 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 | |||
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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, |
| Properties | convex |
The rectified 10-simplex is the vertex figure of the 11-demicube.
Alternate names
- Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)
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 | |||
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | |||
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry |
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, |
| Properties | convex |
Alternate names
- Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)
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 |
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| Dihedral symmetry | |||
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | |||
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry |
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, |
| Properties | convex |
Alternate names
- Trirectified hendecaxennon (Jonathan Bowers)
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 |
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|
| Dihedral symmetry | |||
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | |||
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry |
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, |
| Properties | convex |
Alternate names
- Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)
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 |
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|
| Dihedral symmetry | |||
| Ak Coxeter plane | A7 | A6 | A5 |
| Graph | 
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| Dihedral symmetry | |||
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| Dihedral symmetry |
Notes
- Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
- Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
- Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
- Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)
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 Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
- 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)". x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru
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 polychoron | Pentachoron | 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 | ||||||||||||