Runcinated 7-orthoplexes#Runcinated 7-orthoplex
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| colspan=3|Orthogonal projections in B6 Coxeter plane |
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In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.
There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube.
These polytopes are among 127 uniform 7-polytopes with B7 symmetry.
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Runcinated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Runcinated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,3{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|4|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 | 23520 |
| style="background:#e7dcc3;"|Vertices | 2240 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Small prismated hecatonicosoctaexon (acronym: spaz) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o4x - spaz)
= Images =
{{B7 Coxeter plane graphs|t36|150}}
Biruncinated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Biruncinated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,4{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|4|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 | 60480 |
| style="background:#e7dcc3;"|Vertices | 6720 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Small biprismated hecatonicosoctaexon (Acronym sibpaz) (Jonathan Bowers)Klitzing, (o3x3o3o3x3o4o - sibpaz)
= Images =
{{B7 Coxeter plane graphs|t25|150}}
Runcitruncated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Runcitruncated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,1,3{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|4|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 | 50400 |
| style="background:#e7dcc3;"|Vertices | 6720 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)Klitzing, (o3o3o3x3x3o4x - potaz)
= Images =
{{B7 Coxeter plane graphs|t356|150|NOB7A6}}
Biruncitruncated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Biruncitruncated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,2,4{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|4|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 | 120960 |
| style="background:#e7dcc3;"|Vertices | 20160 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (Jonathan Bowers)Klitzing, (o3o3x3o3x3x4o - baptize)
= Images =
{{B7 Coxeter plane graphs|t245|150}}
Runcicantellated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Runcicantellated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,2,3{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|4|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 | 33600 |
| style="background:#e7dcc3;"|Vertices | 6720 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)Klitzing, (o3o3o3x3x3o4x - parz)
= Images =
{{B7 Coxeter plane graphs|t346|150|NOB7A6}}
Biruncicantellated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|biruncicantellated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,3,4{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|4|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 | 100800 |
| style="background:#e7dcc3;"|Vertices | 20160 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (Jonathan Bowers)Klitzing, (o3x3o3x3x3o4o - boparz)
= Images =
{{B7 Coxeter plane graphs|t235|150}}
Runcicantitruncated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Runcicantitruncated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,3{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|4|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 | 60480 |
| style="background:#e7dcc3;"|Vertices | 13440 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)Klitzing, (o3o3o3x3x3x4x - gopaz)
= Images =
{{B7 Coxeter plane graphs|t3456|150|NOB7A6}}
Biruncicantitruncated 7-orthoplex
| class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|biruncicantitruncated 7-orthoplex | |
| style="background:#e7dcc3;"|Type | uniform 7-polytope |
| style="background:#e7dcc3;"|Schläfli symbol | t1,2,3,4{35,4} |
| style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|4|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 | 161280 |
| style="background:#e7dcc3;"|Vertices | 40320 |
| style="background:#e7dcc3;"|Vertex figure | |
| style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
| style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Great biprismated hecatonicosoctaexon (acronym: gibpaz) (Jonathan Bowers)Klitzing, (o3o3x3x3x3x3o - gibpaz)
= Images =
{{B7 Coxeter plane graphs|t1234|150}}
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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 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)
- {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}} o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz
External links
- [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}