# Truncated 8-simplexes

(Redirected from Tritruncated 8-simplex)
 Orthogonal projections in A8 Coxeter plane 8-simplex Truncated 8-simplex Rectified 8-simplex Quadritruncated 8-simplex Tritruncated 8-simplex Bitruncated 8-simplex

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
Type uniform 8-polytope
Schläfli symbol t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 288
Vertices 72
Vertex figure Elongated 6-simplex pyramid
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Truncated enneazetton (Acronym: tene) (Jonathan Bowers)[1]

### 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
Type uniform 8-polytope
Schläfli symbol 2t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1008
Vertices 252
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)[2]

### 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
Type uniform 8-polytope
Schläfli symbol 3t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 2016
Vertices 504
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)[3]

### 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]

Type uniform 8-polytope
Schläfli symbol 4t{37}
Coxeter-Dynkin diagrams
or
6-faces 18 3t{3,3,3,3,3,3}
7-faces
5-faces
4-faces
Cells
Faces
Edges 2520
Vertices 630
Vertex figure
Coxeter group A8, [[37]], order 725760
Properties convex, isotopic

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

### Alternate names

• Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)[4]

### 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}
${\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}$
Decachoron

2t{33}
Dodecateron

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

3t{35}

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

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

• 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)". x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be