# Truncated 7-orthoplexes

In seven-dimensional geometry, a **truncated 7-orthoplex** is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

7-orthoplex |
Truncated 7-orthoplex |
Bitruncated 7-orthoplex |
Tritruncated 7-orthoplex |

7-cube |
Truncated 7-cube |
Bitruncated 7-cube |
Tritruncated 7-cube |

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

There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

## Truncated 7-orthoplex

Truncated 7-orthoplex | |
---|---|

Type | uniform 7-polytope |

Schläfli symbol | t{3^{5},4} |

Coxeter-Dynkin diagrams | |

6-faces | |

5-faces | |

4-faces | |

Cells | 3920 |

Faces | 2520 |

Edges | 924 |

Vertices | 168 |

Vertex figure | ( )v{3,3,4} |

Coxeter groups | B_{7}, [3^{5},4]D _{7}, [3^{4,1,1}] |

Properties | convex |

### Alternate names

- Truncated heptacross
- Truncated hecatonicosoctaexon (Jonathan Bowers)[1]

### Coordinates

Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of

- (±2,±1,0,0,0,0,0)

### Images

Coxeter plane | B_{7} / A_{6} |
B_{6} / D_{7} |
B_{5} / D_{6} / A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [14] | [12] | [10] |

Coxeter plane | B_{4} / D_{5} |
B_{3} / D_{4} / A_{2} |
B_{2} / D_{3} |

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

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

Graph | |||

Dihedral symmetry | [6] | [4] |

### Construction

There are two Coxeter groups associated with the *truncated 7-orthoplex*, one with the C_{7} or [4,3^{5}] Coxeter group, and a lower symmetry with the D_{7} or [3^{4,1,1}] Coxeter group.

## Bitruncated 7-orthoplex

Bitruncated 7-orthoplex | |
---|---|

Type | uniform 7-polytope |

Schläfli symbol | 2t{3^{5},4} |

Coxeter-Dynkin diagrams | |

6-faces | |

5-faces | |

4-faces | |

Cells | |

Faces | |

Edges | 4200 |

Vertices | 840 |

Vertex figure | { }v{3,3,4} |

Coxeter groups | B_{7}, [3^{5},4]D _{7}, [3^{4,1,1}] |

Properties | convex |

### Alternate names

- Bitruncated heptacross
- Bitruncated hecatonicosoctaexon (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

- (±2,±2,±1,0,0,0,0)

### Images

Coxeter plane | B_{7} / A_{6} |
B_{6} / D_{7} |
B_{5} / D_{6} / A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [14] | [12] | [10] |

Coxeter plane | B_{4} / D_{5} |
B_{3} / D_{4} / A_{2} |
B_{2} / D_{3} |

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

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

Graph | |||

Dihedral symmetry | [6] | [4] |

## Tritruncated 7-orthoplex

The **tritruncated 7-orthoplex** can tessellation space in the quadritruncated 7-cubic honeycomb.

Tritruncated 7-orthoplex | |
---|---|

Type | uniform 7-polytope |

Schläfli symbol | 3t{3^{5},4} |

Coxeter-Dynkin diagrams | |

6-faces | |

5-faces | |

4-faces | |

Cells | |

Faces | |

Edges | 10080 |

Vertices | 2240 |

Vertex figure | {3}v{3,4} |

Coxeter groups | B_{7}, [3^{5},4]D _{7}, [3^{4,1,1}] |

Properties | convex |

### Alternate names

- Tritruncated heptacross
- Tritruncated hecatonicosoctaexon (Jonathan Bowers)[3]

### Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of

- (±2,±2,±2,±1,0,0,0)

### Images

Coxeter plane | B_{7} / A_{6} |
B_{6} / D_{7} |
B_{5} / D_{6} / A_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [14] | [12] | [10] |

Coxeter plane | B_{4} / D_{5} |
B_{3} / D_{4} / A_{2} |
B_{2} / D_{3} |

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

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

Graph | |||

Dihedral symmetry | [6] | [4] |

## Notes

- Klitzing, (x3x3o3o3o3o4o - tez)
- Klitzing, (o3x3x3o3o3o4o - botaz)
- Klitzing, (o3o3x3x3o3o4o - totaz)

## 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)". x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz