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Pentellated 6-simplexes

(Redirected from Expanded 6-simplex)

6-simplex
Pentellated 6-simplex
Pentitruncated 6-simplex
Penticantellated 6-simplex

Penticantitruncated 6-simplex
Pentiruncitruncated 6-simplex
Pentiruncicantellated 6-simplex
Pentiruncicantitruncated 6-simplex

Pentisteritruncated 6-simplex
Pentistericantitruncated 6-simplex
Pentisteriruncicantitruncated 6-simplex
(Omnitruncated 6-simplex)
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed.

Pentellated 6-simplex

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Pentellated 6-simplex
Type Uniform 6-polytope
Schläfli symbol t0,5{3,3,3,3,3}
Coxeter-Dynkin diagram            
5-faces 126:
7+7 {34}  
21+21 {}×{3,3,3}
35+35 {3}×{3,3}
4-faces 434
Cells 630
Faces 490
Edges 210
Vertices 42
Vertex figure 5-cell antiprism
Coxeter group A6×2, [[3,3,3,3,3]], order 10080
Properties convex

Alternate names

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  • Expanded 6-simplex
  • Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)[1]

Cross-sections

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The maximal cross-section of the pentellated 6-simplex with a 5-dimensional hyperplane is a stericated hexateron. This cross-section divides the pentellated 6-simplex into two hexateral hypercupolas consisting of 7 5-simplexes, 21 5-cell prisms and 35 Tetrahedral-Triangular duoprisms each.

Coordinates

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The vertices of the pentellated 6-simplex can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex.

A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0)

Root vectors

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Its 42 vertices represent the root vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph    
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Configuration

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This configuration matrix represents the expanded 6-simplex, with 12 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[2]

Element fk f0 f1 f2 f3 f4 f5
            f0 42 10 20 20 20 60 10 40 30 2 10 20
              f1 2 210 4 4 6 18 4 16 12 1 5 10
              f2 3 3 280 * 3 3 3 6 3 1 3 4
            4 4 * 210 0 6 0 6 6 0 2 6
              f3 4 6 4 0 210 * 2 2 0 1 2 1
              6 9 2 3 * 420 0 2 2 0 1 3
              f4 5 10 10 0 5 0 84 * * 1 1 0
              8 16 8 6 2 4 * 210 * 0 1 1
            9 18 6 9 0 6 * * 140 0 0 2
              f5 6 15 20 0 15 0 6 0 0 14 * *
              10 25 20 10 10 10 2 5 0 * 42 *
              12 30 16 18 3 18 0 3 4 * * 70

Pentitruncated 6-simplex

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Pentitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 826
Cells 1785
Faces 1820
Edges 945
Vertices 210
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)[3]

Coordinates

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The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Penticantellated 6-simplex

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Penticantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1246
Cells 3570
Faces 4340
Edges 2310
Vertices 420
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)[4]

Coordinates

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The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the penticantellated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Penticantitruncated 6-simplex

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penticantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1351
Cells 4095
Faces 5390
Edges 3360
Vertices 840
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)[5]

Coordinates

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The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Pentiruncitruncated 6-simplex

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pentiruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1491
Cells 5565
Faces 8610
Edges 5670
Vertices 1260
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)[6]

Coordinates

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The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the pentiruncitruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Pentiruncicantellated 6-simplex

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Pentiruncicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1596
Cells 5250
Faces 7560
Edges 5040
Vertices 1260
Vertex figure
Coxeter group A6, [[3,3,3,3,3]], order 10080
Properties convex

Alternate names

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  • Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)[7]

Coordinates

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The vertices of the pentiruncicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph    
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Pentiruncicantitruncated 6-simplex

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Pentiruncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1701
Cells 6825
Faces 11550
Edges 8820
Vertices 2520
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)[8]

Coordinates

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The vertices of the pentiruncicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Pentisteritruncated 6-simplex

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Pentisteritruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1176
Cells 3780
Faces 5250
Edges 3360
Vertices 840
Vertex figure
Coxeter group A6, [[3,3,3,3,3]], order 10080
Properties convex

Alternate names

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  • Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)[9]

