Pentellated 6-simplexes

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6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t05.svg
Pentellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t015.svg
Pentitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t025.svg
Penticantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0125.svg
Penticantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0135.svg
Pentiruncitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0235.svg
Pentiruncicantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t01235.svg
Pentiruncicantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-simplex t0145.svg
Pentisteritruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-simplex t01245.svg
Pentistericantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-simplex t012345.svg
Pentisteriruncicantitruncated 6-simplex
(Omnitruncated 6-simplex)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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 polypeton
Schläfli symbol t0,5{3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-faces 126:
7+7 {34} 6-simplex t0.svg
21+21 {}x{3,3,3}
35+35 {3}x{3,3}
4-faces 434
Cells 630
Faces 490
Edges 210
Vertices 42
Vertex figure 5-cell antiprism
Coxeter group A6 [[3,3,3,3,3]], order 10080
Properties convex

Alternate names[edit]

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

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 6-simplex t05.svg 6-simplex t05 A5.svg 6-simplex t05 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t05 A3.svg 6-simplex t05 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.


Pentitruncated 6-simplex[edit]

Pentitruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[2]

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 6-simplex t015.svg 6-simplex t015 A5.svg 6-simplex t015 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t015 A3.svg 6-simplex t015 A2.svg
Dihedral symmetry [4] [3]

Penticantellated 6-simplex[edit]

Penticantellated 6-simplex
Type uniform polypeton
Schläfli symbol t0,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[3]

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 6-simplex t025.svg 6-simplex t025 A5.svg 6-simplex t025 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t025 A3.svg 6-simplex t025 A2.svg
Dihedral symmetry [4] [3]

Penticantitruncated 6-simplex[edit]

penticantitruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[4]

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 6-simplex t0125.svg 6-simplex t0125 A5.svg 6-simplex t0125 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0125 A3.svg 6-simplex t0125 A2.svg
Dihedral symmetry [4] [3]

Pentiruncitruncated 6-simplex[edit]

pentiruncitruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[5]

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 6-simplex t0135.svg 6-simplex t0135 A5.svg 6-simplex t0135 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0135 A3.svg 6-simplex t0135 A2.svg
Dihedral symmetry [4] [3]

Pentiruncicantellated 6-simplex[edit]

Pentiruncicantellated 6-simplex
Type uniform polypeton
Schläfli symbol t0,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[6]

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 6-simplex t0235.svg 6-simplex t0235 A5.svg 6-simplex t0235 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t0235 A3.svg 6-simplex t0235 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.


Pentiruncicantitruncated 6-simplex[edit]

Pentiruncicantitruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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)[7]

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 6-simplex t01235.svg 6-simplex t01235 A5.svg 6-simplex t01235 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t01235 A3.svg 6-simplex t01235 A2.svg
Dihedral symmetry [4] [3]

Pentisteritruncated 6-simplex[edit]

Pentisteritruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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)[8]

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 6-simplex t0145.svg 6-simplex t0145 A5.svg 6-simplex t0145 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t0145 A3.svg 6-simplex t0145 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.


Pentistericantitruncated 6-simplex[edit]

pentistericantitruncated 6-simplex
Type uniform polypeton
Schläfli symbol t0,1,2,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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)[9]

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 6-simplex t01245.svg 6-simplex t01245 A5.svg 6-simplex t01245 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t01245 A3.svg 6-simplex t01245 A2.svg
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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-faces 126:
14 t0,1,2,3,4{34}5-simplex t01234.svg

42 {}xt0,1,2,3{33} Complete graph K2.svgx6-simplex t0123.svg

70 {6}xt0,1,2,3{3,3} 2-simplex t01.svgx3-simplex t012.svg

4-faces 1806
Cells 8400
Faces 16800:
4200 {6} 2-simplex t01.svg
1260 {4}Kvadrato.svg
Edges 15120
Vertices 5040
Vertex figure Omnitruncated 6-simplex verf.png
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)[10]

Permutohedron and related tessellation[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 CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png.

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}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

Images[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t012345.svg 6-simplex t012345 A5.svg 6-simplex t012345 A4.svg
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph 6-simplex t012345 A3.svg 6-simplex t012345 A2.svg
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.


Related uniform 6-polytopes[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.

6-simplex t0.svg
t0
6-simplex t1.svg
t1
6-simplex t2.svg
t2
6-simplex t01.svg
t0,1
6-simplex t02.svg
t0,2
6-simplex t12.svg
t1,2
6-simplex t03.svg
t0,3
6-simplex t13.svg
t1,3
6-simplex t23.svg
t2,3
6-simplex t04.svg
t0,4
6-simplex t14.svg
t1,4
6-simplex t05.svg
t0,5
6-simplex t012.svg
t0,1,2
6-simplex t013.svg
t0,1,3
6-simplex t023.svg
t0,2,3
6-simplex t123.svg
t1,2,3
6-simplex t014.svg
t0,1,4
6-simplex t024.svg
t0,2,4
6-simplex t124.svg
t1,2,4
6-simplex t034.svg
t0,3,4
6-simplex t015.svg
t0,1,5
6-simplex t025.svg
t0,2,5
6-simplex t0123.svg
t0,1,2,3
6-simplex t0124.svg
t0,1,2,4
6-simplex t0134.svg
t0,1,3,4
6-simplex t0234.svg
t0,2,3,4
6-simplex t1234.svg
t1,2,3,4
6-simplex t0125.svg
t0,1,2,5
6-simplex t0135.svg
t0,1,3,5
6-simplex t0235.svg
t0,2,3,5
6-simplex t0145.svg
t0,1,4,5
6-simplex t01234.svg
t0,1,2,3,4
6-simplex t01235.svg
t0,1,2,3,5
6-simplex t01245.svg
t0,1,2,4,5
6-simplex t012345.svg
t0,1,2,3,4,5

Notes[edit]

  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[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, editied 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.
  • Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o3o3o3x - staf, x3x3o3o3o3x - tocal, x3o3x3o3o3x - topal, x3x3x3o3o3x - togral, x3x3o3x3o3x - tocral, x3x3x3x3o3x - tagopal, x3x3o3o3x3x - tactaf, x3x3x3o3x3x - tacogral, x3x3x3x3x3x - gotaf

External links[edit]