# Rectified 6-orthoplexes

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

6-orthoplex |
Rectified 6-orthoplex |
Birectified 6-orthoplex | |

Birectified 6-cube |
Rectified 6-cube |
6-cube | |

Orthogonal projections in B_{6} Coxeter plane |
---|

There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.

## Rectified 6-orthoplex

Rectified hexacross | |
---|---|

Type | uniform 6-polytope |

Schläfli symbols | t_{1}{3^{4},4} or r{3^{4},4}r{3,3,3,3 ^{1,1}} |

Coxeter-Dynkin diagrams | |

5-faces | 76 total: 64 rectified 5-simplex 12 5-orthoplex |

4-faces | 576 total: 192 rectified 5-cell 384 5-cell |

Cells | 1200 total: 240 octahedron 960 tetrahedron |

Faces | 1120 total: 160 and 960 triangles |

Edges | 480 |

Vertices | 60 |

Vertex figure | 16-cell prism |

Petrie polygon | Dodecagon |

Coxeter groups | B_{6}, [3,3,3,3,4]D _{6}, [3^{3,1,1}] |

Properties | convex |

The *rectified 6-orthoplex* is the vertex figure for the demihexeractic honeycomb.

or

### Alternate names

- rectified hexacross
- rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the *rectified hexacross*, one with the C_{6} or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D_{6} or [3^{3,1,1}] Coxeter group.

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:

- (±1,±1,0,0,0,0)

### Images

Coxeter plane | B_{6} |
B_{5} |
B_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [12] | [10] | [8] |

Coxeter plane | B_{3} |
B_{2} | |

Graph | |||

Dihedral symmetry | [6] | [4] | |

Coxeter plane | A_{5} |
A_{3} | |

Graph | |||

Dihedral symmetry | [6] | [4] |

### Root vectors

The 60 vertices represent the root vectors of the simple Lie group D_{6}. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B_{6} and C_{6} simple Lie groups.

The 60 roots of D_{6} can be geometrically folded into H_{3} (Icosahedral symmetry), as

Rectified 6-orthoplex | 2 icosidodecahedra | |
---|---|---|

3D (H3 projection) | A_{4}/B_{5}/D_{6} Coxeter plane |
H_{2} Coxeter plane |

## Birectified 6-orthoplex

Birectified 6-orthoplex | |
---|---|

Type | uniform 6-polytope |

Schläfli symbols | t_{2}{3^{4},4} or 2r{3^{4},4}t _{2}{3,3,3,3^{1,1}} |

Coxeter-Dynkin diagrams | |

5-faces | 76 |

4-faces | 636 |

Cells | 2160 |

Faces | 2880 |

Edges | 1440 |

Vertices | 160 |

Vertex figure | {3}×{3,4} duoprism |

Petrie polygon | Dodecagon |

Coxeter groups | B_{6}, [3,3,3,3,4]D _{6}, [3^{3,1,1}] |

Properties | convex |

The **birectified 6-orthoplex** can tessellation space in the trirectified 6-cubic honeycomb.

### Alternate names

- birectified hexacross
- birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:

- (±1,±1,±1,0,0,0)

### Images

Coxeter plane | B_{6} |
B_{5} |
B_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [12] | [10] | [8] |

Coxeter plane | B_{3} |
B_{2} | |

Graph | |||

Dihedral symmetry | [6] | [4] | |

Coxeter plane | A_{5} |
A_{3} | |

Graph | |||

Dihedral symmetry | [6] | [4] |

It can also be projected into 3D-dimensions as

## Related polytopes

These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.

## Notes

- Icosidodecahedron from D6 John Baez, January 1, 2015

## 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)". o3x3o3o3o4o - rag, o3o3x3o3o4o - brag