Stericated 6-simplexes
![]() 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 |
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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
[edit]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
[edit]- Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)[1]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)[2]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)[3]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)[4]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)[5]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)[6]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
- Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)[7]
Coordinates
[edit]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
[edit]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
[edit]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
[edit]- Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)[8]
Coordinates
[edit]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
[edit]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
[edit]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.
Notes
[edit]References
[edit]- 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. "6D uniform polytopes (polypeta)".