Pentellated 7-cubes#Pentiruncicantellated 7-cube
In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique {{not a typo|pentellations}} of the 7-cube with permutations of truncations, cantellations, runcinations, and {{not a typo|sterications}}. 16 are more simply constructed relative to the 7-orthoplex.
class=wikitable style="float:right; margin-left:8px; width:480px" |
style="text-align:center;"
|120px |120px |120px |120px |
style="text-align:center;"
|120px |120px |120px |120px |
style="text-align:center;"
|120px |120px |120px |120px |
style="text-align:center;"
|120px |120px |120px |120px |
style="text-align:center;"
|120px |
{{-}}
Pentellated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Pentellated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Small {{not a typo|terated}} hepteract (acronym: stesa) (Jonathan Bowers)Klitzing, (x4o3o3o3o3x3o - stesa)
= Images =
{{B7 Coxeter plane graphs|t05|150}}
Pentitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Teritruncated hepteract (acronym: tetsa) (Jonathan Bowers)Klitzing, (x4x3o3o3o3x3o - tetsa)
= Images =
{{B7 Coxeter plane graphs|t015|150}}
Penticantellated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Penticantellated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Terirhombated hepteract (acronym: tersa) (Jonathan Bowers)Klitzing, (x4o3x3o3o3x3o - tersa)
= Images =
{{B7 Coxeter plane graphs|t025|150}}
Penticantitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|penticantitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Terigreatorhombated hepteract (acronym: togresa) (Jonathan Bowers)Klitzing, (x4x3x3o3o3x3o - togresa)
= Images =
{{B7 Coxeter plane graphs|t0125|150}}
Pentiruncinated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentiruncinated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,3,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Teriprismated hepteract (acronym: tapsa) (Jonathan Bowers)Klitzing, (x4o3o3x3o3x3o - tapsa)
= Images =
{{B7 Coxeter plane graphs|t035|150}}
Pentiruncitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentiruncitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,3,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Teriprismatotruncated hepteract (acronym: toptasa) (Jonathan Bowers)Klitzing, (x4x3o3x3o3x3o - toptasa)
= Images =
{{B7 Coxeter plane graphs|t0135|150}}
Pentiruncicantellated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentiruncicantellated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,3,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Teriprismatorhombated hepteract (acronym: topresa) (Jonathan Bowers)Klitzing, (x4o3x3x3o3x3o - topresa)
= Images =
{{B7 Coxeter plane graphs|t0235|180}}
Pentiruncicantitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentiruncicantitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,3,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Terigreatoprismated hepteract (acronym: togapsa) (Jonathan Bowers)Klitzing, (x4x3x3x3o3x3o - togapsa)
= Images =
{{B7 Coxeter plane graphs|t01235|150|NOB7A6|NOA5orA3}}
Pentistericated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentistericated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Tericellated hepteract (acronym: tacosa) (Jonathan Bowers)Klitzing, (x4o3o3o3x3x3o - tacosa)
= Images =
{{B7 Coxeter plane graphs|t045|150}}
Pentisteritruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentisteritruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Tericellitruncated hepteract (acronym: tecatsa) (Jonathan Bowers)Klitzing, (x4x3o3o3x3x3o - tecatsa)
= Images =
{{B7 Coxeter plane graphs|t0145|150}}
Pentistericantellated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentistericantellated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Tericellirhombated hepteract (acronym: tecresa) (Jonathan Bowers)Klitzing, (x4o3x3o3x3x3o - tecresa)
= Images =
{{B7 Coxeter plane graphs|t0245|150}}
Pentistericantitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentistericantitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Tericelligreatorhombated hepteract (acronym: tecgresa) (Jonathan Bowers)Klitzing, (x4x3x3o3x3x3o - tecgresa)
= Images =
{{B7 Coxeter plane graphs|t01245|150|NOB7A6}}
Pentisteriruncinated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|Pentisteriruncinated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,3,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
- Tericelliprismated hepteract (acronym: tecpasa) (Jonathan Bowers)Klitzing, (x4o3o3x3x3x3o - tecpasa)
= Images =
{{B7 Coxeter plane graphs|t0345|150}}
Pentisteriruncitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentisteriruncitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,3,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|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 | 40320 |
style="background:#e7dcc3;"|Vertices | 10080 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Tericelliprismatotruncated hepteract (acronym: tecpetsa) (Jonathan Bowers)Klitzing, (x4x3o3x3x3x3o - tecpetsa)
= Images =
{{B7 Coxeter plane graphs|t01345|150|NOB7A6}}
Pentisteriruncicantellated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentisteriruncicantellated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,3,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|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 | 40320 |
style="background:#e7dcc3;"|Vertices | 10080 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
- Tericelliprismatorhombated hepteract (acronym: tocpresa) (Jonathan Bowers)Klitzing, (x4o3x3x3x3x3o - tocpresa)
= Images =
{{B7 Coxeter plane graphs|t02345|150|NOB7A6}}
Pentisteriruncicantitruncated 7-cube
class="wikitable" style="margin-left:10px; float:right; width:250px;"
! style="background:#e7dcc3;" colspan="2"|pentisteriruncicantitruncated 7-cube | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,3,4,5{4,35} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|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 groups | B7, [4,35] |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names =
- Great {{not a typo|terated}} hepteract (acronym: gotesa) (Jonathan Bowers)Klitzing, (x4x3x3x3x3x3o - gotesa)
= Images =
{{B7 Coxeter plane graphs|t012345|150|NOB7A6}}
Related polytopes
These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.
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.
- {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}}
External links
- [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}