Truncated 6-simplexes

  (Redirected from Tritruncated 6-simplex)
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 t01.svg
Truncated 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.png
6-simplex t12.svg
Bitruncated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t23.svg
Tritruncated 6-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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).

ImagesEdit

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 polytopesEdit

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 ( )v( )  
{ }×{ }
 
{ }v{ }
 
{3}×{3}
 
{3}v{3}
{3,3}x{3,3}  
{3,3}v{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-polytopesEdit

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.

NotesEdit

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

ReferencesEdit

  • 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 linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds