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.
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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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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
Summarize
Perspective
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 | ![]() |
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Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B2 | A3 | |
Graph | ![]() |
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Dihedral symmetry | [4] | [4] |
Birectified 5-cube
Summarize
Perspective
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 |
![]() ![]() |
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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} |
Related polytopes
These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
Notes
References
External links
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