# Rectified 5-cubes

In five-dimensional geometry, a **rectified 5-cube** is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

5-cube |
Rectified 5-cube |
Birectified 5-cube Birectified 5-orthoplex | ||

5-orthoplex |
Rectified 5-orthoplex | |||

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

There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-ocube are located in the square face centers of the 5-cube.

## Rectified 5-cube

Rectified 5-cube rectified penteract (rin) | ||
---|---|---|

Type | uniform 5-polytope | |

Schläfli symbol | r{4,3,3,3} | |

Coxeter diagram | ||

4-faces | 42 | |

Cells | 200 | |

Faces | 400 | |

Edges | 320 | |

Vertices | 80 | |

Vertex figure | tetrahedral prism | |

Coxeter group | B_{5}, [4,3^{3}], order 3840 | |

Dual | ||

Base point | (0,1,1,1,1,1)√2 | |

Circumradius | sqrt(2) = 1.414214 | |

Properties | convex, isogonal |

### Alternate names

- Rectified penteract (acronym: rin) (Jonathan Bowers)

### Construction

The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

### Coordinates

The Cartesian coordinates of the vertices of the rectified 5-cube with edge length is given by all permutations of:

### Images

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] |

## Birectified 5-cube

Birectified 5-cube birectified penteract (nit) | ||
---|---|---|

Type | uniform 5-polytope | |

Schläfli symbol | 2r{4,3,3,3} | |

Coxeter diagram | ||

4-faces | 42 | 10 {3,4,3} 32 t1{3,3,3} |

Cells | 280 | |

Faces | 640 | |

Edges | 480 | |

Vertices | 80 | |

Vertex figure | {3}×{4} | |

Coxeter group | B_{5}, [4,3^{3}], order 3840D _{5}, [3^{2,1,1}], order 1920 | |

Dual | ||

Base point | (0,0,1,1,1,1)√2 | |

Circumradius | sqrt(3/2) = 1.224745 | |

Properties | convex, isogonal |

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr_{5}^{2} as a second rectification of a 5-dimensional cross polytope.

### Alternate names

- Birectified 5-cube/penteract
- Birectified pentacross/5-orthoplex/triacontiditeron
- Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
- Rectified 5-demicube/demipenteract

### Construction and coordinates

The *birectified 5-cube* may be constructed by birectifing the vertices of the 5-cube at of the edge length.

The Cartesian coordinates of the vertices of a *birectified 5-cube* having edge length 2 are all permutations of:

### Images

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

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

Name | t{4} | r{4,3} | 2t{4,3,3} | 2r{4,3,3,3} | 3t{4,3,3,3,3} | 3r{4,3,3,3,3,3} | 4t{4,3,3,3,3,3,3} | ... |

Coxeter diagram |
||||||||

Images | ||||||||

Facets | {3} {4} |
t{3,3} t{3,4} |
r{3,3,3} r{3,3,4} |
2t{3,3,3,3} 2t{3,3,3,4} |
2r{3,3,3,3,3} 2r{3,3,3,3,4} |
3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4} | ||

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

## Related polytopes

These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

## 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. "5D uniform polytopes (polytera)". o3x3o3o4o - rin, o3o3x3o4o - nit