Cantellated 8-simplexes

From The Right Wiki
(Redirected from Bicantellated 8-simplex)
Jump to navigationJump to search
File:8-simplex t02.svg
Cantellated
8-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-simplex t13.svg
Bicantellated
8-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-simplex t24.svg
Tricantellated
8-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:8-simplex t012.svg
Cantitruncated
8-simplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-simplex t123.svg
Bicantitruncated
8-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-simplex t234.svg
Tricantitruncated
8-simplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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
Type uniform 8-polytope
Schläfli symbol rr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1764
Vertices 252
Vertex figure 6-simplex prism
Coxeter group A8, [37], order 362880
Properties convex

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 File:8-simplex t02.svg File:8-simplex t02 A7.svg File:8-simplex t02 A6.svg File:8-simplex t02 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t02 A4.svg File:8-simplex t02 A3.svg File:8-simplex t02 A2.svg
Dihedral symmetry [5] [4] [3]

Bicantellated 8-simplex

Bicantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 5292
Vertices 756
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

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 File:8-simplex t13.svg File:8-simplex t13 A7.svg File:8-simplex t13 A6.svg File:8-simplex t13 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t13 A4.svg File:8-simplex t13 A3.svg File:8-simplex t13 A2.svg
Dihedral symmetry [5] [4] [3]

Tricantellated 8-simplex

tricantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 8820
Vertices 1260
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

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 File:8-simplex t13.svg File:8-simplex t13 A7.svg File:8-simplex t13 A6.svg File:8-simplex t13 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t13 A4.svg File:8-simplex t13 A3.svg File:8-simplex t13 A2.svg
Dihedral symmetry [5] [4] [3]

Cantitruncated 8-simplex

Cantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol tr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

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 File:8-simplex t012.svg File:8-simplex t012 A7.svg File:8-simplex t012 A6.svg File:8-simplex t012 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t012 A4.svg File:8-simplex t012 A3.svg File:8-simplex t012 A2.svg
Dihedral symmetry [5] [4] [3]

Bicantitruncated 8-simplex

Bicantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

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 File:8-simplex t123.svg File:8-simplex t123 A7.svg File:8-simplex t123 A6.svg File:8-simplex t123 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t123 A4.svg File:8-simplex t123 A3.svg File:8-simplex t123 A2.svg
Dihedral symmetry [5] [4] [3]

Tricantitruncated 8-simplex

Tricantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex
  • 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 File:8-simplex t234.svg File:8-simplex t234 A7.svg File:8-simplex t234 A6.svg File:8-simplex t234 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph File:8-simplex t234 A4.svg File:8-simplex t234 A3.svg File:8-simplex t234 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

