 10-simplex   
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 Rectified 10-simplex
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 Birectified 10-simplex
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 Trirectified 10-simplex
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 Quadrirectified 10-simplex
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| Orthogonal projections in A9 Coxeter plane | |||
|---|---|---|---|
In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.
These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.
There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.
Rectified 10-simplex
| Rectified 10-simplex | |
|---|---|
| Type | uniform polyxennon |
| Schläfli symbol | t1{3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 9-faces | 22 |
| 8-faces | 165 |
| 7-faces | 660 |
| 6-faces | 1650 |
| 5-faces | 2772 |
| 4-faces | 3234 |
| Cells | 2640 |
| Faces | 1485 |
| Edges | 495 |
| Vertices | 55 |
| Vertex figure | 9-simplex prism |
| Petrie polygon | decagon |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |
The rectified 10-simplex is the vertex figure of the 11-demicube.
Alternate names
- Rectified hendecaxennon (acronym: ru) (Jonathan Bowers)1
Coordinates
The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.
Images
| Ak Coxeter plane | A10 | A9 | A8 |
|---|---|---|---|
| Graph | 
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| Dihedral symmetry | [11] | [10] | [9] |
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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|
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry | [5] | [4] | [3] |
Birectified 10-simplex
| Birectified 10-simplex | |
|---|---|
| Type | uniform 9-polytope |
| Schläfli symbol | t2{3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1980 |
| Vertices | 165 |
| Vertex figure | {3}x{3,3,3,3,3,3} |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Birectified hendecaxennon (acronym: bru) (Jonathan Bowers)2
Coordinates
The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.
Images
| Ak Coxeter plane | A10 | A9 | A8 |
|---|---|---|---|
| Graph | 
|
|
|
| Dihedral symmetry | [11] | [10] | [9] |
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry | [5] | [4] | [3] |
Trirectified 10-simplex
| Trirectified 10-simplex | |
|---|---|
| Type | uniform polyxennon |
| Schläfli symbol | t3{3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4620 |
| Vertices | 330 |
| Vertex figure | {3,3}x{3,3,3,3,3} |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Trirectified hendecaxennon (acronym: tru) (Jonathan Bowers)3
Coordinates
The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.
Images
| Ak Coxeter plane | A10 | A9 | A8 |
|---|---|---|---|
| Graph | 
|
|
|
| Dihedral symmetry | [11] | [10] | [9] |
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry | [5] | [4] | [3] |
Quadrirectified 10-simplex
| Quadrirectified 10-simplex | |
|---|---|
| Type | uniform polyxennon |
| Schläfli symbol | t4{3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 6930 |
| Vertices | 462 |
| Vertex figure | {3,3,3}x{3,3,3,3} |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Quadrirectified hendecaxennon (acronym: teru) (Jonathan Bowers)4
Coordinates
The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.
Images
| Ak Coxeter plane | A10 | A9 | A8 |
|---|---|---|---|
| Graph | 
|
|
|
| Dihedral symmetry | [11] | [10] | [9] |
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry | [5] | [4] | [3] |
Notes
Notes
References
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. (1966)
- Klitzing, Richard. "10D uniform polytopes (polyxenna) with acronyms". o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru