Cantellated8-simplexBicantellated8-simplexTricantellated8-simplexCantitruncated8-simplexBicantitruncated8-simplexTricantitruncated8-simplexOrthogonal projections in A8 Coxeter plane
In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.
There are six unique cantellations for the 8-simplex, including permutations of truncation.
Cantellated 8-simplex Cantellated 8-simplexTypeuniform 8-polytopeSchläfli symbol rr{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdges1764 Vertices252 Vertex figure6-simplex prismCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.
Images
Bicantellated 8-simplex Bicantellated 8-simplexTypeuniform 8-polytopeSchläfli symbol r2r{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdges5292Vertices756Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)Klitizing, (o3x3o3x3o3o3o3o - sabrene)
Coordinates The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.
Images
Tricantellated 8-simplex Tricantellated 8-simplexTypeuniform 8-polytopeSchläfli symbol r3r{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdges8820Vertices1260Vertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)Klitizing, (o3o3x3o3x3o3o3o - satrene)
Coordinates The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.
Images
Cantitruncated 8-simplex Cantitruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol tr{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)Klitizing, (x3x3x3o3o3o3o3o - grene)
Coordinates The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
Bicantitruncated 8-simplex Bicantitruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol t2r{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)Klitizing, (o3x3x3x3o3o3o3o - gabrene)
Coordinates The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
Tricantitruncated 8-simplex Tricantitruncated 8-simplexTypeuniform 8-polytopeSchläfli symbol t3r{3,3,3,3,3,3,3}Coxeter-Dynkin diagram7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupA8, , order 362880Propertiesconvex
Alternate names Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o3o - gatrene)
Coordinates The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
Related polytopes The six presented polytopes are in the family of 135 uniform 8-polytopes with A8 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. x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene
External links Polytopes of Various Dimensions Multi-dimensional Glossary