Stericated 7-simplexes
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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
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! style="background:#e7dcc3;" colspan="2"|Stericated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 2240 |
style="background:#e7dcc3;"|Vertices | 280 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Small cellated octaexon (acronym: sco) (Jonathan Bowers)Klitizing, (x3o3o3o3x3o3o - sco)
= 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 =
{{7-simplex Coxeter plane graphs|t04|150}}
Bistericated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|bistericated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 3360 |
style="background:#e7dcc3;"|Vertices | 420 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7×2, 36, order 80320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)Klitizing, (o3x3o3o3o3x3o - sabach)
= 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 =
{{7-simplex2 Coxeter plane graphs|t15|150}}
Steritruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|steritruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 7280 |
style="background:#e7dcc3;"|Vertices | 1120 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)Klitizing, (x3x3o3o3x3o3o - cato)
= 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 =
{{7-simplex Coxeter plane graphs|t014|150}}
Bisteritruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|bisteritruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,2,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 9240 |
style="background:#e7dcc3;"|Vertices | 1680 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)Klitizing, (o3x3x3o3o3x3o - bacto)
= 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 =
{{7-simplex Coxeter plane graphs|t125|150}}
Stericantellated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|Stericantellated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 10080 |
style="background:#e7dcc3;"|Vertices | 1680 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)Klitizing, (x3o3x3o3x3o3o - caro)
= 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 =
{{7-simplex Coxeter plane graphs|t024|150}}
Bistericantellated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|Bistericantellated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,3,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 15120 |
style="background:#e7dcc3;"|Vertices | 2520 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7×2, 36, order 80320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)Klitizing, (o3x3o3x3o3x3o - bacroh)
= 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 =
{{7-simplex Coxeter plane graphs|t135|150}}
Stericantitruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|stericantitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 16800 |
style="background:#e7dcc3;"|Vertices | 3360 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)Klitizing, (x3x3x3o3x3o3o - cagro)
= 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 =
{{7-simplex Coxeter plane graphs|t0124|150}}
Bistericantitruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|bistericantitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,2,3,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 22680 |
style="background:#e7dcc3;"|Vertices | 5040 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)Klitizing, (o3x3x3x3o3x3o - bacogro)
= 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 =
{{7-simplex Coxeter plane graphs|t1235|150}}
Steriruncinated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|Steriruncinated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,3,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 5040 |
style="background:#e7dcc3;"|Vertices | 1120 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Celliprismated octaexon (acronym: cepo) (Jonathan Bowers)Klitizing, (x3o3o3x3x3o3o - cepo)
= 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 =
{{7-simplex Coxeter plane graphs|t034|150}}
Steriruncitruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|steriruncitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,3,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 13440 |
style="background:#e7dcc3;"|Vertices | 3360 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Celliprismatotruncated octaexon (acronym: capto) (Jonathan Bowers)Klitizing, (x3x3x3o3x3o3o - capto)
= 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 =
{{7-simplex Coxeter plane graphs|t0134|150}}
Steriruncicantellated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|steriruncicantellated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,2,3,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 13440 |
style="background:#e7dcc3;"|Vertices | 3360 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Celliprismatorhombated octaexon (acronym: capro) (Jonathan Bowers)Klitizing, (x3o3x3x3x3o3o - capro)
= 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 =
{{7-simplex Coxeter plane graphs|t0234|150}}
Bisteriruncitruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|bisteriruncitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,2,4,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 20160 |
style="background:#e7dcc3;"|Vertices | 5040 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7×2, 36, order 80320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Bicelliprismatotruncated hexadecaexon (acronym: bicpath) (Jonathan Bowers)Klitizing, (o3x3x3o3x3x3o - bicpath)
= 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 =
{{7-simplex2 Coxeter plane graphs|t1245|150}}
Steriruncicantitruncated 7-simplex
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! style="background:#e7dcc3;" colspan="2"|steriruncicantitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t0,1,2,3,4{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 23520 |
style="background:#e7dcc3;"|Vertices | 6720 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7, [36], order 40320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Great cellated octaexon (acronym: gecco) (Jonathan Bowers)Klitizing, (x3x3x3x3x3o3o - gecco)
= 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 =
{{7-simplex Coxeter plane graphs|t01234|150}}
Bisteriruncicantitruncated 7-simplex
class="wikitable" align="right" style="margin-left:10px" width="320"
! style="background:#e7dcc3;" colspan="2"|bisteriruncicantitruncated 7-simplex | |
style="background:#e7dcc3;"|Type | uniform 7-polytope |
style="background:#e7dcc3;"|Schläfli symbol | t1,2,3,4,5{3,3,3,3,3,3} |
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams | {{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}} |
style="background:#e7dcc3;"|6-faces | |
style="background:#e7dcc3;"|5-faces | |
style="background:#e7dcc3;"|4-faces | |
style="background:#e7dcc3;"|Cells | |
style="background:#e7dcc3;"|Faces | |
style="background:#e7dcc3;"|Edges | 35280 |
style="background:#e7dcc3;"|Vertices | 10080 |
style="background:#e7dcc3;"|Vertex figure | |
style="background:#e7dcc3;"|Coxeter group | A7×2, 36, order 80320 |
style="background:#e7dcc3;"|Properties | convex |
= Alternate names=
- Great bicellated hexadecaexon (gabach) (Jonathan Bowers) Klitizing, (o3x3x3x3x3x3o - gabach)
= 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 =
{{7-simplex2 Coxeter plane graphs|t12345|150}}
Related polytopes
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
{{Octaexon family}}
Notes
{{reflist}}
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
- (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.
- {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}} 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
- [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}