# Rectified 6-simplexes

In six-dimensional geometry, a **rectified 6-simplex** is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

6-simplex |
Rectified 6-simplex |
Birectified 6-simplex |

Orthogonal projections in A_{6} Coxeter plane |
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There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the *rectified 6-simplex* are located at the edge-centers of the *6-simplex*. Vertices of the *birectified 6-simplex* are located in the triangular face centers of the *6-simplex*.

## Rectified 6-simplex

Rectified 6-simplex | |
---|---|

Type | uniform polypeton |

Schläfli symbol | t_{1}{3^{5}}r{3 ^{5}} = {3^{4,1}}or |

Coxeter diagrams | |

Elements |
f = 63, _{4}C = 140, F = 175, E = 105, V = 21(χ=0) |

Coxeter group | A_{6}, [3^{5}], order 5040 |

Bowers name and (acronym) | Rectified heptapeton (ril) |

Vertex figure | 5-cell prism |

Circumradius | 0.845154 |

Properties | convex, isogonal |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}_{6}. It is also called **0 _{4,1}** for its branching Coxeter-Dynkin diagram, shown as

### Alternate names

- Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

### Coordinates

The vertices of the *rectified 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.

### Images

A_{k} Coxeter plane |
A_{6} |
A_{5} |
A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [7] | [6] | [5] |

A_{k} Coxeter plane |
A_{3} |
A_{2} | |

Graph | |||

Dihedral symmetry | [4] | [3] |

## Birectified 6-simplex

Birectified 6-simplex | |
---|---|

Type | uniform 6-polytope |

Class | A6 polytope |

Schläfli symbol | t_{2}{3,3,3,3,3}2r{3 ^{5}} = {3^{3,2}}or |

Coxeter symbol | 0_{32} |

Coxeter diagrams | |

5-faces | 14 total: 7 t _{1}{3,3,3,3}7 t _{2}{3,3,3,3} |

4-faces | 84 |

Cells | 245 |

Faces | 350 |

Edges | 210 |

Vertices | 35 |

Vertex figure | {3}x{3,3} |

Petrie polygon | Heptagon |

Coxeter groups | A_{6}, [3,3,3,3,3] |

Properties | convex |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}_{6}. It is also called **0 _{3,2}** for its branching Coxeter-Dynkin diagram, shown as

### Alternate names

- Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

### Coordinates

The vertices of the *birectified 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.

### Images

A_{k} Coxeter plane |
A_{6} |
A_{5} |
A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [7] | [6] | [5] |

A_{k} Coxeter plane |
A_{3} |
A_{2} | |

Graph | |||

Dihedral symmetry | [4] | [3] |

## Related uniform 6-polytopes

The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2_{41} polytope.

These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.

## Notes

## References

- H.S.M. Coxeter:
- H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973 **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (Paper 22) H.S.M. Coxeter,
*Regular and Semi Regular Polytopes I*, [Math. Zeit. 46 (1940) 380-407, MR 2,10] - (Paper 23) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559-591] - (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 22) H.S.M. Coxeter,

- H.S.M. Coxeter,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D.

- N.W. Johnson:
- Klitzing, Richard. "6D uniform polytopes (polypeta)". o3x3o3o3o3o - ril, o3x3o3o3o3o - bril