Runcinated 7-simplexes
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| colspan=3|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.
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Runcinated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|Runcinated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,3{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 2100 |
| style="background:#e7dcc3;"|Vertices | 280 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Small prismated octaexon (acronym: spo) (Jonathan Bowers)Klitzing, (x3o3o3x3o3o3o - spo)
= 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
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! style="background:#e7dcc3;" colspan="2"|Biruncinated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,4{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 4200 |
| style="background:#e7dcc3;"|Vertices | 560 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Small biprismated octaexon (sibpo) (Jonathan Bowers)Klitzing, (o3x3o3o3x3o3o - sibpo)
= 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
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! style="background:#e7dcc3;" colspan="2"|runcitruncated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,1,3{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 4620 |
| style="background:#e7dcc3;"|Vertices | 840 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)Klitzing, (x3x3o3x3o3o3o - patto)
= 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 ==
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! style="background:#e7dcc3;" colspan="2"|Biruncitruncated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,2,4{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 8400 |
| style="background:#e7dcc3;"|Vertices | 1680 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)Klitzing, (o3x3x3o3x3o3o - bipto)
= 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
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! style="background:#e7dcc3;" colspan="2"|runcicantellated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,2,3{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 3360 |
| style="background:#e7dcc3;"|Vertices | 840 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)Klitzing, (x3o3x3x3o3o3o - paro)
= 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
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! style="background:#e7dcc3;" colspan="2"|biruncicantellated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,3,4{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | |
| style="background:#e7dcc3;"|Vertices | |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|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
| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|runcicantitruncated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,3{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 5880 |
| style="background:#e7dcc3;"|Vertices | 1680 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Great prismated octaexon (acronym: gapo) (Jonathan Bowers)Klitzing, (x3x3x3x3o3o3o - gapo)
= 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
| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|biruncicantitruncated 7-simplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,2,3,4{3,3,3,3,3,3} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}} |
| style="background:#e7dcc3;"|6-faces | |
| style="background:#e7dcc3;"|5-faces | |
| style="background:#e7dcc3;"|4-faces | |
| style="background:#e7dcc3;"|Cells | |
| style="background:#e7dcc3;"|Faces | |
| style="background:#e7dcc3;"|Edges | 11760 |
| style="background:#e7dcc3;"|Vertices | 3360 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)Klitzing, (o3x3x3x3x3o3o- gibpo)
= 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
{{reflist}}
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
- (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]
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}