| 释义 |
- Rectified 10-simplex Alternate names Coordinates Images
- Birectified 10-simplex Alternate names Coordinates Images
- Trirectified 10-simplex Alternate names Coordinates Images
- Quadrirectified 10-simplex Alternate names Coordinates Images
- Notes
- References
- External links
10-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} | Rectified 10-simplex
{{CDD>node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} | Birectified 10-simplex
{{CDD>node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} | Trirectified 10-simplex
{{CDD>node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}} | Quadrirectified 10-simplex
{{CDD>node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}} | | 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 | node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} | | 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 {{A10 Coxeter plane graphs|t1|100}} 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 | node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} | | 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 {{A10 Coxeter plane graphs|t2|100}} Trirectified 10-simplex | Trirectified 10-simplex | | Type | uniform polyxennon | | Schläfli symbol | t3{3,3,3,3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}} | | 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 (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 triirectified 11-orthoplex. Images {{A10 Coxeter plane graphs|t3|100}} Quadrirectified 10-simplex | Quadrirectified 10-simplex | | Type | uniform polyxennon | | Schläfli symbol | t4{3,3,3,3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}} | | 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 {{A10 Coxeter plane graphs|t4|100}} Notes 1. ^Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
2. ^Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
3. ^Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
4. ^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, [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)
- {{KlitzingPolytopes|polyxenna.htm|10D|uniform polytopes (polyxenna)}} x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru
External links - [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- Multi-dimensional Glossary
{{Polytopes}} 1 : 10-polytopes |