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 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	( )v{3,3,3,3,3}
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Truncated enneazetton (Acronym: tene) (Jonathan Bowers)

The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 9-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	{ }v{3,3,3,3}
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 9-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	{3}v{3,3,3}
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 9-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
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	{3,3}v{3,3}
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)

Coordinates

The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 9-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
Vertex figure	( )∨( )	{ }×{ }	{ }∨{ }	{3}×{3}	{3}∨{3}	{3,3}×{3,3}	{3,3}∨{3,3}
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.

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

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary