Pentellated 6-simplexes


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

edit
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

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

Cross-sections

edit

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

edit

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

edit

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

Images

edit
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

edit

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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

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

Coordinates

edit

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

edit
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

edit
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

edit
  • Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
  • Omnitruncated heptapeton
  • Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)[11]
edit

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

edit

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

edit
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

edit

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

edit

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.

edit

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

edit
  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

edit
  • 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
edit
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