# 5-simplex

The **5-simplex**, also commonly called the **hexateron** or **hix**, is the simplest possible non-degenerate polyteron. The full symmetry version has 6 regular pentachora as cells, joining 5 to a vertex, and is one of the 3 regular polytera. It is the 5-dimensional simplex.

5-simplex | |
---|---|

Rank | 5 |

Type | Regular |

Notation | |

Bowers style acronym | Hix |

Coxeter diagram | x3o3o3o3o () |

Schläfli symbol | {3,3,3,3} |

Tapertopic notation | 1^{4} |

Elements | |

Tera | 6 pentachora |

Cells | 15 tetrahedra |

Faces | 20 triangles |

Edges | 15 |

Vertices | 6 |

Vertex figure | Pentachoron, edge length 1 |

Petrie polygons | 60 skew hexagonal-triangular coils |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Face radius | |

Cell radius | |

Inradius | |

Hypervolume | |

Diteral angle | |

Heights | Point atop pen: |

Dyad atop perp tet: | |

Trig atop perp trig: | |

Central density | 1 |

Number of external pieces | 6 |

Level of complexity | 1 |

Related polytopes | |

Army | Hix |

Regiment | Hix |

Dual | 5-simplex |

Conjugate | None |

Abstract & topological properties | |

Flag count | 720 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | A_{5}, order 720 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

It can be viewed as a segmentoteron in three ways: as a pentachoric pyramid, as a dyad atop perpendicular tetrahedron, and as a triangle atop perpendicular triangle. This makes it the triangular member of an infinite family of isogonal polygonal disphenoids.

## Vertex coordinates Edit

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

- ,
- ,
- ,
- ,
- .

Another set of coordinates can be given in B_{3}÷(B_{2}×A_{1}) symmetry as:

- ,
- ,
- .

This represents the polytope as the convex hull of three skew orthogonal dyads.

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

- .

## Representations Edit

A regular hexateron has the following Coxeter diagrams:

- x3o3o3o3o ( ) (full symmetry)
- ox3oo3oo3oo&#x (A
_{4}axial, pentachoric pyramid) - xo ox3oo3oo&#x (A
_{3}×A_{1}symmetry, tetrahedral scalene) - xo3oo ox3oo&#x (A
_{2}≀S_{2}axial, triangular disphenoid) - oxo3ooo3ooo&#x (A
_{3}symmetry, tetrahedral pyramidal pyramid) - oxo oox3ooo&#x (A
_{2}×A_{1}symmetry, triangular scalenic pyramid) - xoo oxo oox&#x (B
_{3}symmetry, digonal trisphenoid or digonal dihedral trigyroprism) - ooox ooxo&#x (K
_{2}symmetry, digonal disphenoidal pyramidal pyramid) - ooox3oooo&#x (A
_{2}symmetry, triangular symmetry only) - oooox&#x (A
_{1}symmetry only) - oooooo&#x (no symmetry, fully irregular)

## Variations Edit

The regular hexateron has 2 subsymmetrical forms that remain isogonal:

- Triangular disphenoid - triangle atop an orthogonal triangle, facets and vertex figures are triangular scalenes
- Digonal trisphenoid - cells and vertex figures are disphenoidal pyramids

## External links Edit

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

- Klitzing, Richard. "hix".
- Wikipedia contributors. "5-simplex".
- Hi.gher.Space Wiki Contributors. "Pyroteron".

- Hartley, Michael. "{3,3,3,3}*720".