# 8-simplex

The **8-simplex** (also called the **enneazetton**, or **ene**) is the simplest possible non-degenerate 8-polytope. The full symmetry version has 9 regular 7-simplices as facets, joining 3 to a 6-simplex peak and 8 to a vertex, and is regular. It is the 8-dimensional simplex.

8-simplex | |
---|---|

Rank | 8 |

Type | Regular |

Notation | |

Bowers style acronym | Ene |

Coxeter diagram | x3o3o3o3o3o3o3o () |

Schläfli symbol | {3,3,3,3,3,3,3} |

Tapertopic notation | 1^{7} |

Elements | |

Zetta | 9 7-simplices |

Exa | 36 6-simplices |

Peta | 84 5-simplices |

Tera | 126 pentachora |

Cells | 126 tetrahedra |

Faces | 84 triangles |

Edges | 36 |

Vertices | 9 |

Vertex figure | 7-simplex, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Hypervolume | |

Dizettal angle | |

Height | |

Central density | 1 |

Number of external pieces | 9 |

Level of complexity | 1 |

Related polytopes | |

Army | Ene |

Regiment | Ene |

Dual | Enneazetton |

Conjugate | None |

Abstract & topological properties | |

Flag count | 362880 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{8}, order 362880 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

## Vertex coordinates edit

The vertices of a regular 8-simplex of edge length 1, centered at the origin, are given by:

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

Much simpler coordinates can be given in nine dimensions, as all permutations of:

- .

## Representations edit

A regular 8-simplex has the following Coxeter diagrams:

- x3o3o3o3o3o3o3o ( ) (full symmetry)
- ox3oo3oo3oo3oo3oo3oo&#x (A
_{7}axial, octaexal pyramid)

## External links edit

- Klitzing, Richard. "ene".
- Wikipedia contributors. "8-simplex".