Rectified 5-cubes
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| 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-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 Rectified tesseract 32 5-cell |
| Cells | 200 | 40 Cuboctahedron 160 Tetrahedron |
| Faces | 400 | 80 Square 320 Triangle |
| 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 2 {\displaystyle {\sqrt {2}}} is given by all permutations of:
( 0 , ± 1 , ± 1 , ± 1 , ± 1 ) {\displaystyle (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)}
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 24-cell 32 Rectified 5-cell |
| Cells | 280 | 40 Octahedron 160 Octahedron 80 Tetrahedron |
| Faces | 640 | 320 Triangle 320 Triangle |
| 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/triacontaditeron
- Penteractitriacontaditeron (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 2 {\displaystyle {\sqrt {2}}} of the edge length.
The Cartesian coordinates of the vertices of a birectified 5-cube having edge length2 are all permutations of:
( 0 , 0 , ± 1 , ± 1 , ± 1 ) {\displaystyle \left(0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1\right)}
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
- 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 Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN978-0-471-01003-6 (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. . o3x3o3o4o - rin, o3o3x3o4o - nit
External links
| vteFundamental convex regular and uniform polytopes in dimensions 2–10 | |||||
|---|---|---|---|---|---|
| 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 • Polytope operations |