6-demicube = Runcic 6-cube = Runcicantic 6-cube = Orthogonal projections in D6 Coxeter plane In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.
Runcic 6-cube Runcic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,2{3,33,1}h3{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges3840Vertices640Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex
Alternate names Cantellated 6-demicube Cantellated demihexeract Small rhombated hemihexeract (Acronym: sirhax) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±3) with an odd number of plus signs.
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Related polytopes
Runcicantic 6-cube Runcicantic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,1,2{3,33,1}h2,3{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges5760Vertices1920Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex
Alternate names Cantitruncated 6-demicube Cantitruncated demihexeract Great rhombated hemihexeract (Acronym: girhax) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations: (±1,±1,±3,±5,±5,±5) with an odd number of plus signs.
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Related polytopes
This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
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. x3o3o *b3x3o3o, x3x3o *b3x3o3o
External links
Polytopes of Various Dimensions Multi-dimensional Glossary