Stericated 5-cubes

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5-cube

Stericated 5-cube

Steritruncated 5-cube

Stericantellated 5-cube

Steritruncated 5-orthoplex

Stericantitruncated 5-cube

Steriruncitruncated 5-cube

Stericantitruncated 5-orthoplex

Omnitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.

There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the sterirunci​cantitruncated​ 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

Stericated 5-cube

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Stericated 5-cube
Type Uniform 5-polytope
Schläfli symbol 2r2r{4,3,3,3}
Coxeter-Dynkin diagram          
     
4-faces 242
Cells 800
Faces 1040
Edges 640
Vertices 160
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

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  • Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
  • Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
  • Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)[1]

Coordinates

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

 

Images

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The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

Dissections

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The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.

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]

Steritruncated 5-cube

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Steritruncated 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,4{4,3,3,3}
Coxeter-Dynkin diagrams          
4-faces 242
Cells 1600
Faces 2960
Edges 2240
Vertices 640
Vertex figure  
Coxeter groups B5, [3,3,3,4]
Properties convex

Alternate names

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  • Steritruncated penteract
  • Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)[2]

Construction and coordinates

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The Cartesian coordinates of the vertices of a steritruncated 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]

Stericantellated 5-cube

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Stericantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,4{4,3,3,3}
Coxeter-Dynkin diagram          
     
4-faces 242
Cells 2080
Faces 4720
Edges 3840
Vertices 960
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

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  • Stericantellated penteract
  • Stericantellated 5-orthoplex, stericantellated pentacross
  • Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)[3]

Coordinates

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The Cartesian coordinates of the vertices of a stericantellated 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]

Stericantitruncated 5-cube

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Stericantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,4{4,3,3,3}
Coxeter-Dynkin
diagram
         
4-faces 242
Cells 2400
Faces 6000
Edges 5760
Vertices 1920
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

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  • Stericantitruncated penteract
  • Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
  • Celligreatorhombated penteract (cogrin) (Jonathan Bowers)[4]

Coordinates

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The Cartesian coordinates of the vertices of a stericantitruncated 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]

Steriruncitruncated 5-cube

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Steriruncitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol 2t2r{4,3,3,3}
Coxeter-Dynkin
diagram
         
     
4-faces 242
Cells 2160
Faces 5760
Edges 5760
Vertices 1920
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

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  • Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
  • Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)[5]

Coordinates

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The Cartesian coordinates of the vertices of a steriruncitruncated penteract 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]

Steritruncated 5-orthoplex

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Steritruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,4}
Coxeter-Dynkin diagrams          
4-faces 242
Cells 1520
Faces 2880
Edges 2240
Vertices 640
Vertex figure  
Coxeter group B5, [3,3,3,4]
Properties convex

Alternate names

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  • Steritruncated pentacross
  • Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)[6]

Coordinates

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Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, 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]

Stericantitruncated 5-orthoplex

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Stericantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3,4{4,3,3,3}
Coxeter-Dynkin
diagram
         
4-faces 242
Cells 2320
Faces 5920
Edges 5760
Vertices 1920
Vertex figure  
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

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  • Stericantitruncated pentacross
  • Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)[7]

Coordinates

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The Cartesian coordinates of the vertices of a stericantitruncated 5-orthoplex 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]

Omnitruncated 5-cube

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Omnitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr2r{4,3,3,3}
Coxeter-Dynkin
diagram
         
     
4-faces 242
Cells 2640
Faces 8160
Edges 9600
Vertices 3840
Vertex figure  
irr. {3,3,3}
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

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  • Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
  • Omnitruncated penteract
  • Omnitruncated triacontiditeron / omnitruncated pentacross
  • Great cellated penteractitriacontiditeron (Jonathan Bowers)[8]

Coordinates

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The Cartesian coordinates of the vertices of an omnitruncated 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]

Full snub 5-cube

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The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram           and symmetry [4,3,3,3]+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.

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This polytope is one 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

Notes

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  1. ^ Klitzing, (x3o3o3o4x - scant)
  2. ^ Klitzing, (x3o3o3x4x - capt)
  3. ^ Klitzing, (x3o3x3o4x - carnit)
  4. ^ Klitzing, (x3o3x3x4x - cogrin)
  5. ^ Klitzing, (x3x3o3x4x - captint)
  6. ^ Klitzing, (x3x3o3o4x - cappin)
  7. ^ Klitzing, (x3x3x3o4x - cogart)
  8. ^ Klitzing, (x3x3x3x4x - gacnet)

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)". x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart
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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