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Truncated 6-simplexes

April 09, 2024


6-simplex
Truncated 6-simplex

Bitruncated 6-simplex
Tritruncated 6-simplex
Orthogonal projections in A7 Coxeter plane

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

Truncated 6-simplex edit

Truncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol t{3,3,3,3,3}
Coxeter-Dynkin diagram            
         
5-faces 14:
7 {3,3,3,3}  
7 t{3,3,3,3}  
4-faces 63:
42 {3,3,3}  
21 t{3,3,3}  
Cells 140:
105 {3,3}  
35 t{3,3}  
Faces 175:
140 {3}
35 {6}
Edges 126
Vertices 42
Vertex figure  
( )v{3,3,3}
Coxeter group A6, [35], order 5040
Dual ?
Properties convex

Alternate names edit

  • Truncated heptapeton (Acronym: til) (Jonathan Bowers)[1]

Coordinates edit

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

Bitruncated 6-simplex edit

Bitruncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol 2t{3,3,3,3,3}
Coxeter-Dynkin diagram            
       
5-faces 14
4-faces 84
Cells 245
Faces 385
Edges 315
Vertices 105
Vertex figure  
{ }v{3,3}
Coxeter group A6, [35], order 5040
Properties convex

Alternate names edit

  • Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)[2]

Coordinates edit

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

Tritruncated 6-simplex edit

Tritruncated 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol 3t{3,3,3,3,3}
Coxeter-Dynkin diagram            
or      
5-faces 14 2t{3,3,3,3}
4-faces 84
Cells 280
Faces 490
Edges 420
Vertices 140
Vertex figure  
{3}v{3}
Coxeter group A6, [[35]], order 10080
Properties convex, isotopic

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration:       and      .

Alternate names edit

  • Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)[3]

Coordinates edit

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

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 polytopes edit

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter 		Hexagon
  =    
t{3} = {6} 		Octahedron
    =      
r{3,3} = {31,1} = {3,4}
  		Decachoron
   
2t{33} 		Dodecateron
     
2r{34} = {32,2}
  		Tetradecapeton
     
3t{35} 		Hexadecaexon
       
3r{36} = {33,3}
  		Octadecazetton
       
4t{37}
Images                    
Vertex figure ( )∨( )  
{ }×{ } 		 
{ }∨{ } 		 
{3}×{3} 		 
{3}∨{3} 		{3,3}×{3,3}  
{3,3}∨{3,3}
Facets {3}   t{3,3}   r{3,3,3}   2t{3,3,3,3}   2r{3,3,3,3,3}   3t{3,3,3,3,3,3}  
As
intersecting
dual
simplexes 		 
  ∩   		 
    ∩     		 
    ∩     		  
      ∩       		      ∩               ∩                 ∩        

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, (o3x3o3o3o3o - til)
  2. ^ Klitzing, (o3x3x3o3o3o - batal)
  3. ^ Klitzing, (o3o3x3x3o3o - fe)

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)". o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe

