Rectified 5-cubes
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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 A5 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-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 | |
| Schläfli symbol | r{4,3,3,3} | |
| Coxeter diagram | =       | |
| 4-faces | 42 | 10  32  |
| Cells | 200 | 40  160  |
| Faces | 400 | 80  320  |
| 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 is given by all permutations of:
Images
| 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  32  |
| Cells | 280 | 40  160  80 |
| Faces | 640 | 320 320 |
| 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 birectifying 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 | B5 | B4 / D5 | B3 / D4 / A2 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B2 | A3 | |
| 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 Archived 2016-07-11 at the Wayback Machine
- (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)". o3x3o3o4o - rin, o3o3x3o4o - nit