# Rectified 5-cubes

| 5-cube | Rectified 5-cube | Birectified 5-cube Birectified 5-orthoplex |
| 5-orthoplex | Rectified 5-orthoplex | Birectified 5-cube Birectified 5-orthoplex |
| Orthogonal projections in A$_5$ 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.

## Rectified 5-cube

Rectified 5-cube rectified penteract (rin) :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | r{4,3,3,3} | r{4,3,3,3} |
| Coxeter diagram | = | = |
| 4-faces | 42 | 10 32 |
| Cells | 200 | 40 160 |
| Faces | 400 | 80 320 |
| Edges | 320 | 320 |
| Vertices | 80 | 80 |
| Vertex figure | Tetrahedral prism | Tetrahedral prism |
| Coxeter group | B$_5$, [4,3$^3$], order 3840 | B$_5$, [4,3$^3$], order 3840 |
| Dual | | |
| Base point | (0,1,1,1,1,1)√2 | (0,1,1,1,1,1)√2 |
| Circumradius | sqrt(2) = 1.414214 | sqrt(2) = 1.414214 |
| Properties | convex, isogonal | 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 ${\sqrt {2}}$ is given by all permutations of:
    $(0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)$

### Images

orthographic projections :
| 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 | uniform 5-polytope |
| Schläfli symbol | 2r{4,3,3,3} | 2r{4,3,3,3} |
| Coxeter diagram | = | = |
| 4-faces | 42 | 10 32 |
| Cells | 280 | 40 160 80 |
| Faces | 640 | 320 320 |
| Edges | 480 | 480 |
| Vertices | 80 | 80 |
| Vertex figure | {3}×{4} | {3}×{4} |
| Coxeter group | B$_5$, [4,3$^3$], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 | B$_5$, [4,3$^3$], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 |
| Dual | | |
| Base point | (0,0,1,1,1,1)√2 | (0,0,1,1,1,1)√2 |
| Circumradius | sqrt(3/2) = 1.224745 | sqrt(3/2) = 1.224745 |
| Properties | convex, isogonal | 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 birectifying the vertices of the 5-cube at ${\sqrt {2}}$ of the edge length.
The Cartesian coordinates of the vertices of a birectified 5-cube having edge length 2 are all permutations of:
    $\left(0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1\right)$

### Images

orthographic projections :
| 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

2-isotopic hypercubes :
| 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.
B5 polytopes :
| β$_5$ | t$_1$β$_5$ | t$_2$γ$_5$ | t$_1$γ$_5$ | γ$_5$ | t$_{0,1}$β$_5$ | t$_{0,2}$β$_5$ | t$_{1,2}$β$_5$ |
| t$_{0,3}$β$_5$ | t$_{1,3}$γ$_5$ | t$_{1,2}$γ$_5$ | t$_{0,4}$γ$_5$ | t$_{0,3}$γ$_5$ | t$_{0,2}$γ$_5$ | t$_{0,1}$γ$_5$ | t$_{0,1,2}$β$_5$ |
| t$_{0,1,3}$β$_5$ | t$_{0,2,3}$β$_5$ | t$_{1,2,3}$γ$_5$ | t$_{0,1,4}$β$_5$ | t$_{0,2,4}$γ$_5$ | t$_{0,2,3}$γ$_5$ | t$_{0,1,4}$γ$_5$ | t$_{0,1,3}$γ$_5$ |
| t$_{0,1,2}$γ$_5$ | t$_{0,1,2,3}$β$_5$ | t$_{0,1,2,4}$β$_5$ | t$_{0,1,3,4}$γ$_5$ | t$_{0,1,2,4}$γ$_5$ | t$_{0,1,2,3}$γ$_5$ | t$_{0,1,2,3,4}$γ$_5$ |