| 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 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)
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
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 |
| 7-faces | 18 3t{3,3,3,3,3,3} |
| 6-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} { 3 3 } {\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}} | Decachoron 2t{33} | Dodecateron 2r{34} = {32,2} { 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}} | Tetradecapeton 3t{35} | Hexadecaexon 3r{36} = {33,3} { 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \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 | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ |
The four presented polytopes are in the family 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
- 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 (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. . x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be
External links