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5-cube |
Rectified 5-cube |
Birectified 5-cube Birectified 5-orthoplex | ||
5-orthoplex |
Rectified 5-orthoplex | |||
Orthogonal projections in A5 Coxeter plane |
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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.
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
editRectified 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
edit- Rectified penteract (acronym: rin) (Jonathan Bowers)
Construction
editThe rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.
Coordinates
editThe Cartesian coordinates of the vertices of the rectified 5-cube with edge length is given by all permutations of:
Images
editCoxeter 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
editBirectified 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
edit- Birectified 5-cube/penteract
- Birectified pentacross/5-orthoplex/triacontiditeron
- Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
- Rectified 5-demicube/demipenteract
Construction and coordinates
editThe 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
editCoxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | |||
Dihedral symmetry | [4] | [4] |
Related polytopes
editDim. | 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
editThese polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
Notes
editReferences
edit- 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.
- Klitzing, Richard. "5D uniform polytopes (polytera)". o3x3o3o4o - rin, o3o3x3o4o - nit