5-cube

Rectified 5-cube

Birectified 5-cube
Birectified 5-orthoplex

5-orthoplex

Rectified 5-orthoplex
Orthogonal projections in A5 Coxeter plane

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

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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

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  • Rectified penteract (acronym: rin) (Jonathan Bowers)

Construction

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The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

Coordinates

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The Cartesian coordinates of the vertices of the rectified 5-cube with edge length   is given by all permutations of:

 

Images

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orthographic projections
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

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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

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  • Birectified 5-cube/penteract
  • Birectified pentacross/5-orthoplex/triacontiditeron
  • Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
  • Rectified 5-demicube/demipenteract

Construction and coordinates

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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

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orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph      
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph    
Dihedral symmetry [4] [4]
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2-isotopic hypercubes
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}
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These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

B5 polytopes
 
β5
 
t1β5
 
t2γ5
 
t1γ5
 
γ5
 
t0,1β5
 
t0,2β5
 
t1,2β5
 
t0,3β5
 
t1,3γ5
 
t1,2γ5
 
t0,4γ5
 
t0,3γ5
 
t0,2γ5
 
t0,1γ5
 
t0,1,2β5
 
t0,1,3β5
 
t0,2,3β5
 
t1,2,3γ5
 
t0,1,4β5
 
t0,2,4γ5
 
t0,2,3γ5
 
t0,1,4γ5
 
t0,1,3γ5
 
t0,1,2γ5
 
t0,1,2,3β5
 
t0,1,2,4β5
 
t0,1,3,4γ5
 
t0,1,2,4γ5
 
t0,1,2,3γ5
 
t0,1,2,3,4γ5

Notes

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References

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  • 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
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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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds