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)[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 | ( )∨( ) | { }×{ } | { }∨{ } | {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.
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
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
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 polychoron | Pentachoron | 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 |