# Hexateron

Hexateron | |
---|---|

Rank | 5 |

Type | Regular |

Space | Spherical |

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 |

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 | Hexateron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 720 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | A_{5}, order 720 |

Convex | Yes |

Nature | Tame |

The **hexateron**, **hix** or **triangular disphenoid**, also commonly called the **5-simplex**, 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.

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 | edit source]

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

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

## Representations[edit | edit source]

A regular hexateron has the following Coxeter diagrams:

- x3o3o3o3o (full symmetry)
- ox3oo3oo3oo&#x (A4 axial, pentachoric pyramid)
- xo ox3oo3oo&#x (A3×A1 symmetry, tetrahedral scalene)
- xo3oo ox3oo&#x (A2×A2 axial, triangular disphenoid)
- oxo3ooo3ooo&#x (A3 symmetry, tetrahedral pyramidal pyramid)
- oxo oox3ooo&#x A2×A1 symmetry, triangular scalene pyramid)
- xoo oxo oox&#x (A1×A1×A1 symmetry, digonal trisphenoid)
- ooox ooxo&#x (A1×A1 symmetry, disphenoidal pyramidal pyramid)
- ooox3oooo&#x (A2 symmetry, triangular symmetry only)
- oooox&#x (A1 symmetrry only)
- oooooo&#x (no symmetry, fully irregular)

## Variations[edit | edit source]

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

## Related polytopes[edit | edit source]

Name | OBSA | CD diagram | Picture |
---|---|---|---|

Hexateron | hix | ||

Rectified hexateron | rix | ||

Dodecateron | dot | ||

Rectified hexateron | rix | ||

Hexateron | hix | ||

Truncated hexateron | tix | ||

Bitruncated hexateron | bittix | ||

Bitruncated hexateron | bittix | ||

Truncated hexateron | tix | ||

Small rhombated hexateron | sarx | ||

Small birhombidodecateron | sibrid | ||

Small rhombated hexateron | sarx | ||

Great rhombated hexateron | garx | ||

Great birhombidodecateron | gibrid | ||

Great rhombated hexateron | garx | ||

Small prismated hexateron | spix | ||

Small prismated hexateron | spix | ||

Prismatotruncated hexateron | pattix | ||

Prismatorhombated hexateron | pirx | ||

Prismatorhombated hexateron | pirx | ||

Prismatotruncated hexateron | pattix | ||

Great prismated hexateron | gippix | ||

Great prismated hexateron | gippix | ||

Small cellidodecateron | scad | ||

Celliprismated hexateron | cappix | ||

Cellirhombidodecateron | card | ||

Celligreatorhombated hexateron | cograx | ||

Celliprismated hexateron | cappix | ||

Celliprismatotruncatododecateron | captid | ||

Celligreatorhombated hexateron | cograx | ||

Great cellidodecateron | gocad |

## External links[edit | edit source]

- 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".