7-simplexCantellated 7-simplexBicantellated 7-simplexTricantellated 7-simplexBirectified 7-simplexCantitruncated 7-simplexBicantitruncated 7-simplexTricantitruncated 7-simplexOrthogonal projections in A7 Coxeter plane
In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.
There are unique 6 degrees of cantellation for the 7-simplex, including truncations.
Cantellated 7-simplex Cantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol rr{3,3,3,3,3,3}or Coxeter-Dynkin diagramor 6-faces5-faces4-facesCellsFacesEdges1008 Vertices168 Vertex figure5-simplex prismCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Small rhombated octaexon (acronym: saro) (Jonathan Bowers)
Coordinates The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.
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Bicantellated 7-simplex Bicantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol r2r{3,3,3,3,3,3}or Coxeter-Dynkin diagramsor 6-faces5-faces4-facesCellsFacesEdges2520 Vertices420 Vertex figureCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)
Coordinates The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.
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Tricantellated 7-simplex Tricantellated 7-simplexTypeuniform 7-polytopeSchläfli symbol r3r{3,3,3,3,3,3}or Coxeter-Dynkin diagramsor 6-faces5-faces4-facesCellsFacesEdges3360 Vertices560 Vertex figureCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)
Coordinates The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.
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Cantitruncated 7-simplex Cantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol tr{3,3,3,3,3,3}or Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges1176 Vertices336 Vertex figureCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Great rhombated octaexon (acronym: garo) (Jonathan Bowers)Klitizing, (x3x3x3o3o3o3o - garo)
Coordinates The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.
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Bicantitruncated 7-simplex Bicantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t2r{3,3,3,3,3,3}or Coxeter-Dynkin diagramsor 6-faces5-faces4-facesCellsFacesEdges2940 Vertices840 Vertex figureCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)Klitizing, (o3x3x3x3o3o3o - gabro)
Coordinates The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.
Images
Tricantitruncated 7-simplex Tricantitruncated 7-simplexTypeuniform 7-polytopeSchläfli symbol t3r{3,3,3,3,3,3}or Coxeter-Dynkin diagramsor 6-faces5-faces4-facesCellsFacesEdges3920 Vertices1120 Vertex figureCoxeter groupsA7, [3,3,3,3,3,3]Propertiesconvex
Alternate names Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o - gatroh)
Coordinates The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.
Images
Related polytopes This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
See also List of A7 polytopes
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. x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh
External links Polytopes of Various Dimensions Multi-dimensional Glossary