Rectified 8-simplexes

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8-simplex

Rectified 8-simplex

Birectified 8-simplex

Trirectified 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

Rectified 8-simplex

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Rectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 061
Schläfli symbol t1{37}
r{37} = {36,1}
or  
Coxeter-Dynkin diagrams                
or              
7-faces 18
6-faces 108
5-faces 336
4-faces 630
Cells 756
Faces 588
Edges 252
Vertices 36
Vertex figure 7-simplex prism, {}×{3,3,3,3,3}
Petrie polygon enneagon
Coxeter group A8, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
8
. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as              .

Coordinates

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The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph        
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Birectified 8-simplex

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Birectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 052
Schläfli symbol t2{37}
2r{37} = {35,2} or
 
Coxeter-Dynkin diagrams                
or            
7-faces 18
6-faces 144
5-faces 588
4-faces 1386
Cells 2016
Faces 1764
Edges 756
Vertices 84
Vertex figure {3}×{3,3,3,3}
Coxeter group A8, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
8
. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as            .

The birectified 8-simplex is the vertex figure of the 152 honeycomb.

Coordinates

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The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph        
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Trirectified 8-simplex

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Trirectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 043
Schläfli symbol t3{37}
3r{37} = {34,3} or
 
Coxeter-Dynkin diagrams                
or          
7-faces 9 + 9
6-faces 36 + 72 + 36
5-faces 84 + 252 + 252 + 84
4-faces 126 + 504 + 756 + 504
Cells 630 + 1260 + 1260
Faces 1260 + 1680
Edges 1260
Vertices 126
Vertex figure {3,3}×{3,3,3}
Petrie polygon enneagon
Coxeter group A7, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
8
. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as          .

Coordinates

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The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph        
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]
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This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.

It is also one of 135 uniform 8-polytopes with A8 symmetry.

A8 polytopes
 
t0
 
t1
 
t2
 
t3
 
t01
 
t02
 
t12
 
t03
 
t13
 
t23
 
t04
 
t14
 
t24
 
t34
 
t05
 
t15
 
t25
 
t06
 
t16
 
t07
 
t012
 
t013
 
t023
 
t123
 
t014
 
t024
 
t124
 
t034
 
t134
 
t234
 
t015
 
t025
 
t125
 
t035
 
t135
 
t235
 
t045
 
t145
 
t016
 
t026
 
t126
 
t036
 
t136
 
t046
 
t056
 
t017
 
t027
 
t037
 
t0123
 
t0124
 
t0134
 
t0234
 
t1234
 
t0125
 
t0135
 
t0235
 
t1235
 
t0145
 
t0245
 
t1245
 
t0345
 
t1345
 
t2345
 
t0126
 
t0136
 
t0236
 
t1236
 
t0146
 
t0246
 
t1246
 
t0346
 
t1346
 
t0156
 
t0256
 
t1256
 
t0356
 
t0456
 
t0127
 
t0137
 
t0237
 
t0147
 
t0247
 
t0347
 
t0157
 
t0257
 
t0167
 
t01234
 
t01235
 
t01245
 
t01345
 
t02345
 
t12345
 
t01236
 
t01246
 
t01346
 
t02346
 
t12346
 
t01256
 
t01356
 
t02356
 
t12356
 
t01456
 
t02456
 
t03456
 
t01237
 
t01247
 
t01347
 
t02347
 
t01257
 
t01357
 
t02357
 
t01457
 
t01267
 
t01367
 
t012345
 
t012346
 
t012356
 
t012456
 
t013456
 
t023456
 
t123456
 
t012347
 
t012357
 
t012457
 
t013457
 
t023457
 
t012367
 
t012467
 
t013467
 
t012567
 
t0123456
 
t0123457
 
t0123467
 
t0123567
 
t01234567

Notes

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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. "8D Uniform polytopes (polyzetta)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene
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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