5-cube

Cantellated 5-cube

Bicantellated 5-cube

Cantellated 5-orthoplex

5-orthoplex

Cantitruncated 5-cube

Bicantitruncated 5-cube

Cantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

Cantellated 5-cube

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Cantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol rr{4,3,3,3} =  
Coxeter-Dynkin diagram           =        
4-faces 122 10          
80          
32          
Cells 680 40        
320        
160        
160        
Faces 1520 80      
480      
320      
640      
Edges 1280 320+960
Vertices 320
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex, uniform

Alternate names

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  • Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a cantellated 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]

Bicantellated 5-cube

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Bicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbols 2rr{4,3,3,3} =  
r{32,1,1} =  
Coxeter-Dynkin diagrams           =      
       
4-faces 122 10          
80          
32          
Cells 840 40        
240        
160        
320        
80        
Faces 2160 240      
320      
960      
320      
320      
Edges 1920 960+960
Vertices 480
Vertex figure  
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex, uniform

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

Alternate names

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  • Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
  • Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:

(0,1,1,2,2)

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]




Cantitruncated 5-cube

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Cantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr{4,3,3,3} =  
Coxeter-Dynkin
diagram
          =        
4-faces 122 10          
80          
32          
Cells 680 40        
320        
160        
160        
Faces 1520 80      
480      
320      
640      
Edges 1600 320+320+960
Vertices 640
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex, uniform

Alternate names

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  • Tricantitruncated 5-orthoplex / tricantitruncated pentacross
  • Great rhombated penteract (girn) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign 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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It is third in a series of cantitruncated hypercubes:

Petrie polygon projections
                 
Truncated cuboctahedron Cantitruncated tesseract Cantitruncated 5-cube Cantitruncated 6-cube Cantitruncated 7-cube Cantitruncated 8-cube
                                                                 

Bicantitruncated 5-cube

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Bicantitruncated 5-cube
Type uniform 5-polytope
Schläfli symbol 2tr{3,3,3,4} =  
t{32,1,1} =  
Coxeter-Dynkin diagrams           =      
       
4-faces 122 10          
80          
32          
Cells 840 40        
240        
160        
320        
80        
Faces 2160 240      
320      
960      
320      
320      
Edges 2400 960+480+960
Vertices 960
Vertex figure  
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex, uniform

Alternate names

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  • Bicantitruncated penteract
  • Bicantitruncated pentacross
  • Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

Coordinates

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Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

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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These polytopes are from a set of 31 uniform 5-polytopes 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

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, editied 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)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant
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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