Truncated 7-simplexes
| 150px 7-simplex   
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150px Truncated 7-simplex
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| 150px Bitruncated 7-simplex
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150px Tritruncated 7-simplex
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| Orthogonal projections in A7 Coxeter plane | ||
|---|---|---|
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
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 | 
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|
|
| 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
- List of A7 polytopes
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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. "7D uniform polytopes (polyexa) with acronyms". https://bendwavy.org/klitzing/dimensions/polyexa.htm. x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - bittoc, o3o3x3x3o3o3o - tattoc
External links
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||
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