![]() 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
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)[1]
Coordinates
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
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)[2]
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
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)[3]
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
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)[4]
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
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
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 | ( )v( ) | ![]() { }×{ } | ![]() { }v{ } | ![]() {3}×{3} | ![]() {3}v{3} | {3,3}x{3,3} | ![]() {3,3}v{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.
A8 polytopes | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
![]() t0 | ![]() t1 | ![]() t2 | ![]() t3 | ![]() t01 | ![]() t02 | ![]() t12 | ![]() t03 | ![]() t13 | ![]() t23 | ![]() t04 | ![]() t14 | ![]() t24 | ![]() t34 | ![]() t05 |
![]() t15 | ![]() t25 | ![]() t06 | ![]() t16 | ![]() t07 | ![]() t012 | ![]() t013 | ![]() t023 | ![]() t123 | ![]() t014 | ![]() t024 | ![]() t124 | ![]() t034 | ![]() t134 | ![]() t234 |
![]() t015 | ![]() t025 | ![]() t125 | ![]() t035 | ![]() t135 | ![]() t235 | ![]() t045 | ![]() t145 | ![]() t016 | ![]() t026 | ![]() t126 | ![]() t036 | ![]() t136 | ![]() t046 | ![]() t056 |
![]() t017 | ![]() t027 | ![]() t037 | ![]() t0123 | ![]() t0124 | ![]() t0134 | ![]() t0234 | ![]() t1234 | ![]() t0125 | ![]() t0135 | ![]() t0235 | ![]() t1235 | ![]() t0145 | ![]() t0245 | ![]() t1245 |
![]() t0345 | ![]() t1345 | ![]() t2345 | ![]() t0126 | ![]() t0136 | ![]() t0236 | ![]() t1236 | ![]() t0146 | ![]() t0246 | ![]() t1246 | ![]() t0346 | ![]() t1346 | ![]() t0156 | ![]() t0256 | ![]() t1256 |
![]() t0356 | ![]() t0456 | ![]() t0127 | ![]() t0137 | ![]() t0237 | ![]() t0147 | ![]() t0247 | ![]() t0347 | ![]() t0157 | ![]() t0257 | ![]() t0167 | ![]() t01234 | ![]() t01235 | ![]() t01245 | ![]() t01345 |
![]() t02345 | ![]() t12345 | ![]() t01236 | ![]() t01246 | ![]() t01346 | ![]() t02346 | ![]() t12346 | ![]() t01256 | ![]() t01356 | ![]() t02356 | ![]() t12356 | ![]() t01456 | ![]() t02456 | ![]() t03456 | ![]() t01237 |
![]() t01247 | ![]() t01347 | ![]() t02347 | ![]() t01257 | ![]() t01357 | ![]() t02357 | ![]() t01457 | ![]() t01267 | ![]() t01367 | ![]() t012345 | ![]() t012346 | ![]() t012356 | ![]() t012456 | ![]() t013456 | ![]() t023456 |
![]() t123456 | ![]() t012347 | ![]() t012357 | ![]() t012457 | ![]() t013457 | ![]() t023457 | ![]() t012367 | ![]() t012467 | ![]() t013467 | ![]() t012567 | ![]() t0123456 | ![]() t0123457 | ![]() t0123467 | ![]() t0123567 | ![]() t01234567 |
Notes
- ^ Klitizing, (x3x3o3o3o3o3o3o - tene)
- ^ Klitizing, (o3x3x3o3o3o3o3o - batene)
- ^ Klitizing, (o3o3x3x3o3o3o3o - tatene)
- ^ 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
External links
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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 polychoron | 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 |