Cantellated
8-simplex
Bicantellated
8-simplex
Tricantellated
8-simplex

Cantitruncated
8-simplex
Bicantitruncated
8-simplex
Tricantitruncated
8-simplex
Orthogonal projections in A8 Coxeter plane

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

There are six unique cantellations for the 8-simplex, including permutations of truncation.

Cantellated 8-simplex

Cantellated 8-simplex
Typeuniform 8-polytope
Schläfli symbolrr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges1764
Vertices252
Vertex figure6-simplex prism
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)[1]

Coordinates

The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.

Images

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]

Bicantellated 8-simplex

Bicantellated 8-simplex
Typeuniform 8-polytope
Schläfli symbolr2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges5292
Vertices756
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)[2]

Coordinates

The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.

Images

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]

Tricantellated 8-simplex

tricantellated 8-simplex
Typeuniform 8-polytope
Schläfli symbolr3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges8820
Vertices1260
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)[3]

Coordinates

The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.

Images

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]

Cantitruncated 8-simplex

Cantitruncated 8-simplex
Typeuniform 8-polytope
Schläfli symboltr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)[4]

Coordinates

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

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]

Bicantitruncated 8-simplex

Bicantitruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

  • Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)[5]

Coordinates

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

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]

Tricantitruncated 8-simplex

Tricantitruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex
  • Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)[6]

Coordinates

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

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]

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

  1. ↑ Klitizing, (x3o3x3o3o3o3o3o - srene)
  2. ↑ Klitizing, (o3x3o3x3o3o3o3o - sabrene)
  3. ↑ Klitizing, (o3o3x3o3x3o3o3o - satrene)
  4. ↑ Klitizing, (x3x3x3o3o3o3o3o - grene)
  5. ↑ Klitizing, (o3x3x3x3o3o3o3o - gabrene)
  6. ↑ Klitizing, (o3o3x3x3x3o3o3o - gatrene)

References

  • 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
      • (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)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.