# Rectified 8-simplexes

In eight-dimensional geometry, a **rectified 8-simplex** is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

8-simplex |
Rectified 8-simplex | ||

Birectified 8-simplex |
Trirectified 8-simplex | ||

Orthogonal projections in A_{8} Coxeter plane |
---|

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

## Rectified 8-simplex

Rectified 8-simplex | |
---|---|

Type | uniform 8-polytope |

Coxeter symbol | 0_{61} |

Schläfli symbol | t_{1}{3^{7}}r{3 ^{7}} = {3^{6,1}}or |

Coxeter-Dynkin diagrams | or |

7-faces | 18 |

6-faces | 108 |

5-faces | 336 |

4-faces | 630 |

Cells | 756 |

Faces | 588 |

Edges | 252 |

Vertices | 36 |

Vertex figure | 7-simplex prism, {}×{3,3,3,3,3} |

Petrie polygon | enneagon |

Coxeter group | A_{8}, [3^{7}], order 362880 |

Properties | convex |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}_{8}. It is also called **0 _{6,1}** for its branching Coxeter-Dynkin diagram, shown as

### Coordinates

The Cartesian coordinates of the vertices of the *rectified 8-simplex* can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.

### Images

A_{k} Coxeter plane |
A_{8} |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|---|

Graph | ||||

Dihedral symmetry | [9] | [8] | [7] | [6] |

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

Graph | ||||

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

## Birectified 8-simplex

Birectified 8-simplex | |
---|---|

Type | uniform 8-polytope |

Coxeter symbol | 0_{52} |

Schläfli symbol | t_{2}{3^{7}}2r{3 ^{7}} = {3^{5,2}} or |

Coxeter-Dynkin diagrams | or |

7-faces | 18 |

6-faces | 144 |

5-faces | 588 |

4-faces | 1386 |

Cells | 2016 |

Faces | 1764 |

Edges | 756 |

Vertices | 84 |

Vertex figure | {3}×{3,3,3,3} |

Coxeter group | A_{8}, [3^{7}], order 362880 |

Properties | convex |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}_{8}. It is also called **0 _{5,2}** for its branching Coxeter-Dynkin diagram, shown as

The *birectified 8-simplex* is the vertex figure of the 1_{52} honeycomb.

### Coordinates

The Cartesian coordinates of the vertices of the *birectified 8-simplex* can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.

### Images

A_{k} Coxeter plane |
A_{8} |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|---|

Graph | ||||

Dihedral symmetry | [9] | [8] | [7] | [6] |

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

Graph | ||||

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

## Trirectified 8-simplex

Trirectified 8-simplex | |
---|---|

Type | uniform 8-polytope |

Coxeter symbol | 0_{43} |

Schläfli symbol | t_{3}{3^{7}}3r{3 ^{7}} = {3^{4,3}} or |

Coxeter-Dynkin diagrams | or |

7-faces | 9 + 9 |

6-faces | 36 + 72 + 36 |

5-faces | 84 + 252 + 252 + 84 |

4-faces | 126 + 504 + 756 + 504 |

Cells | 630 + 1260 + 1260 |

Faces | 1260 + 1680 |

Edges | 1260 |

Vertices | 126 |

Vertex figure | {3,3}×{3,3,3} |

Petrie polygon | enneagon |

Coxeter group | A_{7}, [3^{7}], order 362880 |

Properties | convex |

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{3}_{8}. It is also called **0 _{4,3}** for its branching Coxeter-Dynkin diagram, shown as

### Coordinates

The Cartesian coordinates of the vertices of the *trirectified 8-simplex* can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.

### Images

A_{k} Coxeter plane |
A_{8} |
A_{7} |
A_{6} |
A_{5} |
---|---|---|---|---|

Graph | ||||

Dihedral symmetry | [9] | [8] | [7] | [6] |

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

Graph | ||||

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

## Related polytopes

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 2_{61} honeycomb.

It is also one of 135 uniform 8-polytopes with A_{8} symmetry.

A8 polytopes | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

t _{0} |
t _{1} |
t _{2} |
t _{3} |
t _{01} |
t _{02} |
t _{12} |
t _{03} |
t _{13} |
t _{23} |
t _{04} |
t _{14} |
t _{24} |
t _{34} |
t _{05} |

t _{15} |
t _{25} |
t _{06} |
t _{16} |
t _{07} |
t _{012} |
t _{013} |
t _{023} |
t _{123} |
t _{014} |
t _{024} |
t _{124} |
t _{034} |
t _{134} |
t _{234} |

t _{015} |
t _{025} |
t _{125} |
t _{035} |
t _{135} |
t _{235} |
t _{045} |
t _{145} |
t _{016} |
t _{026} |
t _{126} |
t _{036} |
t _{136} |
t _{046} |
t _{056} |

t _{017} |
t _{027} |
t _{037} |
t _{0123} |
t _{0124} |
t _{0134} |
t _{0234} |
t _{1234} |
t _{0125} |
t _{0135} |
t _{0235} |
t _{1235} |
t _{0145} |
t _{0245} |
t _{1245} |

t _{0345} |
t _{1345} |
t _{2345} |
t _{0126} |
t _{0136} |
t _{0236} |
t _{1236} |
t _{0146} |
t _{0246} |
t _{1246} |
t _{0346} |
t _{1346} |
t _{0156} |
t _{0256} |
t _{1256} |

t _{0356} |
t _{0456} |
t _{0127} |
t _{0137} |
t _{0237} |
t _{0147} |
t _{0247} |
t _{0347} |
t _{0157} |
t _{0257} |
t _{0167} |
t _{01234} |
t _{01235} |
t _{01245} |
t _{01345} |

t _{02345} |
t _{12345} |
t _{01236} |
t _{01246} |
t _{01346} |
t _{02346} |
t _{12346} |
t _{01256} |
t _{01356} |
t _{02356} |
t _{12356} |
t _{01456} |
t _{02456} |
t _{03456} |
t _{01237} |

t _{01247} |
t _{01347} |
t _{02347} |
t _{01257} |
t _{01357} |
t _{02357} |
t _{01457} |
t _{01267} |
t _{01367} |
t _{012345} |
t _{012346} |
t _{012356} |
t _{012456} |
t _{013456} |
t _{023456} |

t _{123456} |
t _{012347} |
t _{012357} |
t _{012457} |
t _{013457} |
t _{023457} |
t _{012367} |
t _{012467} |
t _{013467} |
t _{012567} |
t _{0123456} |
t _{0123457} |
t _{0123467} |
t _{0123567} |
t _{01234567} |

## 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. "8D Uniform polytopes (polyzetta)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene