- Pentellated 6-simplex
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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 Type Uniform polypeton Schläfli symbol t0,5{3,3,3,3,3} Coxeter-Dynkin diagram 5-faces 126 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
- Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)[1]
Coordinates
The vertices of the pentellated 6-simplex can be most simply 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.
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]](*) [6] [[5]](*) Ak Coxeter plane A3 A2 Graph 
Symmetry [4] [[3]](*) - Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.
Pentitruncated 6-simplex
Pentitruncated 6-simplex Type uniform polypeton 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 polypeton 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 polypeton 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 polypeton 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 polypeton 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]](*) [6] [[5]](*) Ak Coxeter plane A3 A2 Graph 
Symmetry [4] [[3]](*) - Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.
Pentiruncicantitruncated 6-simplex
Pentiruncicantitruncated 6-simplex Type uniform polypeton 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 polypeton 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]](*) [6] [[5]](*) Ak Coxeter plane A3 A2 Graph 
Symmetry [4] [[3]](*) - Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-Dynkin diagram.
Pentistericantitruncated 6-simplex
pentistericantitruncated 6-simplex Type uniform polypeton 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 {}xt0,1,2,3{33}
x
70 {6}xt0,1,2,3{3,3}
x
4-faces 1806 Cells 8400 Faces 16800:
4200 {6}
1260 {4}
Edges 15120 Vertices 5040 Vertex figure 
irregular 5-simplexCoxeter group A6 [[3,3,3,3,3]], 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
- Pentisteriruncicantituncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
- Omnitruncated heptapeton
- Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)[10]
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]](*) [6] [[5]](*) Ak Coxeter plane A3 A2 Graph 
Symmetry [4] [[3]](*) - Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxter-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
- ^ Klitzing, (x3o3o3o3o3x - staf)
- ^ Klitzing, (x3x3o3o3o3x - tocal)
- ^ Klitzing, (x3o3x3o3o3x - topal)
- ^ Klitzing, (x3x3x3o3o3x - togral)
- ^ Klitzing, (x3x3o3x3o3x - tocral)
- ^ Klitzing, (x3o3x3x3o3x - taporf)
- ^ Klitzing, (x3x3x3o3x3x - tagopal)
- ^ Klitzing, (x3x3o3o3x3x - tactaf)
- ^ Klitzing, (x3x3x3o3x3x - gatocral)
- ^ 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, 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
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An BCn Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube n-polytopes n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories:- 6-polytopes
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