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 a 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.
Contents |
Truncated 8-simplex | |
---|---|
Type | uniform polyzetton |
Schläfli symbol | t0,1{37} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 288 |
Vertices | 72 |
Vertex figure | Elongated 6-simplex pyramid |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.
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 | |
---|---|
Type | uniform polyzetton |
Schläfli symbol | t1,2{37} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1008 |
Vertices | 252 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 9-orthoplex.
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 | |
---|---|
Type | uniform polyzetton |
Schläfli symbol | t2,3{37} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 2016 |
Vertices | 504 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.
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 | |
---|---|
Type | uniform polyzetton |
Schläfli symbol | t3,4{37} |
Coxeter-Dynkin diagrams | |
6-faces | 18 t2,3{3,3,3,3,3,3} |
7-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 2520 |
Vertices | 630 |
Vertex figure | |
Coxeter group | A8, [[37]], order 725760 |
Properties | convex, isotopic |
The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.
The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.
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]] |
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
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 |