Stericated_6-simplexes
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
 6-simplex    |
 Stericated 6-simplex |
 Steritruncated 6-simplex |
 Stericantellated 6-simplex |
 Stericantitruncated 6-simplex |
 Steriruncinated 6-simplex |
 Steriruncitruncated 6-simplex |
 Steriruncicantellated 6-simplex |
 Steriruncicantitruncated 6-simplex |
| Orthogonal projections in A6 Coxeter plane | ||
|---|---|---|
There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.
| Stericated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 700 |
| Cells | 1470 |
| Faces | 1400 |
| Edges | 630 |
| Vertices | 105 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)[1]
Coordinates
The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph |  |
 | |
| Dihedral symmetry | [4] | [3] |
| Steritruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 945 |
| Cells | 2940 |
| Faces | 3780 |
| Edges | 2100 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)[2]
Coordinates
The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| Stericantellated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,2,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1050 |
| Cells | 3465 |
| Faces | 5040 |
| Edges | 3150 |
| Vertices | 630 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)[3]
Coordinates
The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| stericantitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,2,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1155 |
| Cells | 4410 |
| Faces | 7140 |
| Edges | 5040 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)[4]
Coordinates
The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| steriruncinated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,3,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 700 |
| Cells | 1995 |
| Faces | 2660 |
| Edges | 1680 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)[5]
Coordinates
The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| steriruncitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,3,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 945 |
| Cells | 3360 |
| Faces | 5670 |
| Edges | 4410 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)[6]
Coordinates
The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| steriruncicantellated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,2,3,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1050 |
| Cells | 3675 |
| Faces | 5880 |
| Edges | 4410 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
- Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)[7]
Coordinates
The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| Steriuncicantitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,2,3,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1155 |
| Cells | 4620 |
| Faces | 8610 |
| Edges | 7560 |
| Vertices | 2520 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)[8]
Coordinates
The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
- 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. "6D uniform polytopes (polypeta)".
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 | ||||||||||||