Coordinates

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The vertices of the pentisteritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,2,3,4). This construction is based on facets of the pentisteritruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph    
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Pentistericantitruncated 6-simplex

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pentistericantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 126
4-faces 1596
Cells 6510
Faces 11340
Edges 8820
Vertices 2520
Vertex figure
Coxeter group A6, [3,3,3,3,3], order 5040
Properties convex

Alternate names

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  • Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)[10]

Coordinates

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The vertices of the pentistericantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Omnitruncated 6-simplex

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Omnitruncated 6-simplex
Type Uniform 6-polytope
Schläfli symbol t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams            
5-faces 126:
14 t0,1,2,3,4{34} 
42 {}×t0,1,2,3{33}  × 
70 {6}×t0,1,2{3,3}  × 
4-faces 1806
Cells 8400
Faces 16800:
4200 {6}  
1260 {4} 
Edges 15120
Vertices 5040
Vertex figure  
irregular 5-simplex
Coxeter group A6, [[35]], order 10080
Properties convex, isogonal, zonotope

The omnitruncated 6-simplex has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex.

Alternate names

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  • Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
  • Omnitruncated heptapeton
  • Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)[11]
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The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. It has Coxeter-Dynkin diagram of        .

Coordinates

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The vertices of the omnitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 7-orthoplex, t0,1,2,3,4,5{35,4},              .

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph    
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.


Configuration

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This configuration matrix represents the omnitruncated 6-simplex, with 35 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[12]

Full snub 6-simplex

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The full snub 6-simplex or omnisnub 6-simplex, defined as an alternation of the omnitruncated 6-simplex is not uniform, but it can be given Coxeter diagram             and symmetry [[3,3,3,3,3]]+, and constructed from 14 snub 5-simplexes, 42 snub 5-cell antiprisms, 70 3-s{3,4} duoantiprisms, and 2520 irregular 5-simplexes filling the gaps at the deleted vertices.

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The pentellated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

A6 polytopes
 
t0 		 
t1 		 
t2 		 
t0,1 		 
t0,2 		 
t1,2 		 
t0,3 		 
t1,3 		 
t2,3
 
t0,4 		 
t1,4 		 
t0,5 		 
t0,1,2 		 
t0,1,3 		 
t0,2,3 		 
t1,2,3 		 
t0,1,4 		 
t0,2,4
 
t1,2,4 		 
t0,3,4 		 
t0,1,5 		 
t0,2,5 		 
t0,1,2,3 		 
t0,1,2,4 		 
t0,1,3,4 		 
t0,2,3,4 		 
t1,2,3,4
 
t0,1,2,5 		 
t0,1,3,5 		 
t0,2,3,5 		 
t0,1,4,5 		 
t0,1,2,3,4 		 
t0,1,2,3,5 		 
t0,1,2,4,5 		 
t0,1,2,3,4,5

Notes

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  1. ^ Klitzing, (x3o3o3o3o3x - staf)
  2. ^ "Staf".
  3. ^ Klitzing, (x3x3o3o3o3x - tocal)
  4. ^ Klitzing, (x3o3x3o3o3x - topal)
  5. ^ Klitzing, (x3x3x3o3o3x - togral)
  6. ^ Klitzing, (x3x3o3x3o3x - tocral)
  7. ^ Klitzing, (x3o3x3x3o3x - taporf)
  8. ^ Klitzing, (x3x3x3o3x3x - tagopal)
  9. ^ Klitzing, (x3x3o3o3x3x - tactaf)
  10. ^ Klitzing, (x3x3x3o3x3x - gatocral)
  11. ^ Klitzing, (x3x3x3x3x3x - gotaf)
  12. ^ "Gotaf".

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. "6D uniform polytopes (polypeta)". x3o3o3o3o3x - staf, x3x3o3o3o3x - tocal, x3o3x3o3o3x - topal, x3x3x3o3o3x - togral, x3x3o3x3o3x - tocral, x3x3x3x3o3x - tagopal, x3x3o3o3x3x - tactaf, x3x3x3o3x3x - tacogral, x3x3x3x3x3x - gotaf
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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