7-cubeRectified 7-cubeBirectified 7-cubeTrirectified 7-cubeBirectified 7-orthoplexRectified 7-orthoplex7-orthoplexOrthogonal projections in B7 Coxeter plane In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.
There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.
Rectified 7-cube Rectified 7-cubeTypeuniform 7-polytopeSchläfli symbol r{4,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces 128 + 145-faces 896 + 844-faces 2688 + 280Cells 4480 + 560Faces 4480 + 672Edges 2688Vertices448 Vertex figure5-simplex prismCoxeter groupsB7, [3,3,3,3,3,4]Propertiesconvex
Alternate names rectified hepteract (acronym: rasa) (Jonathan Bowers)
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Cartesian coordinates Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,0)
Birectified 7-cube Birectified 7-cubeTypeuniform 7-polytopeCoxeter symbol 0411Schläfli symbol 2r{4,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces 128 + 145-faces 448 + 896 + 844-faces 2688 + 2688 + 280Cells 6720 + 4480 + 560Faces 8960 + 4480Edges 6720Vertices672Vertex figure{3}x{3,3,3}Coxeter groupsB7, [3,3,3,3,3,4]Propertiesconvex
Alternate names Birectified hepteract (acronym: bersa) (Jonathan Bowers)
Images
Cartesian coordinates Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,0,0)
Trirectified 7-cube Trirectified 7-cubeTypeuniform 7-polytopeSchläfli symbol 3r{4,3,3,3,3,3}Coxeter-Dynkin diagrams6-faces128 + 145-faces 448 + 896 + 844-faces 672 + 2688 + 2688 + 280 Cells 3360 + 6720 + 4480Faces 6720 + 8960Edges 6720Vertices 560Vertex figure{3,3}x{3,3}Coxeter groupsB7, [3,3,3,3,3,4]Propertiesconvex
Alternate names Trirectified hepteract Trirectified 7-orthoplex Trirectified heptacross (acronym: sez) (Jonathan Bowers)
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Cartesian coordinates Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,0,0,0)
Related polytopes
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. o3o3o3o3o3x4o - rasa, o3o3o3o3x3o4o - bersa, o3o3o3x3o3o4o - sez
External links Polytopes of Various Dimensions Multi-dimensional Glossary