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

Pentellated 6-simplex
TypeUniform 6-polytope
Schläfli symbolt0,5{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces126:
7+7 {34}
21+21 {}×{3,3,3}
35+35 {3}×{3,3}
4-faces434
Cells630
Faces490
Edges210
Vertices42
Vertex figure5-cell antiprism
Coxeter groupA6×2, [[3,3,3,3,3]], order 10080
Propertiesconvex

Alternate names

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

Coordinates

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

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

Images

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

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]

Elementfk 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

Pentitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces826
Cells1785
Faces1820
Edges945
Vertices210
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

Penticantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1246
Cells3570
Faces4340
Edges2310
Vertices420
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

penticantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1351
Cells4095
Faces5390
Edges3360
Vertices840
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

pentiruncitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1491
Cells5565
Faces8610
Edges5670
Vertices1260
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

Pentiruncicantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1596
Cells5250
Faces7560
Edges5040
Vertices1260
Vertex figure
Coxeter groupA6, [[3,3,3,3,3]], order 10080
Propertiesconvex

Alternate names

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

Coordinates

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

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

Pentiruncicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1701
Cells6825
Faces11550
Edges8820
Vertices2520
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

Pentisteritruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1176
Cells3780
Faces5250
Edges3360
Vertices840
Vertex figure
Coxeter groupA6, [[3,3,3,3,3]], order 10080
Propertiesconvex

Alternate names

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

Coordinates

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

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

pentistericantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces126
4-faces1596
Cells6510
Faces11340
Edges8820
Vertices2520
Vertex figure
Coxeter groupA6, [3,3,3,3,3], order 5040
Propertiesconvex

Alternate names

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

Coordinates

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

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

Omnitruncated 6-simplex
Type Uniform 6-polytope
Schläfli symbolt0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams
5-faces126:
14 t0,1,2,3,4{34}
42 {}×t0,1,2,3{33} ×
70 {6}×t0,1,2{3,3} ×
4-faces1806
Cells8400
Faces16800:
4200 {6}
1260 {4}
Edges15120
Vertices5040
Vertex figure
irregular 5-simplex
Coxeter groupA6, [[35]], order 10080
Propertiesconvex, 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

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

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

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

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

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

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.

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

  1. Klitzing, (x3o3o3o3o3x - staf)
  2. https://bendwavy.org/klitzing/incmats/staf.htm
  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. https://bendwavy.org/klitzing/incmats/gotaf.htm

References

  • 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
      • (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
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
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