Rectified 5-cubes

5-cube

Rectified 5-cube

Birectified 5-cube
Birectified 5-orthoplex

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5-orthoplex

Rectified 5-orthoplex

Orthogonal projections in A5 Coxeter plane

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

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-cube are located in the square face centers of the 5-cube.

## Contents

* 1 Rectified 5-cube
* 1.1 Alternate names
* 1.2 Construction
* 1.3 Coordinates
* 1.4 Images
* 2 Birectified 5-cube
* 2.1 Alternate names
* 2.2 Construction and coordinates
* 2.3 Images
* 2.4 Related polytopes
* 3 Related polytopes
* 4 Notes
* 5 References
* 6 External links

## Rectified 5-cube

### 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 [math]\displaystyle{ \sqrt{2} }[/math] is given by all permutations of:

[math]\displaystyle{ (0,\ \pm1,\ \pm1,\ \pm1,\ \pm1) }[/math]

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

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Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Birectified 5-cube

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 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 birectifying the vertices of the 5-cube at [math]\displaystyle{ \sqrt{2} }[/math] of the edge length.

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

[math]\displaystyle{ \left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right) }[/math]

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

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Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

### Related polytopes

## 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 [1]
* (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]
* Norman Johnson Uniform Polytopes, Manuscript (1991)
* N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
* Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm. o3x3o3o4o - rin, o3o3x3o4o - nit

## External links

* Polytopes of Various Dimensions
* Multi-dimensional Glossary

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* t
* e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube |  | Dodecahedron • Icosahedron
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube |  |
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 |
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 |
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 |
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube |  |
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube |  |
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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