6-simplexCantellated 6-simplexBicantellated 6-simplexBirectified 6-simplexCantitruncated 6-simplexBicantitruncated 6-simplexOrthogonal projections in A6 Coxeter plane
In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.
There are unique 4 degrees of cantellation for the 6-simplex, including truncations.
Cantellated 6-simplex Cantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol rr{3,3,3,3,3}or Coxeter-Dynkin diagrams5-faces354-faces210Cells560Faces805Edges525Vertices105Vertex figure5-cell prismCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)
Coordinates The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.
Images
Bicantellated 6-simplex Bicantellated 6-simplexTypeuniform 6-polytopeSchläfli symbol 2rr{3,3,3,3,3}or Coxeter-Dynkin diagrams5-faces494-faces329Cells980Faces1540Edges1050Vertices210Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)
Coordinates The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.
Images
Cantitruncated 6-simplex Cantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol tr{3,3,3,3,3}or Coxeter-Dynkin diagrams5-faces35 4-faces210 Cells560 Faces805 Edges630 Vertices210 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)
Coordinates The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.
Images
Bicantitruncated 6-simplex Bicantitruncated 6-simplexTypeuniform 6-polytopeSchläfli symbol 2tr{3,3,3,3,3}or Coxeter-Dynkin diagrams5-faces49 4-faces329 Cells980 Faces1540 Edges1260 Vertices420 Vertex figureCoxeter groupA6, , order 5040Propertiesconvex
Alternate names Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)
Coordinates The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.
Images
Related uniform 6-polytopes The cantellated 6-simplexes are in a set 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
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. x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril
External links Polytopes of Various Dimensions Multi-dimensional Glossary