5-orthoplexCantellated 5-orthoplexBicantellated 5-cubeCantellated 5-cube5-cubeCantitruncated 5-orthoplexBicantitruncated 5-cubeCantitruncated 5-cubeOrthogonal projections in B5 Coxeter plane In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.
There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.
Cantellated 5-orthoplex Cantellated 5-orthoplexTypeUniform 5-polytopeSchläfli symbol rr{3,3,3,4}rr{3,3,31,1}Coxeter-Dynkin diagrams4-faces8210 40 32 Cells64080 160 320 80 Faces1520640 320 480 80 Edges1200960 240 Vertices240Vertex figureSquare pyramidal prism Coxeter group B5, [4,3,3,3], order 3840D5, [32,1,1], order 1920Propertiesconvex
Alternate names Cantellated 5-orthoplex Bicantellated 5-demicube Small rhombated triacontaditeron (Acronym: sart) (Jonathan Bowers)
Coordinates The vertices of the can be made in 5-space, as permutations and sign combinations of: (0,0,1,1,2)
Images The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.
Cantitruncated 5-orthoplex Cantitruncated 5-orthoplexTypeuniform 5-polytopeSchläfli symbol tr{3,3,3,4}tr{3,31,1}Coxeter-Dynkin diagrams4-faces8210 40 32 Cells64080 160 320 80 Faces1520640 320 480 80 Edges1440960 240 240 Vertices480Vertex figureSquare pyramidal pyramid Coxeter groupsB5, [3,3,3,4], order 3840D5, [32,1,1], order 1920Propertiesconvex
Alternate names Cantitruncated pentacross Cantitruncated triacontaditeron (Acronym: gart) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of (±3,±2,±1,0,0)
Images
Related polytopes These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-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. x3o3x3o4o - sart, x3x3x3o4o - gart
External links
Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary