7-simplexStericated 7-simplexBistericated 7-simplexSteritruncated 7-simplexBisteritruncated 7-simplexStericantellated 7-simplexBistericantellated 7-simplexStericantitruncated 7-simplexBistericantitruncated 7-simplexSteriruncinated 7-simplexSteriruncitruncated 7-simplexSteriruncicantellated 7-simplexBisteriruncitruncated 7-simplexSteriruncicantitruncated 7-simplexBisteriruncicantitruncated 7-simplexOrthogonal projections in A7 Coxeter plane 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-simplexTypeuniform 7-polytopeSchläfli symbol t0,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges2240 Vertices280 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
Alternate names Small cellated octaexon (acronym: sco) (Jonathan Bowers)
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.
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Bistericated 7-simplex Bistericated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges3360 Vertices420 Vertex figureCoxeter groupA7×2, [], order 80320Propertiesconvex
Alternate names Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)
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.
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Steritruncated 7-simplex Steritruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges7280 Vertices1120 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Bisteritruncated 7-simplex Bisteritruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges9240 Vertices1680 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Stericantellated 7-simplex Stericantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges10080 Vertices1680 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Bistericantellated 7-simplex Bistericantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,3,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges15120Vertices2520Vertex figureCoxeter groupA7×2, [], order 80320Propertiesconvex
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.
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Stericantitruncated 7-simplex Stericantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges16800 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Bistericantitruncated 7-simplex Bistericantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,3,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges22680 Vertices5040 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Steriruncinated 7-simplex Steriruncinated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges5040 Vertices1120 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Steriruncitruncated 7-simplex Steriruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges13440 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Steriruncicantellated 7-simplex Steriruncicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,2,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges13440 Vertices3360 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Bisteriruncitruncated 7-simplex Bisteriruncitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,4,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges20160 Vertices5040 Vertex figureCoxeter groupA7×2, [], order 80320Propertiesconvex
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.
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Steriruncicantitruncated 7-simplex Steriruncicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t0,1,2,3,4{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges23520 Vertices6720 Vertex figureCoxeter groupA7, , order 40320Propertiesconvex
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.
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Bisteriruncicantitruncated 7-simplex Bisteriruncicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t1,2,3,4,5{3,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges35280 Vertices10080 Vertex figureCoxeter groupA7×2, [], order 80320Propertiesconvex
Alternate names Great bicellated hexadecaexon (acronym: gabach) (Jonathan Bowers)
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.
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Related polytopes This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
Notes
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (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. 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
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