Cantellated 8-simplexes

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8-simplex t02.svg
Cantellated
8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t13.svg
Bicantellated
8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t24.svg
Tricantellated
8-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t012.svg
Cantitruncated
8-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t123.svg
Bicantitruncated
8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t234.svg
Tricantitruncated
8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

Cantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol rr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t02.svg 8-simplex t02 A7.svg 8-simplex t02 A6.svg 8-simplex t02 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t02 A4.svg 8-simplex t02 A3.svg 8-simplex t02 A2.svg
Dihedral symmetry [5] [4] [3]

Bicantellated 8-simplex[edit]

Bicantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t13.svg 8-simplex t13 A7.svg 8-simplex t13 A6.svg 8-simplex t13 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t13 A4.svg 8-simplex t13 A3.svg 8-simplex t13 A2.svg
Dihedral symmetry [5] [4] [3]

Tricantellated 8-simplex[edit]

tricantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t13.svg 8-simplex t13 A7.svg 8-simplex t13 A6.svg 8-simplex t13 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t13 A4.svg 8-simplex t13 A3.svg 8-simplex t13 A2.svg
Dihedral symmetry [5] [4] [3]

Cantitruncated 8-simplex[edit]

Cantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol tr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t012.svg 8-simplex t012 A7.svg 8-simplex t012 A6.svg 8-simplex t012 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t012 A4.svg 8-simplex t012 A3.svg 8-simplex t012 A2.svg
Dihedral symmetry [5] [4] [3]

Bicantitruncated 8-simplex[edit]

Bicantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t123.svg 8-simplex t123 A7.svg 8-simplex t123 A6.svg 8-simplex t123 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t123 A4.svg 8-simplex t123 A3.svg 8-simplex t123 A2.svg
Dihedral symmetry [5] [4] [3]

Tricantitruncated 8-simplex[edit]

Tricantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 8-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[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t234.svg 8-simplex t234 A7.svg 8-simplex t234 A6.svg 8-simplex t234 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t234 A4.svg 8-simplex t234 A3.svg 8-simplex t234 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes[edit]

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

Notes[edit]

  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[edit]

  • 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[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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