# Heptapeton

Heptapeton | |
---|---|

Rank | 6 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | Hop |

Coxeter diagram | x3o3o3o3o3o () |

Schläfli symbol | {3,3,3,3,3} |

Tapertopic notation | 1^{5} |

Elements | |

Peta | 7 hexatera |

Tera | 21 pentachora |

Cells | 35 tetrahedra |

Faces | 35 triangles |

Edges | 21 |

Vertices | 7 |

Vertex figure | Hexateron, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Face radius | |

Cell radius | |

Teron radius | |

Inradius | |

Hypervolume | |

Dipetal angle | |

Heights | Point atop hix: |

Dyad atop perp pen: | |

Trig atop perp tet: | |

Central density | 1 |

Number of pieces | 7 |

Level of complexity | 1 |

Related polytopes | |

Army | Hop |

Regiment | Hop |

Dual | Heptapeton |

Conjugate | None |

Abstract properties | |

Flag count | 5040 |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | A_{6}, order 5040 |

Convex | Yes |

Nature | Tame |

The **heptapeton**, or **hop**, also commonly called the **6-simplex**, is the simplest possible non-degenerate polypeton. The full symmetry version has 7 regular hexatera as facets, joining 3 to a tetrahedron peak and 6 to a vertex, and is one of the 3 regular polypeta. It is the 6-dimensional simplex. It is one of two uniform self-dual polypeta, the other being the great icosiheptapeton. It is also the 7-2-3 step prism and gyropeton, making it the simplest 6D step prism.

It can be obtained as a segmentopeton in three ways: as a hexateric pyramid, dyad atop perpendicular pentachoron, or triangle atop perpendicular tetrahedron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

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

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

## Related polytopes[edit | edit source]

## Representations[edit | edit source]

A regular heptapeton has the following Coxeter diagrams:

- x3o3o3o3o3o (full symmetry)
- ox3oo3oo3oo3oo&#x (A
_{5}axial, hexateric pyramid) - xo ox3oo3oo3oo&#x (A
_{4}×A_{1}axial, pentachric scalene) - xo3oo ox3oo3oo&#x (A
_{3}×A_{2}axial, tetrahedral tettene) - oxo3ooo3ooo3ooo&#x (A
_{4}only, pentachoric pyramidal pyramid) - oxo oox3ooo3ooo&#xt (A
_{3}×A_{1}axial, tetrahedral scalenic pyramid) - oxo3ooo oox3ooo&#x (A
_{2}×A_{2}axial, triangular disphenoidal pyramid) - xoo oxo oox3ooo&#x (A
_{1}×A_{2}×A_{1}axial, triangular scalenic scalene)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Primary Polypeta" (#1).

- Klitzing, Richard. "hop".

- Wikipedia Contributors. "6-simplex".
- Hi.gher.Space Wiki Contributors. "Pyropeton".