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

Pentellated 6-simplex
TypeUniform 6-polytope
Schläfli symbol t0,5{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 group A6×2, 3,3,3,3,3, order 10080
Propertiesconvex

Alternate names

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.

Pentitruncated 6-simplex

Pentitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol t0,1,5{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

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 symbol t0,2,5{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

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 symbol t0,1,2,5{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

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 symbol t0,1,3,5{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

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 symbol t0,2,3,5{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

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 symbol t0,1,2,3,5{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

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 symbol t0,1,4,5{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

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 symbol t0,1,2,4,5{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

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 symbol t0,1,2,3,4,5{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{3,3} ×
4-faces1806
Cells8400
Faces16800:
4200 {6}
1260 {4}
Edges15120
Vertices5040
Vertex figure
irregular 5-simplex
Coxeter groupA6, [[3<sup>5</sup>]], 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

Permutohedron and related tessellation

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.

Related uniform 6-polytopes

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.

Notes

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

References

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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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