Runcinated 8-simplexes


8-simplex
Runcinated 8-simplex
Biruncinated 8-simplex
Triruncinated 8-simplex

Runcitruncated 8-simplex
Biruncitruncated 8-simplex
Triruncitruncated 8-simplex
Runcicantellated 8-simplex

Biruncicantellated 8-simplex
Runcicantitruncated 8-simplex
Biruncicantitruncated 8-simplex
Triruncicantitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.

Runcinated 8-simplex

Runcinated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t0,3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges4536
Vertices504
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

Coordinates

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

Biruncinated 8-simplex

Biruncinated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges11340
Vertices1260
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

Coordinates

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

Triruncinated 8-simplex

Triruncinated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges15120
Vertices1680
Vertex figure
Coxeter groupA8×2, [[37]], order 725760
Propertiesconvex

Alternate names

Coordinates

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

Images

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]

Runcitruncated 8-simplex

Images

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]

Biruncitruncated 8-simplex

Images

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]

Triruncitruncated 8-simplex

Images

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]

Runcicantellated 8-simplex

Images

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]

Biruncicantellated 8-simplex

Images

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]

Runcicantitruncated 8-simplex

Images

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]

Biruncicantitruncated 8-simplex

Images

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]

Triruncicantitruncated 8-simplex

Images

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]

Related polytopes

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

Notes

  1. ↑ Klitzing (x3o3o3x3o3o3o3o - spene)
  2. ↑ Klitzing (o3x3o3o3x3o3o3o - sabpene)
  3. ↑ Klitzing (o3o3x3o3o3x3o3o - satpeb)

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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