Pentellated 7-cubes
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 pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-orthoplex.
7-cube |
Pentellated 7-cube |
Pentitruncated 7-cube |
Penticantellated 7-cube |
Penticantitruncated 7-cube |
Pentiruncinated 7-cube |
Pentiruncitruncated 7-cube |
Pentiruncicantellated 7-cube |
Pentiruncicantitruncated 7-cube |
Pentistericated 7-cube |
Pentisteritruncated 7-cube |
Pentistericantellated 7-cube |
Pentistericantitruncated 7-cube |
Pentisteriruncinated 7-cube |
Pentisteriruncitruncated 7-cube |
Pentisteriruncicantellated 7-cube |
Pentisteriruncicantitruncated 7-cube |
Pentellated 7-cube
Pentellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Small terated hepteract (acronym: ) (Jonathan Bowers)[1]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentitruncated 7-cube
pentitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Teritruncated hepteract (acronym: ) (Jonathan Bowers)[2]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Penticantellated 7-cube
Penticantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,2,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Terirhombated hepteract (acronym: ) (Jonathan Bowers)[3]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Penticantitruncated 7-cube
penticantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,2,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Terigreatorhombated hepteract (acronym: ) (Jonathan Bowers)[4]
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentiruncinated 7-cube
pentiruncinated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,3,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Teriprismated hepteract (acronym: ) (Jonathan Bowers)[5]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentiruncitruncated 7-cube
pentiruncitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,3,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Teriprismatotruncated hepteract (acronym: ) (Jonathan Bowers)[6]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentiruncicantellated 7-cube
pentiruncicantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,2,3,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Teriprismatorhombated hepteract (acronym: ) (Jonathan Bowers)[7]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentiruncicantitruncated 7-cube
pentiruncicantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,2,3,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Terigreatoprismated hepteract (acronym: ) (Jonathan Bowers)[8]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | too complex | too complex | |
Dihedral symmetry | [6] | [4] |
Pentistericated 7-cube
pentistericated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Tericellated hepteract (acronym: ) (Jonathan Bowers)[9]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentisteritruncated 7-cube
pentisteritruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Tericellitruncated hepteract (acronym: ) (Jonathan Bowers)[10]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentistericantellated 7-cube
pentistericantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,2,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Tericellirhombated hepteract (acronym: ) (Jonathan Bowers)[11]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentistericantitruncated 7-cube
pentistericantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,2,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Tericelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)[12]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentisteriruncinated 7-cube
Pentisteriruncinated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,3,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
- Tericelliprismated hepteract (acronym: ) (Jonathan Bowers)[13]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentisteriruncitruncated 7-cube
pentisteriruncitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,3,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 40320 |
Vertices | 10080 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Tericelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)[14]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentisteriruncicantellated 7-cube
pentisteriruncicantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,2,3,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 40320 |
Vertices | 10080 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
- Tericelliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)[15]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Pentisteriruncicantitruncated 7-cube
pentisteriruncicantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,2,3,4,5{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
- Great terated hepteract (acronym: ) (Jonathan Bowers)[16]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Related polytopes
These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.
Notes
- ↑ Klitzing, (x3o3o3o3o3x4o - )
- ↑ Klitzing, (x3x3o3o3o3x4o - )
- ↑ Klitzing, (x3o3x3o3o3x4o - )
- ↑ Klitzing, (x3x3x3oxo3x4o - )
- ↑ Klitzing, (x3o3o3x3o3x4o - )
- ↑ Klitzing, (x3x3o3x3o3x4o - )
- ↑ Klitzing, (x3o3x3x3o3x4o - )
- ↑ Klitzing, (x3x3x3x3o3x4o - )
- ↑ Klitzing, (x3o3o3o3x3x4o - )
- ↑ Klitzing, (x3x3o3o3x3x4o - )
- ↑ Klitzing, (x3o3x3o3x3x4o - )
- ↑ Klitzing, (x3x3x3o3x3x4o - )
- ↑ Klitzing, (x3o3o3x3x3x4o - )
- ↑ Klitzing, (x3x3o3x3x3x4o - )
- ↑ Klitzing, (x3o3x3x3x3x4o - )
- ↑ Klitzing, (x3x3x3x3x3x4o - )
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 Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (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.
- Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3x4o, x3x3o3o3o3x4o, x3o3x3o3o3x4o, x3x3x3oxo3x4o, x3o3o3x3o3x4o, x3x3o3x3o3x4o, x3o3x3x3o3x4o, x3x3x3x3o3x4o, x3o3o3o3x3x4o, x3x3o3o3x3x4o, x3o3x3o3x3x4o, x3x3x3o3x3x4o, x3o3o3x3x3x4o, x3x3o3x3x3x4o, x3o3x3x3x3x4o, x3x3x3x3x3x3o
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |
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