6-orthoplexCantellated 6-orthoplexBicantellated 6-orthoplex6-cubeCantellated 6-cubeBicantellated 6-cubeCantitruncated 6-orthoplexBicantitruncated 6-orthoplexBicantitruncated 6-cubeCantitruncated 6-cubeOrthogonal projections in B6 Coxeter plane In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.
There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 6-cube
Cantellated 6-orthoplex Cantellated 6-orthoplexTypeuniform 6-polytopeSchläfli symbol t0,2{3,3,3,3,4}rr{3,3,3,3,4}Coxeter-Dynkin diagrams = 5-faces1364-faces1656Cells5040Faces6400Edges3360Vertices480Vertex figureCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Cantellated hexacross Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of (2,1,1,0,0,0)
Images
Bicantellated 6-orthoplex Bicantellated 6-orthoplexTypeuniform 6-polytopeSchläfli symbol t1,3{3,3,3,3,4}2rr{3,3,3,3,4}Coxeter-Dynkin diagrams5-faces4-facesCellsFacesEdges8640Vertices1440Vertex figureCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Bicantellated hexacross, bicantellated hexacontatetrapeton Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of (2,2,1,1,0,0)
Images
Cantitruncated 6-orthoplex Cantitruncated 6-orthoplexTypeuniform 6-polytopeSchläfli symbol t0,1,2{3,3,3,3,4}tr{3,3,3,3,4}Coxeter-Dynkin diagrams5-faces4-facesCellsFacesEdges3840Vertices960Vertex figureCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Cantitruncated hexacross, cantitruncated hexacontatetrapeton Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of (3,2,1,0,0,0)
Images
Bicantitruncated 6-orthoplex Bicantitruncated 6-orthoplexTypeuniform 6-polytopeSchläfli symbol t1,2,3{3,3,3,3,4}2tr{3,3,3,3,4}Coxeter-Dynkin diagrams5-faces4-facesCellsFacesEdges10080Vertices2880Vertex figureCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of (3,3,2,1,0,0)
Images
Related polytopes These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube and 6-orthoplex.
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. x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg
External links Polytopes of Various Dimensions Multi-dimensional Glossary