# Pentellated 6-simplexes

(Redirected from Pentellated 6-simplex)
 Orthogonal projections in A6 Coxeter plane 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)

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
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

• 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.

## Pentitruncated 6-simplex

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

• Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)[2]

### 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
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

• Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)[3]

### 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
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

• Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)[4]

### 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
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

• Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)[5]

### 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
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

• Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)[6]

### 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
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

• Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)[7]

### 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
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

• Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)[8]

### 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
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

• Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)[9]

### 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-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

• Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
• Omnitruncated heptapeton
• Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)[10]

### 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.

### 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.

## 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

• 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