Truncated 8-simplexes


8-simplex
Truncated 8-simplex
Rectified 8-simplex

Quadritruncated 8-simplex
Tritruncated 8-simplex
Bitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

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

There are a four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.

Truncated 8-simplex

Truncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges288
Vertices72
Vertex figureElongated 6-simplex pyramid
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

Coordinates

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

Bitruncated 8-simplex

Bitruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol 2t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges1008
Vertices252
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

Coordinates

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

Tritruncated 8-simplex

tritruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol 3t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges2016
Vertices504
Vertex figure
Coxeter groupA8, [37], order 362880
Propertiesconvex

Alternate names

Coordinates

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

Quadritruncated 8-simplex

Quadritruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol 4t{37}
Coxeter-Dynkin diagrams
or
6-faces18 3t{3,3,3,3,3,3}
7-faces
5-faces
4-faces
Cells
Faces
Edges2520
Vertices630
Vertex figure
Coxeter groupA8, [[3<sup>7</sup>]], order 725760
Propertiesconvex, isotopic

The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.

Alternate names

Coordinates

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

Related polytopes

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter		 Hexagon
=
t{3} = {6}		 Octahedron
=
r{3,3} = {31,1} = {3,4}
\left\{\begin{array}{l}3\\3\end{array}\right\}		 Decachoron

2t{33}		 Dodecateron

2r{34} = {32,2}
\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\}		 Tetradecapeton

3t{35}		 Hexadecaexon

3r{36} = {33,3}
\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}		 Octadecazetton

4t{37}
Images
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩ 		∩ ∩ ∩

Related polytopes

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

Notes

  1. ↑ Klitizing, (x3x3o3o3o3o3o3o - tene)
  2. ↑ Klitizing, (o3x3x3o3o3o3o3o - batene)
  3. ↑ Klitizing, (o3o3x3x3o3o3o3o - tatene)
  4. ↑ Klitizing, (o3o3o3x3x3o3o3o - be)

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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