| 释义 |
- Runcinated 7-simplex Alternate names Coordinates Images
- Biruncinated 7-simplex Alternate names Coordinates Images
- Runcitruncated 7-simplex Alternate names Coordinates Images
- Biruncitruncated 7-simplex Alternate names Coordinates Images
- Runcicantellated 7-simplex Alternate names Coordinates Images
- Biruncicantellated 7-simplex Alternate names Coordinates Images
- Runcicantitruncated 7-simplex Alternate names Coordinates Images
- Biruncicantitruncated 7-simplex Alternate names Coordinates Images
- Related polytopes
- Notes
- References
- External links
7-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node}} | Runcinated 7-simplex
{{CDD>node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}} | Biruncinated 7-simplex
{{CDD>node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}} | Runcitruncated 7-simplex
{{CDD>node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}} | Biruncitruncated 7-simplex
{{CDD>node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}} | Runcicantellated 7-simplex
{{CDD>node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}} | Biruncicantellated 7-simplex
{{CDD>node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node}} | Runcicantitruncated 7-simplex
{{CDD>node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}} | Biruncicantitruncated 7-simplex
{{CDD>node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}} | | Orthogonal projections in A7 Coxeter plane |
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In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex. There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations. Runcinated 7-simplex| Runcinated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t0,3{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 2100 | | Vertices | 280 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Small prismated octaexon (acronym: spo) (Jonathan Bowers)[1]
Coordinates The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t03|150}}Biruncinated 7-simplex | Biruncinated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t1,4{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 4200 | | Vertices | 560 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Small biprismated octaexon (sibpo) (Jonathan Bowers)[2]
Coordinates The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t14|150}}Runcitruncated 7-simplex| runcitruncated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t0,1,3{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 4620 | | Vertices | 840 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)[3]
Coordinates The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t013|150}}Biruncitruncated 7-simplex| Biruncitruncated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t1,2,4{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 8400 | | Vertices | 1680 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)[4]
Coordinates The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t124|150}}Runcicantellated 7-simplex | runcicantellated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t0,2,3{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 3360 | | Vertices | 840 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)[5]
Coordinates The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t023|150}}Biruncicantellated 7-simplex | biruncicantellated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t1,3,4{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | | Vertices | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)
Coordinates The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t134|150}}Runcicantitruncated 7-simplex | runcicantitruncated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t0,1,2,3{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 5880 | | Vertices | 1680 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Great prismated octaexon (acronym: gapo) (Jonathan Bowers)[6]
Coordinates The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t0123|150}}Biruncicantitruncated 7-simplex| biruncicantitruncated 7-simplex | | Type | uniform 7-polytope | | Schläfli symbol | t1,2,3,4{3,3,3,3,3,3} | | Coxeter-Dynkin diagrams | node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}} | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | 11760 | | Vertices | 3360 | | Vertex figure | | Coxeter group | A7, [36], order 40320 | | Properties | convex |
Alternate names- Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)[7]
Coordinates The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex. Images {{7-simplex Coxeter plane graphs|t1234|150}} Related polytopes These polytopes are among 71 uniform 7-polytopes with A7 symmetry. {{Octaexon family}} Notes 1. ^Klitzing, (x3o3o3x3o3o3o - spo)
2. ^Klitzing, (o3x3o3o3x3o3o - sibpo)
3. ^Klitzing, (x3x3o3x3o3o3o - patto)
4. ^Klitzing, (o3x3x3o3x3o3o - bipto)
5. ^Klitzing, (x3o3x3x3o3o3o - paro)
6. ^Klitzing, (x3x3x3x3o3o3o - gapo)
7. ^Klitzing, (o3x3x3x3x3o3o- gibpo)
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.
- {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}} x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo
External links - [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- Multi-dimensional Glossary
{{Polytopes}} 1 : 7-polytopes |