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 |
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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
[edit]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
[edit]- Truncated enneazetton (Acronym: tene) (Jonathan Bowers)[1]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)[2]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)[3]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)[4]
Coordinates
[edit]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
[edit]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
[edit]Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
Name Coxeter | Hexagon![]() ![]() ![]() ![]() t{3} = {6} | Octahedron![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() r{3,3} = {31,1} = {3,4} | Decachoron![]() ![]() ![]() 2t{33} | Dodecateron![]() ![]() ![]() ![]() ![]() 2r{34} = {32,2} | Tetradecapeton![]() ![]() ![]() ![]() ![]() 3t{35} | Hexadecaexon![]() ![]() ![]() ![]() ![]() ![]() ![]() 3r{36} = {33,3} | 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
[edit]This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
Notes
[edit]References
[edit]- 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