Heptellated 8-simplexes

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8-simplex

Heptellated 8-simplex

Heptihexipentisteriruncicantitruncated 8-simplex
(Omnitruncated 8-simplex)
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation)

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.

Heptellated 8-simplex

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Heptellated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,7{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams                
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 504
Vertices 72
Vertex figure 6-simplex antiprism
Coxeter group A8×2, [[37]], order 725760
Properties convex

Alternate names

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  • Expanded 8-simplex
  • Small exated enneazetton (soxeb) (Jonathan Bowers)[1]

Coordinates

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The vertices of the heptellated 8-simplex can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.

A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0,0,0)

Root vectors

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Its 72 vertices represent the root vectors of the simple Lie group A8.

Images

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orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph        
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Omnitruncated 8-simplex

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Omnitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,1,2,3,4,5,6,7{37}
Coxeter-Dynkin diagrams                
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1451520
Vertices 362880
Vertex figure irr. 7-simplex
Coxeter group A8, [[37]], order 725760
Properties convex

The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names

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  • Heptihexipentisteriruncicantitruncated 8-simplex
  • Great exated enneazetton (goxeb) (Jonathan Bowers)[2]

Coordinates

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The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t0,1,2,3,4,5,6,7{37,4}

Images

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orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph        
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]
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The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of          .

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This polytope is 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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  1. ^ Klitzing, (x3o3o3o3o3o3o3x - soxeb)
  2. ^ Klitzing, (x3x3x3x3x3x3x3x - goxeb)

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)". x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb
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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