A8 polytopes
File:8-simplex t0.svg
t0 		File:8-simplex t1.svg
t1 		File:8-simplex t2.svg
t2 		File:8-simplex t3.svg
t3 		File:8-simplex t01.svg
t01 		File:8-simplex t02.svg
t02 		File:8-simplex t12.svg
t12 		File:8-simplex t03.svg
t03 		File:8-simplex t13.svg
t13 		File:8-simplex t23.svg
t23 		File:8-simplex t04.svg
t04 		File:8-simplex t14.svg
t14 		File:8-simplex t24.svg
t24 		File:8-simplex t34.svg
t34 		File:8-simplex t05.svg
t05
File:8-simplex t15.svg
t15 		File:8-simplex t25.svg
t25 		File:8-simplex t06.svg
t06 		File:8-simplex t16.svg
t16 		File:8-simplex t07.svg
t07 		File:8-simplex t012.svg
t012 		File:8-simplex t013.svg
t013 		File:8-simplex t023.svg
t023 		File:8-simplex t123.svg
t123 		File:8-simplex t014.svg
t014 		File:8-simplex t024.svg
t024 		File:8-simplex t124.svg
t124 		File:8-simplex t034.svg
t034 		File:8-simplex t134.svg
t134 		File:8-simplex t234.svg
t234
File:8-simplex t015.svg
t015 		File:8-simplex t025.svg
t025 		File:8-simplex t125.svg
t125 		File:8-simplex t035.svg
t035 		File:8-simplex t135.svg
t135 		File:8-simplex t235.svg
t235 		File:8-simplex t045.svg
t045 		File:8-simplex t145.svg
t145 		File:8-simplex t016.svg
t016 		File:8-simplex t026.svg
t026 		File:8-simplex t126.svg
t126 		File:8-simplex t036.svg
t036 		File:8-simplex t136.svg
t136 		File:8-simplex t046.svg
t046 		File:8-simplex t056.svg
t056
File:8-simplex t017.svg
t017 		File:8-simplex t027.svg
t027 		File:8-simplex t037.svg
t037 		File:8-simplex t0123.svg
t0123 		File:8-simplex t0124.svg
t0124 		File:8-simplex t0134.svg
t0134 		File:8-simplex t0234.svg
t0234 		File:8-simplex t1234.svg
t1234 		File:8-simplex t0125.svg
t0125 		File:8-simplex t0135.svg
t0135 		File:8-simplex t0235.svg
t0235 		File:8-simplex t1235.svg
t1235 		File:8-simplex t0145.svg
t0145 		File:8-simplex t0245.svg
t0245 		File:8-simplex t1245.svg
t1245
File:8-simplex t0345.svg
t0345 		File:8-simplex t1345.svg
t1345 		File:8-simplex t2345.svg
t2345 		File:8-simplex t0126.svg
t0126 		File:8-simplex t0136.svg
t0136 		File:8-simplex t0236.svg
t0236 		File:8-simplex t1236.svg
t1236 		File:8-simplex t0146.svg
t0146 		File:8-simplex t0246.svg
t0246 		File:8-simplex t1246.svg
t1246 		File:8-simplex t0346.svg
t0346 		File:8-simplex t1346.svg
t1346 		File:8-simplex t0156.svg
t0156 		File:8-simplex t0256.svg
t0256 		File:8-simplex t1256.svg
t1256
File:8-simplex t0356.svg
t0356 		File:8-simplex t0456.svg
t0456 		File:8-simplex t0127.svg
t0127 		File:8-simplex t0137.svg
t0137 		File:8-simplex t0237.svg
t0237 		File:8-simplex t0147.svg
t0147 		File:8-simplex t0247.svg
t0247 		File:8-simplex t0347.svg
t0347 		File:8-simplex t0157.svg
t0157 		File:8-simplex t0257.svg
t0257 		File:8-simplex t0167.svg
t0167 		File:8-simplex t01234.svg
t01234 		File:8-simplex t01235.svg
t01235 		File:8-simplex t01245.svg
t01245 		File:8-simplex t01345.svg
t01345
File:8-simplex t02345.svg
t02345 		File:8-simplex t12345.svg
t12345 		File:8-simplex t01236.svg
t01236 		File:8-simplex t01246.svg
t01246 		File:8-simplex t01346.svg
t01346 		File:8-simplex t02346.svg
t02346 		File:8-simplex t12346.svg
t12346 		File:8-simplex t01256.svg
t01256 		File:8-simplex t01356.svg
t01356 		File:8-simplex t02356.svg
t02356 		File:8-simplex t12356.svg
t12356 		File:8-simplex t01456.svg
t01456 		File:8-simplex t02456.svg
t02456 		File:8-simplex t03456.svg
t03456 		File:8-simplex t01237.svg
t01237
File:8-simplex t01247.svg
t01247 		File:8-simplex t01347.svg
t01347 		File:8-simplex t02347.svg
t02347 		File:8-simplex t01257.svg
t01257 		File:8-simplex t01357.svg
t01357 		File:8-simplex t02357.svg
t02357 		File:8-simplex t01457.svg
t01457 		File:8-simplex t01267.svg
t01267 		File:8-simplex t01367.svg
t01367 		File:8-simplex t012345.svg
t012345 		File:8-simplex t012346.svg
t012346 		File:8-simplex t012356.svg
t012356 		File:8-simplex t012456.svg
t012456 		File:8-simplex t013456.svg
t013456 		File:8-simplex t023456.svg
t023456
File:8-simplex t123456.svg
t123456 		File:8-simplex t012347.svg
t012347 		File:8-simplex t012357.svg
t012357 		File:8-simplex t012457.svg
t012457 		File:8-simplex t013457.svg
t013457 		File:8-simplex t023457.svg
t023457 		File:8-simplex t012367.svg
t012367 		File:8-simplex t012467.svg
t012467 		File:8-simplex t013467.svg
t013467 		File:8-simplex t012567.svg
t012567 		File:8-simplex t0123456 A7.svg
t0123456 		File:8-simplex t0123457 A7.svg
t0123457 		File:8-simplex t0123467 A7.svg
t0123467 		File:8-simplex t0123567 A7.svg
t0123567 		File:8-simplex t01234567 A7.svg
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 [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)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene

External links

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
Retrieved from "https://rightwiki.org/index.php?title=Cantellated_8-simplexes&oldid=13294896#Bicantellated_8-simplex"