oooxo ooxoo3ooooo4ooooor (B
_{3}×A_{2}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".