Truncated 7-simplexes
Uniform 7-polytope From Wikipedia, the free encyclopedia
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
![]() 7-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Truncated 7-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() Bitruncated 7-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Tritruncated 7-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Orthogonal projections in A7 Coxeter plane |
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There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.
Truncated 7-simplex
Truncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t{3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | 16 |
5-faces | |
4-faces | |
Cells | 350 |
Faces | 336 |
Edges | 196 |
Vertices | 56 |
Vertex figure | ( )v{3,3,3,3} |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex, Vertex-transitive |
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
Alternate names
- Truncated octaexon (Acronym: toc) (Jonathan Bowers)[1]
Coordinates
The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [5] | [4] | [3] |
Bitruncated 7-simplex
Bitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | 2t{3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 588 |
Vertices | 168 |
Vertex figure | { }v{3,3,3} |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex, Vertex-transitive |
Alternate names
- Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)[2]
Coordinates
The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [5] | [4] | [3] |
Tritruncated 7-simplex
Tritruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | 3t{3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 980 |
Vertices | 280 |
Vertex figure | {3}v{3,3} |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex, Vertex-transitive |
Alternate names
- Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)[3]
Coordinates
The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [5] | [4] | [3] |
Related polytopes
These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.
See also
Notes
References
External links
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