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 | |
---|---|
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 |
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.
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 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 |
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.
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 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 |
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.
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 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.
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.
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] |
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 |
∩ |
∩ |
∩ |
∩ |
∩ | ∩ | ∩ |
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
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 | |
---|---|
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 |
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.
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 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 |
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.
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 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 |
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.
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 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.
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.
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] |
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 |
∩ |
∩ |
∩ |
∩ |
∩ | ∩ | ∩ |
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.