# Rectified 7-simplexes

In seven-dimensional geometry, a **rectified 7-simplex** is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

7-simplex |
Rectified 7-simplex | |

Birectified 7-simplex |
Trirectified 7-simplex | |

Orthogonal projections in A_{7} Coxeter plane |
---|

There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the *rectified 7-simplex* are located at the edge-centers of the *7-simplex*. Vertices of the *birectified 7-simplex* are located in the triangular face centers of the *7-simplex*. Vertices of the *trirectified 7-simplex* are located in the tetrahedral cell centers of the *7-simplex*.

## Rectified 7-simplex

Rectified 7-simplex | |
---|---|

Type | uniform 7-polytope |

Coxeter symbol | 0_{51} |

Schläfli symbol | r{3^{6}} = {3^{5,1}}or |

Coxeter diagrams | Or |

6-faces | 16 |

5-faces | 84 |

4-faces | 224 |

Cells | 350 |

Faces | 336 |

Edges | 168 |

Vertices | 28 |

Vertex figure | 6-simplex prism |

Petrie polygon | Octagon |

Coxeter group | A_{7}, [3^{6}], order 40320 |

Properties | convex |

The rectified 7-simplex is the edge figure of the 2_{51} honeycomb. It is called **0 _{5,1}** for its branching Coxeter-Dynkin diagram, shown as

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}_{7}.

### Alternate names

- Rectified octaexon (Acronym: roc) (Jonathan Bowers)

### Coordinates

The vertices of the *rectified 7-simplex* can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.

### Images

A_{k} Coxeter plane |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [7] | [6] |

A_{k} Coxeter plane |
A_{4} |
A_{3} |
A_{2} |

Graph | |||

Dihedral symmetry | [5] | [4] | [3] |

## Birectified 7-simplex

Birectified 7-simplex | |
---|---|

Type | uniform 7-polytope |

Coxeter symbol | 0_{42} |

Schläfli symbol | 2r{3,3,3,3,3,3} = {3^{4,2}}or |

Coxeter diagrams | Or |

6-faces | 16: 8 r{3 ^{5}} 8 2r{3 ^{5}} |

5-faces | 112: 28 {3 ^{4}} 56 r{3 ^{4}} 28 2r{3 ^{4}} |

4-faces | 392: 168 {3 ^{3}} (56+168) r{3 ^{3}} |

Cells | 770: (420+70) {3,3} 280 {3,4} |

Faces | 840: (280+560) {3} |

Edges | 420 |

Vertices | 56 |

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

Coxeter group | A_{7}, [3^{6}], order 40320 |

Properties | convex |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}_{7}. It is also called **0 _{4,2}** for its branching Coxeter-Dynkin diagram, shown as

### Alternate names

- Birectified octaexon (Acronym: broc) (Jonathan Bowers)

### Coordinates

The vertices of the *birectified 7-simplex* can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.

### Images

A_{k} Coxeter plane |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [7] | [6] |

A_{k} Coxeter plane |
A_{4} |
A_{3} |
A_{2} |

Graph | |||

Dihedral symmetry | [5] | [4] | [3] |

## Trirectified 7-simplex

Trirectified 7-simplex | |
---|---|

Type | uniform 7-polytope |

Coxeter symbol | 0_{33} |

Schläfli symbol | 3r{3^{6}} = {3^{3,3}}or |

Coxeter diagrams | Or |

6-faces | 16 2r{3^{5}} |

5-faces | 112 |

4-faces | 448 |

Cells | 980 |

Faces | 1120 |

Edges | 560 |

Vertices | 70 |

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

Coxeter group | A_{7}×2, [[3^{6}]], order 80640 |

Properties | convex, isotopic |

The *trirectified 7-simplex* is the intersection of two regular 7-simplexes in dual configuration.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{3}_{7}.

This polytope is the vertex figure of the 1_{33} honeycomb. It is called **0 _{3,3}** for its branching Coxeter-Dynkin diagram, shown as

### Alternate names

- Hexadecaexon (Acronym: he) (Jonathan Bowers)

### Coordinates

The vertices of the *trirectified 7-simplex* can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.

The *trirectified 7-simplex* is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

### Images

A_{k} Coxeter plane |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [[7]] | [6] |

A_{k} Coxeter plane |
A_{4} |
A_{3} |
A_{2} |

Graph | |||

Dihedral symmetry | [[5]] | [4] | [[3]] |

### Related polytopes

Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

Name Coxeter |
Hexagon t{3} = {6} |
Octahedron r{3,3} = {3 ^{1,1}} = {3,4} |
Decachoron 2t{3 ^{3}} |
Dodecateron 2r{3 ^{4}} = {3^{2,2}} |
Tetradecapeton 3t{3 ^{5}} |
Hexadecaexon 3r{3 ^{6}} = {3^{3,3}} |
Octadecazetton 4t{3 ^{7}} |

Images | |||||||

Vertex figure | ( )v( ) | { }×{ } |
{ }v{ } |
{3}×{3} |
{3}v{3} |
{3,3}x{3,3} | {3,3}v{3,3} |

Facets | {3} |
t{3,3} |
r{3,3,3} |
2t{3,3,3,3} |
2r{3,3,3,3,3} |
3t{3,3,3,3,3,3} | |

As intersecting dual simplexes |

## Related polytopes

These polytopes are three of 71 uniform 7-polytopes with A_{7} symmetry.

## See also

## 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. "7D uniform polytopes (polyexa)". o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he