# Rectified 5-cubes

(Redirected from Rectified demipenteract)
 Orthogonal projections in A5 Coxeter plane 5-cube Rectified 5-cube Birectified 5-cube Birectified 5-orthoplex 5-orthoplex Rectified 5-orthoplex

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-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 BC5, [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:

$(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)$

### Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
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 duoprism
Coxeter group BC5, [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

### 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:

$\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right)$

### Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

### Related polytopes

2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8
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

Rectangle

Disphenoid

{3}×{4} duoprism
{3,3}×{3,4} duoprism

## Related polytopes

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

 β5 t1β5 t2γ5 t1γ5 γ5 t0,1β5 t0,2β5 t1,2β5 t0,3β5 t1,3γ5 t1,2γ5 t0,4γ5 t0,3γ5 t0,2γ5 t0,1γ5 t0,1,2β5 t0,1,3β5 t0,2,3β5 t1,2,3γ5 t0,1,4β5 t0,2,4γ5 t0,2,3γ5 t0,1,4γ5 t0,1,3γ5 t0,1,2γ5 t0,1,2,3β5 t0,1,2,4β5 t0,1,3,4γ5 t0,1,2,4γ5 t0,1,2,3γ5 t0,1,2,3,4γ5

## 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.
• Richard Klitzing, 5D, uniform polytopes (polytera) o3x3o3o4o - rin, o3o3x3o4o - nit