# Hepteract

(Redirected from 7-hypercube)

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Hepteract | |
---|---|

Rank | 7 |

Type | Regular |

Space | Spherical |

Bowers style acronym | Hept |

Info | |

Coxeter diagram | x4o3o3o3o3o3o |

Schläfli symbol | {4,3,3,3,3,3} |

Symmetry | BC7, order 645120 |

Army | Hept |

Regiment | Hept |

Elements | |

Vertex figure | Heptapeton, edge length √2 |

Exa | 14 hexeracts |

Peta | 84 penteracts |

Tera | 280 tesseracts |

Cells | 560 cubes |

Faces | 672 squares |

Edges | 448 |

Vertices | 128 |

Measures (edge length 1) | |

Circumradius | √7/2 ≈ 1.32288 |

Inradius | 1/2 = 0.5 |

Hypervolume | 1 |

Diexal angle | 90° |

Central density | 1 |

Euler characteristic | 2 |

Related polytopes | |

Dual | Hecatonicosoctaexon |

Conjugate | Hepteract |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

The **hepteract**, or **hept**, also called the **7-cube**, or **tetradecaexon**, is one of the 3 regular polyexa. It has 14 hexeracts as facets, joining 7 to a vertex. It is the 7-dimensional hypercube.

It can be alternated into a demihepteract, which is uniform.

A regular octaexon of edge length 2 can be inscribed in the unit hepteract.^{[1]} The next largest simplex that can be inscribed in a hypercube is the dodecadakon.^{[2]}

## Vertex coordinates[edit | edit source]

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

- (±1/2, ±1/2, ±1/2, ±1/2, ±1/2, ±1/2, ±1/2).

## Representations[edit | edit source]

A hepteract has the following Coxeter diagrams:

- x4o3o3o3o3o3o (full symmetry)
- x x4o3o3o3o3o (BC6×A1 symmetry, hexxeractic prism)
- x4o x4o3o3o3o (BC5×BC2 symmetry, square-penteractic duoprism)
- x4o3o x4o3o3o (BC4×BC3 symmetry, cubic-tesseractic duoprism)
- xx4oo3oo3oo3oo3oo&#x (BC6 axial)
- oqoooooo3ooqooooo3oooqoooo3ooooqooo3oooooqoo3ooooooqo&#xt (A6 axial, vertex-first)

## References[edit | edit source]

- ↑ Adams, Joshua; Zvengrowski, Peter; Laird, Philip (2003). "Vertex Embeddings of Regular Polytopes".
*Expositiones Mathematicae*. - ↑ Sloane, N. J. A. (ed.). "Sequence A019442".
*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.

## External links[edit | edit source]

- Klitzing, Richard. "Hept".

- Wikipedia Contributors. "7-cube".