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

April 09, 2024
truncated, simplexes, simplex, truncated, simplexbitruncated, simplex, tritruncated, simplexorthogonal, projections, coxeter, planein, dimensional, geometry, truncated, simplex, convex, uniform, polytope, being, truncation, regular, simplex, there, unique, deg. 6 simplex Truncated 6 simplexBitruncated 6 simplex Tritruncated 6 simplexOrthogonal projections in A7 Coxeter planeIn six dimensional geometry a truncated 6 simplex is a convex uniform 6 polytope being a truncation of the regular 6 simplex There are unique 3 degrees of truncation Vertices of the truncation 6 simplex are located as pairs on the edge of the 6 simplex Vertices of the bitruncated 6 simplex are located on the triangular faces of the 6 simplex Vertices of the tritruncated 6 simplex are located inside the tetrahedral cells of the 6 simplex Contents 1 Truncated 6 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bitruncated 6 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Tritruncated 6 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 3 4 Related polytopes 4 Related uniform 6 polytopes 5 Notes 6 References 7 External linksTruncated 6 simplex editTruncated 6 simplexType uniform 6 polytopeClass A6 polytopeSchlafli symbol t 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 14 7 3 3 3 3 nbsp 7 t 3 3 3 3 nbsp 4 faces 63 42 3 3 3 nbsp 21 t 3 3 3 nbsp Cells 140 105 3 3 nbsp 35 t 3 3 nbsp Faces 175 140 3 35 6 Edges 126Vertices 42Vertex figure nbsp v 3 3 3 Coxeter group A6 35 order 5040Dual Properties convexAlternate names edit Truncated heptapeton Acronym til Jonathan Bowers 1 Coordinates edit The vertices of the truncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 0 0 1 2 This construction is based on facets of the truncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Bitruncated 6 simplex editBitruncated 6 simplexType uniform 6 polytopeClass A6 polytopeSchlafli symbol 2t 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 144 faces 84Cells 245Faces 385Edges 315Vertices 105Vertex figure nbsp v 3 3 Coxeter group A6 35 order 5040Properties convexAlternate names edit Bitruncated heptapeton Acronym batal Jonathan Bowers 2 Coordinates edit The vertices of the bitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 0 1 2 2 This construction is based on facets of the bitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Tritruncated 6 simplex editTritruncated 6 simplexType uniform 6 polytopeClass A6 polytopeSchlafli symbol 3t 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp 5 faces 14 2t 3 3 3 3 4 faces 84Cells 280Faces 490Edges 420Vertices 140Vertex figure nbsp 3 v 3 Coxeter group A6 35 order 10080Properties convex isotopicThe tritruncated 6 simplex is an isotopic uniform polytope with 14 identical bitruncated 5 simplex facets The tritruncated 6 simplex is the intersection of two 6 simplexes in dual configuration nbsp nbsp nbsp nbsp nbsp and nbsp nbsp nbsp nbsp nbsp Alternate names edit Tetradecapeton as a 14 facetted 6 polytope Acronym fe Jonathan Bowers 3 Coordinates edit The vertices of the tritruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 2 2 2 This construction is based on facets of the bitruncated 7 orthoplex Alternately it can be centered on the origin as permutations of 1 1 1 0 1 1 1 Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Symmetry 7 14 6 5 10 Ak Coxeter plane A3 A2Graph nbsp nbsp Symmetry 4 3 6 Note Symmetry doubled for Ak graphs with even k due to symmetrically ringed Coxeter Dynkin diagram Related polytopes edit Isotopic uniform truncated simplices Dim 2 3 4 5 6 7 8NameCoxeter Hexagon nbsp nbsp nbsp nbsp t 3 6 Octahedron nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp r 3 3 31 1 3 4 33 displaystyle left begin array l 3 3 end array right nbsp Decachoron nbsp nbsp nbsp 2t 33 Dodecateron nbsp nbsp nbsp nbsp nbsp 2r 34 32 2 3 33 3 displaystyle left begin array l 3 3 3 3 end array right nbsp Tetradecapeton nbsp nbsp nbsp nbsp nbsp 3t 35 Hexadecaexon nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3r 36 33 3 3 3 33 3 3 displaystyle left begin array l 3 3 3 3 3 3 end array right nbsp Octadecazetton nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4t 37 Images nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Vertex figure nbsp nbsp nbsp 3 3 nbsp 3 3 3 3 3 3 nbsp 3 3 3 3 Facets 3 nbsp t 3 3 nbsp r 3 3 3 nbsp 2t 3 3 3 3 nbsp 2r 3 3 3 3 3 nbsp 3t 3 3 3 3 3 3 nbsp Asintersectingdualsimplexes nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 o3x3o3o3o3o til Klitzing o3x3x3o3o3o batal Klitzing o3o3x3x3o3o fe 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 o3x3o3o3o3o til o3x3x3o3o3o batal o3o3x3x3o3o feExternal links editPolytopes of Various Dimensions Multi dimensional GlossaryvteFundamental convex regular and uniform polytopes in dimensions 2 10Family An Bn I2 p Dn E6 E7 E8 F4 G2 HnRegular polygon Triangle Square p gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron IcosahedronUniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cellUniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicubeUniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicubeUniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicubeUniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytopeTopics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Truncated 6 simplexes amp oldid 1148111088 Tritruncated 6 simplex, wikipedia, wiki, book, books, library,

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