Pentellated_7-simplex
In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.
 7-simplex    |
 Pentellated 7-simplex |
 Pentitruncated 7-simplex |
 Penticantellated 7-simplex |
 Penticantitruncated 7-simplex |
 Pentiruncinated 7-simplex |
 Pentiruncitruncated 7-simplex |
 Pentiruncicantellated 7-simplex |
 Pentiruncicantitruncated 7-simplex |
 Pentistericated 7-simplex |
 Pentisteritruncated 7-simplex |
 Pentistericantellated 7-simplex |
 Pentistericantitruncated 7-simplex |
 Pentisteriruncinated 7-simplex |
 Pentisteriruncitruncated 7-simplex |
 Pentisteriruncicantellated 7-simplex |
 Pentisteriruncicantitruncated 7-simplex | |||
There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications.
| Pentellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1260 |
| Vertices | 168 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Small terated octaexon (acronym: seto) (Jonathan Bowers)[1]
Coordinates
The vertices of the pentellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |
 |
 |
| Dihedral symmetry | [5] | [4] | [3] |
| pentitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5460 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Teritruncated octaexon (acronym: teto) (Jonathan Bowers)[2]
Coordinates
The vertices of the pentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| Penticantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11760 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Terirhombated octaexon (acronym: tero) (Jonathan Bowers)[3]
Coordinates
The vertices of the penticantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,2,3). This construction is based on facets of the penticantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| penticantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Terigreatorhombated octaexon (acronym: tegro) (Jonathan Bowers)[4]
Coordinates
The vertices of the penticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 8-orthoplex.
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentiruncinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10920 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Teriprismated octaexon (acronym: tepo) (Jonathan Bowers)[5]
Coordinates
The vertices of the pentiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the pentiruncinated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentiruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 27720 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Teriprismatotruncated octaexon (acronym: tapto) (Jonathan Bowers)[6]
Coordinates
The vertices of the pentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,4). This construction is based on facets of the pentiruncitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentiruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 25200 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Teriprismatorhombated octaexon (acronym: tapro) (Jonathan Bowers)[7]
Coordinates
The vertices of the pentiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentiruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Terigreatoprismated octaexon (acronym: tegapo) (Jonathan Bowers)[8]
Coordinates
The vertices of the pentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentistericated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4200 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Tericellated octaexon (acronym: teco) (Jonathan Bowers)[9]
Coordinates
The vertices of the pentistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the pentistericated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentisteritruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Tericellitruncated octaexon (acronym: tecto) (Jonathan Bowers)[10]
Coordinates
The vertices of the pentisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the pentisteritruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentistericantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 25200 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Tericellirhombated octaexon (acronym: tecro) (Jonathan Bowers)[11]
Coordinates
The vertices of the pentistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,3,4). This construction is based on facets of the pentistericantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentistericantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Tericelligreatorhombated octaexon (acronym: tecagro) (Jonathan Bowers)[12]
Coordinates
The vertices of the pentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| Pentisteriruncinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,3,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Bipenticantitruncated 7-simplex as t1,2,3,6{3,3,3,3,3,3}
- Tericelliprismated octaexon (acronym: tacpo) (Jonathan Bowers)[13]
Coordinates
The vertices of the pentisteriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,3,4). This construction is based on facets of the pentisteriruncinated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [5] | [4] | [3] |
| pentisteriruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Tericelliprismatotruncated octaexon (acronym: tacpeto) (Jonathan Bowers)[14]
Coordinates
The vertices of the pentisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,4,5). This construction is based on facets of the pentisteriruncitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| pentisteriruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Bipentiruncicantitruncated 7-simplex as t1,2,3,4,6{3,3,3,3,3,3}
- Tericelliprismatorhombated octaexon (acronym: tacpro) (Jonathan Bowers)[15]
Coordinates
The vertices of the pentisteriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,4,5). This construction is based on facets of the pentisteriruncicantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [[5]] | [4] | [[3]] |
| pentisteriruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 70560 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter groups | A7, [3,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Great terated octaexon (acronym: geto) (Jonathan Bowers)[16]
Coordinates
The vertices of the pentisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [8] | [[7]] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | |||
| Dihedral symmetry | [[5]] | [4] | [[3]] |
These polytopes are a part of a set of 71 uniform 7-polytopes with A7 symmetry.
- Klitzing, (x3o3o3o3o3x3o - seto)
- Klitzing, (x3x3o3o3o3x3o - teto)
- Klitzing, (x3o3x3o3o3x3o - tero)
- Klitzing, (x3x3x3oxo3x3o - tegro)
- Klitzing, (x3o3o3x3o3x3o - tepo)
- Klitzing, (x3x3o3x3o3x3o - tapto)
- Klitzing, (x3o3x3x3o3x3o - tapro)
- Klitzing, (x3x3x3x3o3x3o - tegapo)
- Klitzing, (x3o3o3o3x3x3o - teco)
- Klitzing, (x3x3o3o3x3x3o - tecto)
- Klitzing, (x3o3x3o3x3x3o - tecro)
- Klitzing, (x3x3x3o3x3x3o - tecagro)
- Klitzing, (x3o3o3x3x3x3o - tacpo)
- Klitzing, (x3x3o3x3x3x3o - tacpeto)
- Klitzing, (x3o3x3x3x3x3o - tacpro)
- Klitzing, (x3x3x3x3x3x3o - geto)
- 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.
- Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3x3o - seto, x3x3o3o3o3x3o - teto, x3o3x3o3o3x3o - tero, x3x3x3oxo3x3o - tegro, x3o3o3x3o3x3o - tepo, x3x3o3x3o3x3o - tapto, x3o3x3x3o3x3o - tapro, x3x3x3x3o3x3o - tegapo, x3o3o3o3x3x3o - teco, x3x3o3o3x3x3o - tecto, x3o3x3o3x3x3o - tecro, x3x3x3o3x3x3o - tecagro, x3o3o3x3x3x3o - tacpo, x3x3o3x3x3x3o - tacpeto, x3o3x3x3x3x3o - tacpro, x3x3x3x3x3x3o - geto
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | Pentachoron | 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 | ||||||||||||