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Runcinated 6-simplexes

January 01, 1970


6-simplex
Runcinated 6-simplex
Biruncinated 6-simplex

Runcitruncated 6-simplex
Biruncitruncated 6-simplex
Runcicantellated 6-simplex

Runcicantitruncated 6-simplex
Biruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.

There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.

Runcinated 6-simplex edit

Runcinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 70
4-faces 455
Cells 1330
Faces 1610
Edges 840
Vertices 140
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)[1]

Coordinates edit

The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.

Images edit

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Biruncinated 6-simplex edit

biruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 84
4-faces 714
Cells 2100
Faces 2520
Edges 1260
Vertices 210
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex

Alternate names edit

  • Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)[2]

Coordinates edit

The vertices of the biruncinted 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.

Images edit

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.


Runcitruncated 6-simplex edit

Runcitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 70
4-faces 560
Cells 1820
Faces 2800
Edges 1890
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)[3]

Coordinates edit

The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.

Images edit

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Biruncitruncated 6-simplex edit

biruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 84
4-faces 714
Cells 2310
Faces 3570
Edges 2520
Vertices 630
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)[4]

Coordinates edit

The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.

Images edit

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Runcicantellated 6-simplex edit

Runcicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 70
4-faces 455
Cells 1295
Faces 1960
Edges 1470
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)[5]

Coordinates edit

The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.

Images edit

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Runcicantitruncated 6-simplex edit

Runcicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 70
4-faces 560
Cells 1820
Faces 3010
Edges 2520
Vertices 840
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Runcicantitruncated heptapeton
  • Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)[6]

Coordinates edit

The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.

Images edit

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Biruncicantitruncated 6-simplex edit

biruncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 84
4-faces 714
Cells 2520
Faces 4410
Edges 3780
Vertices 1260
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex

Alternate names edit

  • Biruncicantitruncated heptapeton
  • Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)[7]

Coordinates edit

The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.

Images edit

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 edit

The truncated 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 edit

  1. ^ Klitzing, (x3o3o3x3o3o - spil)
  2. ^ Klitzing, (o3x3o3o3x3o - sibpof)
  3. ^ Klitzing, (x3x3o3x3o3o - patal)
  4. ^ Klitzing, (o3x3x3o3x3o - bapril)
  5. ^ Klitzing, (x3o3x3x3o3o - pril)
  6. ^ Klitzing, (x3x3x3x3o3o - gapil)
  7. ^ Klitzing, (o3x3x3x3x3o - gibpof)

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, 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)". x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof

External links edit

  • Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 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
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

