5-cubeCantellated 5-cubeBicantellated 5-cubeCantellated 5-orthoplex5-orthoplexCantitruncated 5-cubeBicantitruncated 5-cubeCantitruncated 5-orthoplexOrthogonal projections in B5 Coxeter plane In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.
There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex
Cantellated 5-cube Cantellated 5-cubeTypeUniform 5-polytopeSchläfli symbol rr{4,3,3,3} = Coxeter-Dynkin diagram = 4-faces12210 80 32 Cells68040 320 160 160 Faces152080 480 320 640 Edges1280320+960Vertices320Vertex figureCoxeter group B5 [4,3,3,3]Propertiesconvex, uniform
Alternate names Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:
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Bicantellated 5-cube Bicantellated 5-cubeTypeUniform 5-polytopeSchläfli symbols 2rr{4,3,3,3} = r{32,1,1} = Coxeter-Dynkin diagrams = 4-faces12210 80 32 Cells84040 240 160 320 80 Faces2160240 320 960 320 320 Edges1920960+960Vertices480Vertex figureCoxeter groupsB5, [3,3,3,4]D5, [32,1,1]Propertiesconvex, uniform In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.
Alternate names Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross Small birhombated penteractitriacontaditeron (Acronym: sibrant) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of: (0,1,1,2,2)
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Cantitruncated 5-cube Cantitruncated 5-cubeTypeUniform 5-polytopeSchläfli symboltr{4,3,3,3} = Coxeter-Dynkindiagram = 4-faces12210 80 32 Cells68040 320 160 160 Faces152080 480 320 640 Edges1600320+320+960Vertices640Vertex figureCoxeter groupB5 [4,3,3,3]Propertiesconvex, uniform
Alternate names Tricantitruncated 5-orthoplex / tricantitruncated pentacross Great rhombated penteract (girn) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
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Related polytopes It is third in a series of cantitruncated hypercubes:
Bicantitruncated 5-cube Bicantitruncated 5-cubeTypeuniform 5-polytopeSchläfli symbol 2tr{3,3,3,4} = t{32,1,1} = Coxeter-Dynkin diagrams = 4-faces12210 80 32 Cells84040 240 160 320 80 Faces2160240 320 960 320 320 Edges2400960+480+960Vertices960Vertex figureCoxeter groupsB5, [3,3,3,4]D5, [32,1,1]Propertiesconvex, uniform
Alternate names Bicantitruncated penteract Bicantitruncated pentacross Great birhombated penteractitriacontaditeron (Acronym: gibrant) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of (±3,±3,±2,±1,0)
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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, editied 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. o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant
External links
Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary