5-cube

Runcinated 5-cube

Runcinated 5-orthoplex

Runcitruncated 5-cube

Runcicantellated 5-cube

Runcicantitruncated 5-cube

Runcitruncated 5-orthoplex

Runcicantellated 5-orthoplex

Runcicantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Runcinated 5-cube

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Runcinated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,3{4,3,3,3}
Coxeter diagram          
4-faces 202 10          
80          
80          
32          
Cells 1240 40        
240        
320        
160        
320        
160        
Faces 2160 240      
960      
640      
320      
Edges 1440 480+960
Vertices 320
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

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  • Small prismated penteract (Acronym: span) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a runcinated 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]




Runcitruncated 5-cube

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Runcitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams          
4-faces 202 10          
80          
80          
32          
Cells 1560 40        
240        
320        
320        
160        
320        
160        
Faces 3760 240      
960      
320      
960      
640      
640      
Edges 3360 480+960+1920
Vertices 960
Vertex figure  
Coxeter group B5, [3,3,3,4]
Properties convex

Alternate names

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  • Runcitruncated penteract
  • Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates

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The Cartesian coordinates of the vertices of a runcitruncated 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]





Runcicantellated 5-cube

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Runcicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram          
4-faces 202 10          
80          
80          
32          
Cells 1240 40        
240        
320        
320        
160        
160        
Faces 2960 240      
480      
960      
320      
640      
320      
Edges 2880 960+960+960
Vertices 960
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

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  • Runcicantellated penteract
  • Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a runcicantellated 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]





Runcicantitruncated 5-cube

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Runcicantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{4,3,3,3}
Coxeter-Dynkin
diagram
         
4-faces 202
Cells 1560
Faces 4240
Edges 4800
Vertices 1920
Vertex figure  
Irregular 5-cell
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

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  • Runcicantitruncated penteract
  • Biruncicantitruncated pentacross
  • great prismated penteract (gippin) (Jonathan Bowers)

Coordinates

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The Cartesian coordinates of the vertices of a runcicantitruncated 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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These polytopes are a part of a set 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

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)". o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
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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