January 01, 1970
runcinated, simplexes, simplex, runcinated, simplex, biruncinated, simplex, runcitruncated, simplex, biruncitruncated, simplex, runcicantellated, simplex, runcicantitruncated, simplex, biruncicantitruncated, simplex, orthogonal, projections, coxeter, plane, di. 6 simplex Runcinated 6 simplex Biruncinated 6 simplex Runcitruncated 6 simplex Biruncitruncated 6 simplex Runcicantellated 6 simplex Runcicantitruncated 6 simplex Biruncicantitruncated 6 simplex Orthogonal projections in A6 Coxeter plane In six dimensional geometry a runcinated 6 simplex is a convex uniform 6 polytope constructed as a runcination 3rd order truncations of the regular 6 simplex There are 8 unique runcinations of the 6 simplex with permutations of truncations and cantellations Contents 1 Runcinated 6 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Biruncinated 6 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Runcitruncated 6 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Biruncitruncated 6 simplex 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Runcicantellated 6 simplex 5 1 Alternate names 5 2 Coordinates 5 3 Images 6 Runcicantitruncated 6 simplex 6 1 Alternate names 6 2 Coordinates 6 3 Images 7 Biruncicantitruncated 6 simplex 7 1 Alternate names 7 2 Coordinates 7 3 Images 8 Related uniform 6 polytopes 9 Notes 10 References 11 External linksRuncinated 6 simplex editRuncinated 6 simplex Type uniform 6 polytope Schlafli symbol t0 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 70 4 faces 455 Cells 1330 Faces 1610 Edges 840 Vertices 140 Vertex figure Coxeter group A6 35 order 5040 Properties convex Alternate names edit Small prismated heptapeton Acronym spil Jonathan Bowers 1 Coordinates edit The vertices of the runcinated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 1 1 2 This construction is based on facets of the runcinated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Biruncinated 6 simplex editbiruncinated 6 simplex Type uniform 6 polytope Schlafli symbol t1 4 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 84 4 faces 714 Cells 2100 Faces 2520 Edges 1260 Vertices 210 Vertex figure Coxeter group A6 35 order 10080 Properties convex Alternate names edit Small biprismated tetradecapeton Acronym sibpof Jonathan Bowers 2 Coordinates edit The vertices of the biruncinted 6 simplex can be most simply positioned in 7 space as permutations of 0 0 1 1 1 2 2 This construction is based on facets of the biruncinated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Symmetry 7 14 6 5 10 Ak Coxeter plane A3 A2 Graph nbsp nbsp Symmetry 4 3 6 Note Symmetry doubled for Ak graphs with even k due to symmetrically ringed Coxeter Dynkin diagram Runcitruncated 6 simplex editRuncitruncated 6 simplex Type uniform 6 polytope Schlafli symbol t0 1 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 70 4 faces 560 Cells 1820 Faces 2800 Edges 1890 Vertices 420 Vertex figure Coxeter group A6 35 order 5040 Properties convex Alternate names edit Prismatotruncated heptapeton Acronym patal Jonathan Bowers 3 Coordinates edit The vertices of the runcitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 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 nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Biruncitruncated 6 simplex editbiruncitruncated 6 simplex Type uniform 6 polytope Schlafli symbol t1 2 4 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 84 4 faces 714 Cells 2310 Faces 3570 Edges 2520 Vertices 630 Vertex figure Coxeter group A6 35 order 5040 Properties convex Alternate names edit Biprismatorhombated heptapeton Acronym bapril Jonathan Bowers 4 Coordinates edit The vertices of the biruncitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 1 1 2 3 3 This construction is based on facets of the biruncitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Runcicantellated 6 simplex editRuncicantellated 6 simplex Type uniform 6 polytope Schlafli symbol t0 2 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 70 4 faces 455 Cells 1295 Faces 1960 Edges 1470 Vertices 420 Vertex figure Coxeter group A6 35 order 5040 Properties convex Alternate names edit Prismatorhombated heptapeton Acronym pril Jonathan Bowers 5 Coordinates edit The vertices of the runcicantellated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 2 2 3 This construction is based on facets of the runcicantellated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Runcicantitruncated 6 simplex editRuncicantitruncated 6 simplex Type uniform 6 polytope Schlafli symbol t0 1 2 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 70 4 faces 560 Cells 1820 Faces 3010 Edges 2520 Vertices 840 Vertex figure Coxeter group A6 35 order 5040 Properties convex Alternate names edit Runcicantitruncated heptapeton Great prismated heptapeton Acronym gapil Jonathan Bowers 6 Coordinates edit The vertices of the runcicantitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 2 3 4 This construction is based on facets of the runcicantitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Biruncicantitruncated 6 simplex editbiruncicantitruncated 6 simplex Type uniform 6 polytope Schlafli symbol t1 2 3 4 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 84 4 faces 714 Cells 2520 Faces 4410 Edges 3780 Vertices 1260 Vertex figure Coxeter group A6 35 order 10080 Properties convex Alternate names edit Biruncicantitruncated heptapeton Great biprismated tetradecapeton Acronym gibpof Jonathan Bowers 7 Coordinates edit The vertices of the biruncicantittruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 1 2 3 4 4 This construction is based on facets of the biruncicantitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4 Graph nbsp nbsp nbsp Symmetry 7 14 6 5 10 Ak Coxeter plane A3 A2 Graph nbsp nbsp Symmetry 4 3 6 Note Symmetry doubled for Ak graphs with even k due to symmetrically ringed Coxeter Dynkin diagram Related uniform 6 polytopes editThe truncated 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 nbsp t0 nbsp t1 nbsp t2 nbsp t0 1 nbsp t0 2 nbsp t1 2 nbsp t0 3 nbsp t1 3 nbsp t2 3 nbsp t0 4 nbsp t1 4 nbsp t0 5 nbsp t0 1 2 nbsp t0 1 3 nbsp t0 2 3 nbsp t1 2 3 nbsp t0 1 4 nbsp t0 2 4 nbsp t1 2 4 nbsp t0 3 4 nbsp t0 1 5 nbsp t0 2 5 nbsp t0 1 2 3 nbsp t0 1 2 4 nbsp t0 1 3 4 nbsp t0 2 3 4 nbsp t1 2 3 4 nbsp t0 1 2 5 nbsp t0 1 3 5 nbsp t0 2 3 5 nbsp t0 1 4 5 nbsp t0 1 2 3 4 nbsp t0 1 2 3 5 nbsp t0 1 2 4 5 nbsp t0 1 2 3 4 5Notes edit Klitzing x3o3o3x3o3o spil Klitzing o3x3o3o3x3o sibpof Klitzing x3x3o3x3o3o patal Klitzing o3x3x3o3x3o bapril Klitzing x3o3x3x3o3o pril Klitzing x3x3x3x3o3o gapil Klitzing o3x3x3x3x3o gibpof References editH 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 x3o3o3x3o3o spil o3x3o3o3x3o sibpof x3x3o3x3o3o patal o3x3x3o3x3o bapril x3o3x3x3o3o pril x3x3x3x3o3o gapil o3x3x3x3x3o gibpofExternal links editPolytopes of Various Dimensions Multi dimensional Glossary vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 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 Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Runcinated 6 simplexes amp oldid 1148115543 Runcicantitruncated 6 simplex, wikipedia, wiki, book, books, library,

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