Truncated 7-simplexes

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7-simplex

Truncated 7-simplex

Bitruncated 7-simplex

Tritruncated 7-simplex
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.

Truncated 7-simplex

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Truncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces 16
5-faces
4-faces
Cells 350
Faces 336
Edges 196
Vertices 56
Vertex figure ( )v{3,3,3,3}
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

Alternate names

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  • Truncated octaexon (Acronym: toc) (Jonathan Bowers)[1]

Coordinates

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The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Bitruncated 7-simplex

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Bitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol 2t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 588
Vertices 168
Vertex figure { }v{3,3,3}
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

Alternate names

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  • Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)[2]

Coordinates

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The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Tritruncated 7-simplex

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Tritruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol 3t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 980
Vertices 280
Vertex figure {3}v{3,3}
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

Alternate names

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  • Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)[3]

Coordinates

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The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]
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These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes
 
t0
 
t1
 
t2
 
t3
 
t0,1
 
t0,2
 
t1,2
 
t0,3
 
t1,3
 
t2,3
 
t0,4
 
t1,4
 
t2,4
 
t0,5
 
t1,5
 
t0,6
 
t0,1,2
 
t0,1,3
 
t0,2,3
 
t1,2,3
 
t0,1,4
 
t0,2,4
 
t1,2,4
 
t0,3,4
 
t1,3,4
 
t2,3,4
 
t0,1,5
 
t0,2,5
 
t1,2,5
 
t0,3,5
 
t1,3,5
 
t0,4,5
 
t0,1,6
 
t0,2,6
 
t0,3,6
 
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
 
t1,2,3,5
 
t0,1,4,5
 
t0,2,4,5
 
t1,2,4,5
 
t0,3,4,5
 
t0,1,2,6
 
t0,1,3,6
 
t0,2,3,6
 
t0,1,4,6
 
t0,2,4,6
 
t0,1,5,6
 
t0,1,2,3,4
 
t0,1,2,3,5
 
t0,1,2,4,5
 
t0,1,3,4,5
 
t0,2,3,4,5
 
t1,2,3,4,5
 
t0,1,2,3,6
 
t0,1,2,4,6
 
t0,1,3,4,6
 
t0,2,3,4,6
 
t0,1,2,5,6
 
t0,1,3,5,6
 
t0,1,2,3,4,5
 
t0,1,2,3,4,6
 
t0,1,2,3,5,6
 
t0,1,2,4,5,6
 
t0,1,2,3,4,5,6

See also

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Notes

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  1. ^ Klitizing, (x3x3o3o3o3o3o - toc)
  2. ^ Klitizing, (o3x3x3o3o3o3o - roc)
  3. ^ Klitizing, (o3o3x3x3o3o3o - tattoc)

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. "7D uniform polytopes (polyexa)". x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc
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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