Cantellated 8-simplexes


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 symbol rr{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

Coordinates

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

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 symbol r2r{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

Coordinates

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

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 symbol r3r{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

Coordinates

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

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 symbol tr{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

Coordinates

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

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 symbol t2r{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

Coordinates

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

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 symbol t3r{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

Coordinates

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

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]

Related polytopes

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

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

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 4-polytope 5-cell 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
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