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.
YouTube Encyclopedic
-
1/5Views:11 1179328 236451708
-
N-Simplex
-
Polyhedrons
-
Pentachoron
-
Lecture 5 | Numerical Optimization
-
Truncated 5-Cell
Transcription
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 |
t<sub>015</sub> |
t<sub>025</sub> |
t125 |
t<sub>035</sub> |
t135 |
t<sub>235</sub> |
t<sub>045</sub> |
t145 |
t<sub>016</sub> |
t<sub>026</sub> |
t<sub>126</sub> |
t<sub>036</sub> |
t<sub>136</sub> |
t<sub>046</sub> |
t<sub>056</sub> |
t<sub>017</sub> |
t<sub>027</sub> |
t<sub>037</sub> |
t0123 |
t0124 |
t0134 |
t0234 |
t1234 |
t<sub>0125</sub> |
t<sub>0135</sub> |
t<sub>0235</sub> |
t1235 |
t<sub>0145</sub> |
t<sub>0245</sub> |
t1245 |
t<sub>0345</sub> |
t1345 |
t2345 |
t<sub>0126</sub> |
t<sub>0136</sub> |
t<sub>0236</sub> |
t<sub>1236</sub> |
t<sub>0146</sub> |
t<sub>0246</sub> |
t<sub>1246</sub> |
t<sub>0346</sub> |
t<sub>1346</sub> |
t<sub>0156</sub> |
t<sub>0256</sub> |
t<sub>1256</sub> |
t<sub>0356</sub> |
t<sub>0456</sub> |
t<sub>0127</sub> |
t<sub>0137</sub> |
t<sub>0237</sub> |
t<sub>0147</sub> |
t<sub>0247</sub> |
t<sub>0347</sub> |
t<sub>0157</sub> |
t<sub>0257</sub> |
t<sub>0167</sub> |
t01234 |
t<sub>01235</sub> |
t<sub>01245</sub> |
t<sub>01345</sub> |
t<sub>02345</sub> |
t12345 |
t<sub>01236</sub> |
t<sub>01246</sub> |
t<sub>01346</sub> |
t<sub>02346</sub> |
t<sub>12346</sub> |
t<sub>01256</sub> |
t<sub>01356</sub> |
t<sub>02356</sub> |
t<sub>12356</sub> |
t<sub>01456</sub> |
t<sub>02456</sub> |
t<sub>03456</sub> |
t<sub>01237</sub> |
t<sub>01247</sub> |
t<sub>01347</sub> |
t<sub>02347</sub> |
t<sub>01257</sub> |
t<sub>01357</sub> |
t<sub>02357</sub> |
t<sub>01457</sub> |
t<sub>01267</sub> |
t<sub>01367</sub> |
t<sub>012345</sub> |
t<sub>012346</sub> |
t<sub>012356</sub> |
t<sub>012456</sub> |
t<sub>013456</sub> |
t<sub>023456</sub> |
t<sub>123456</sub> |
t<sub>012347</sub> |
t<sub>012357</sub> |
t<sub>012457</sub> |
t<sub>013457</sub> |
t<sub>023457</sub> |
t<sub>012367</sub> |
t<sub>012467</sub> |
t<sub>013467</sub> |
t<sub>012567</sub> |
t<sub>0123456</sub> |
t<sub>0123457</sub> |
t<sub>0123467</sub> |
t<sub>0123567</sub> |
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