# Truncated 5-orthoplexes

In six-dimensional geometry, a **truncated 5-orthoplex** is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

5-orthoplex |
Truncated 5-orthoplex |
Bitruncated 5-orthoplex | |

5-cube |
Truncated 5-cube |
Bitruncated 5-cube | |

Orthogonal projections in B_{5} Coxeter plane |
---|

There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

## Truncated 5-orthoplex

Truncated 5-orthoplex | |
---|---|

Type | uniform 5-polytope |

Schläfli symbol | t{3,3,3,4} t{3,3 ^{1,1}} |

Coxeter-Dynkin diagrams | |

4-faces | 42 |

Cells | 240 |

Faces | 400 |

Edges | 280 |

Vertices | 80 |

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

Coxeter groups | B_{5}, [3,3,3,4]D _{5}, [3^{2,1,1}] |

Properties | convex |

### Alternate names

- Truncated pentacross
- Truncated triacontiditeron (Acronym: tot) (Jonathan Bowers)[1]

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

- (±2,±1,0,0,0)

### Images

The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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

Graph | |||

Dihedral symmetry | [10] | [8] | [6] |

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

Graph | |||

Dihedral symmetry | [4] | [4] |

## Bitruncated 5-orthoplex

Bitruncated 5-orthoplex | |
---|---|

Type | uniform 5-polytope |

Schläfli symbol | 2t{3,3,3,4} 2t{3,3 ^{1,1}} |

Coxeter-Dynkin diagrams | |

4-faces | 42 |

Cells | 280 |

Faces | 720 |

Edges | 720 |

Vertices | 240 |

Vertex figure | { }v{4} |

Coxeter groups | B_{5}, [3,3,3,4]D _{5}, [3^{2,1,1}] |

Properties | convex |

The **bitruncated 5-orthoplex** can tessellate space in the tritruncated 5-cubic honeycomb.

### Alternate names

- Bitruncated pentacross
- Bitruncated triacontiditeron (acronym: gart) (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of

- (±2,±2,±1,0,0)

### Images

The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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

Graph | |||

Dihedral symmetry | [10] | [8] | [6] |

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

Graph | |||

Dihedral symmetry | [4] | [4] |

## Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

## Notes

- Klitzing, (x3x3o3o4o - tot)
- Klitzing, (x3x3x3o4o - gart)

## 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. "5D uniform polytopes (polytera)". x3x3o3o4o - tot, x3x3x3o4o - gart

## External links

- Weisstein, Eric W. "Hypercube".
*MathWorld*. - Polytopes of Various Dimensions
- Multi-dimensional Glossary