Rectified_penteract
In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (October 2022) |
 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 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 
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 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
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 |
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| Images |  |
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| 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.
- 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
- (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
Fundamental convex regular and uniform polytopes in dimensions 2–10
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|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 polychoron | Pentachoron | 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 | ||||||||||||