Stericated 7-simplexes
 7-simplex   
|
 Stericated 7-simplex
|
 Bistericated 7-simplex
|
 Steritruncated 7-simplex
|
 Bisteritruncated 7-simplex
|
 Stericantellated 7-simplex
|
 Bistericantellated 7-simplex
|
 Stericantitruncated 7-simplex
|
 Bistericantitruncated 7-simplex
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 Steriruncinated 7-simplex
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 Steriruncitruncated 7-simplex
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 Steriruncicantellated 7-simplex
|
 Bisteriruncitruncated 7-simplex
|
 Steriruncicantitruncated 7-simplex
|
 Bisteriruncicantitruncated 7-simplex
|
In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex.
There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations.
Stericated 7-simplex
| Stericated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2240 |
| Vertices | 280 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Small cellated octaexon (acronym: sco) (Jonathan Bowers)[1]
Coordinates
The vertices of the stericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 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] |
Bistericated 7-simplex
| bistericated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A7×2, 36, order 80320 |
| Properties | convex |
Alternate names
- Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)[2]
Coordinates
The vertices of the bistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex.
Images
Steritruncated 7-simplex
| steritruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7280 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)[3]
Coordinates
The vertices of the steritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 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] |
Bisteritruncated 7-simplex
| bisteritruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 9240 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)[4]
Coordinates
The vertices of the bisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,3). This construction is based on facets of the bisteritruncated 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] |
Stericantellated 7-simplex
| Stericantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10080 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)[5]
Coordinates
The vertices of the stericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 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] |
Bistericantellated 7-simplex
| Bistericantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 2520 |
| Vertex figure | |
| Coxeter group | A7×2, 36, order 80320 |
| Properties | convex |
Alternate names
- Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)[6]
Coordinates
The vertices of the bistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,3). This construction is based on facets of the stericantellated 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] |
Stericantitruncated 7-simplex
| stericantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)[7]
Coordinates
The vertices of the stericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,4). This construction is based on facets of the stericantitruncated 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] |
Bistericantitruncated 7-simplex
| bistericantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,3,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 22680 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)[8]
Coordinates
The vertices of the bistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,4). This construction is based on facets of the bistericantitruncated 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] |
Steriruncinated 7-simplex
| Steriruncinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5040 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Celliprismated octaexon (acronym: cepo) (Jonathan Bowers)[9]
Coordinates
The vertices of the steriruncinated 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 steriruncinated 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] |
Steriruncitruncated 7-simplex
| steriruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Celliprismatotruncated octaexon (acronym: capto) (Jonathan Bowers)[10]
Coordinates
The vertices of the steriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,4). This construction is based on facets of the steriruncitruncated 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] |
Steriruncicantellated 7-simplex
| steriruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Celliprismatorhombated octaexon (acronym: capro) (Jonathan Bowers)[11]
Coordinates
The vertices of the steriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,3,4). This construction is based on facets of the steriruncicantellated 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] |
Bisteriruncitruncated 7-simplex
| bisteriruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A7×2, 36, order 80320 |
| Properties | convex |
Alternate names
- Bicelliprismatotruncated hexadecaexon (acronym: bicpath) (Jonathan Bowers)[12]
Coordinates
The vertices of the bisteriruncitruncated 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 bisteriruncitruncated 8-orthoplex.
Images
Steriruncicantitruncated 7-simplex
| steriruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23520 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Great cellated octaexon (acronym: gecco) (Jonathan Bowers)[13]
Coordinates
The vertices of the steriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 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] |
Bisteriruncicantitruncated 7-simplex
| bisteriruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,3,4,5{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 35280 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A7×2, 36, order 80320 |
| Properties | convex |
Alternate names
- Great bicellated hexadecaexon (gabach) (Jonathan Bowers) [14]
Coordinates
The vertices of the bisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,5). This construction is based on facets of the bisteriruncicantitruncated 8-orthoplex.
Images
Related polytopes
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
Notes
- ↑ Klitizing, (x3o3o3o3x3o3o - sco)
- ↑ Klitizing, (o3x3o3o3o3x3o - sabach)
- ↑ Klitizing, (x3x3o3o3x3o3o - cato)
- ↑ Klitizing, (o3x3x3o3o3x3o - bacto)
- ↑ Klitizing, (x3o3x3o3x3o3o - caro)
- ↑ Klitizing, (o3x3o3x3o3x3o - bacroh)
- ↑ Klitizing, (x3x3x3o3x3o3o - cagro)
- ↑ Klitizing, (o3x3x3x3o3x3o - bacogro)
- ↑ Klitizing, (x3o3o3x3x3o3o - cepo)
- ↑ Klitizing, (x3x3x3o3x3o3o - capto)
- ↑ Klitizing, (x3o3x3x3x3o3o - capro)
- ↑ Klitizing, (o3x3x3o3x3x3o - bicpath)
- ↑ Klitizing, (x3x3x3x3x3o3o - gecco)
- ↑ Klitizing, (o3x3x3x3x3x3o - gabach)
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 [1]
- (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)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. x3o3o3o3x3o3o - sco, o3x3o3o3o3x3o - sabach, x3x3o3o3x3o3o - cato, o3x3x3o3o3x3o - bacto, x3o3x3o3x3o3o - caro, o3x3o3x3o3x3o - bacroh, x3x3x3o3x3o3o - cagro, o3x3x3x3o3x3o - bacogro, x3o3o3x3x3o3o - cepo, x3x3x3o3x3o3o - capto, x3o3x3x3x3o3o - capro, o3x3x3o3x3x3o - bicpath, x3x3x3x3x3o3o - gecco, o3x3x3x3x3x3o - gabach
External links
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 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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