7-demicubeRuncic 7-cubeRuncicantic 7-cubeOrthogonal projections in D7 Coxeter plane In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.
Runcic 7-cube Runcic 7-cubeTypeuniform 7-polytopeSchläfli symbol t0,2{3,34,1}h3{4,35}Coxeter-Dynkin diagram6-faces5-faces4-facesCellsFacesEdges16800Vertices2240Vertex figureCoxeter groupsD7, [34,1,1]Propertiesconvex
A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.
Alternate names Small rhombated hemihepteract (acronym: sirhesa) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±3,±3) with an odd number of plus signs.
Images
Runcicantic 7-cube Runcicantic 7-cubeTypeuniform 7-polytopeSchläfli symbol t0,1,2{3,34,1}h2,3{4,35}Coxeter-Dynkin diagram6-faces5-faces4-facesCellsFacesEdges23520Vertices6720Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.
Alternate names Great rhombated hemihepteract (acronym: girhesa) (Jonathan Bowers)Klitzing, (x3x3o *b3x3o3o3o - girhesa)
Cartesian coordinates The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations: (±1,±1,±1,±1,±3,±5,±5) with an odd number of plus signs.
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Related polytopes These polytopes are based on the 7-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 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 *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa
External links
Polytopes of Various Dimensions Multi-dimensional Glossary