| Orthogonal projections in A6 Coxeter plane |
|---|
| 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 |
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.
Stericated 6-simplex
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| 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
| 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)
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
Orthographic projections| Ak Coxeter plane | A6 | A5 | A4 |
|---|
| Graph | | | |
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 |
| Graph | | |
| Dihedral symmetry | [4] | [3] |
Steriruncicantitruncated 6-simplex
| 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)
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
Orthographic projections| Ak Coxeter plane | A6 | A5 | A4 |
|---|
| Graph | | | |
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 |
| Graph | | |
| Dihedral symmetry | [4] | [3] |
Related uniform 6-polytopes
The stericated 6-simplexes are in a set of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
| A6 polytopes |
|---|
| t0 | t1 | t2 | t0,1 | t0,2 | t1,2 | t0,3 | t1,3 | t2,3 |
| t0,4 | t1,4 | t0,5 | t0,1,2 | t0,1,3 | t0,2,3 | t1,2,3 | t0,1,4 | t0,2,4 |
| t1,2,4 | t0,3,4 | t0,1,5 | t0,2,5 | t0,1,2,3 | t0,1,2,4 | t0,1,3,4 | t0,2,3,4 | t1,2,3,4 |
| t0,1,2,5 | t0,1,3,5 | t0,2,3,5 | t0,1,4,5 | t0,1,2,3,4 | t0,1,2,3,5 | t0,1,2,4,5 | t0,1,2,3,4,5 |
Notes
- 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 Ivić 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. .
External links