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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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.
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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
 
  
 
      
 
      
  
          
                                        
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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.

A6 polytopes
 
t0
 
t1
 
t2
 
t0,1
 
t0,2
 
t1,2
 
t0,3
 
t1,3
 
t2,3
 
t0,4
 
t1,4
 
t0,5
 
t0,1,2
 
t0,1,3
 
t0,2,3
 
t1,2,3
 
t0,1,4
 
t0,2,4
 
t1,2,4
 
t0,3,4
 
t0,1,5
 
t0,2,5
 
t0,1,2,3
 
t0,1,2,4
 
t0,1,3,4
 
t0,2,3,4
 
t1,2,3,4
 
t0,1,2,5
 
t0,1,3,5
 
t0,2,3,5
 
t0,1,4,5
 
t0,1,2,3,4
 
t0,1,2,3,5
 
t0,1,2,4,5
 
t0,1,2,3,4,5

Notes

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

References

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  • 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
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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds