# 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.

 Orthogonal projections in A5 Coxeter plane 5-cube Rectified 5-cube Birectified 5-cubeBirectified 5-orthoplex 5-orthoplex Rectified 5-orthoplex

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-faces42
Cells200
Faces400
Edges 320
Vertices 80
Vertex figure
tetrahedral prism
Coxeter group B5, [4,33], 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 ${\sqrt {2}}$ is given by all permutations of:

$(0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)$ ### Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry   
Coxeter plane B2 A3
Graph
Dihedral symmetry  

## Birectified 5-cube

Birectified 5-cube
birectified penteract (nit)
Type uniform 5-polytope
Schläfli symbol 2r{4,3,3,3}
Coxeter diagram =
4-faces4210 {3,4,3}
32 t1{3,3,3}
Cells280
Faces640
Edges 480
Vertices 80
Vertex figure
{3}×{4}
Coxeter group B5, [4,33], order 3840
D5, [32,1,1], 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 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 birectifing 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 B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry   
Coxeter plane B2 A3
Graph
Dihedral symmetry  
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}

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

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