7-cubeTruncated 7-cubeBitruncated 7-cubeTritruncated 7-cube7-orthoplexTruncated 7-orthoplexBitruncated 7-orthoplexTritruncated 7-orthoplexOrthogonal projections in B7 Coxeter plane In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.
There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.
Truncated 7-cube Truncated 7-cubeTypeuniform 7-polytopeSchläfli symbol t{4,35}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges3136Vertices896Vertex figureElongated 5-simplex pyramidCoxeter groupsB7, [35,4]Propertiesconvex
Alternate names Truncated hepteract (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of (1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)
Images
Related polytopes The truncated 7-cube, is sixth in a sequence of truncated hypercubes:
Bitruncated 7-cube Bitruncated 7-cubeTypeuniform 7-polytopeSchläfli symbol 2t{4,35}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges9408Vertices2688Vertex figure{ }v{3,3,3}Coxeter groupsB7, [35,4]D7, [34,1,1]Propertiesconvex
Alternate names Bitruncated hepteract (Jonathan Bowers)Klitzing (o4x3x3o3o3o3o - betsa)
Coordinates Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of (±2,±2,±2,±2,±2,±1,0)
Images
Related polytopes The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:
Tritruncated 7-cube Tritruncated 7-cubeTypeuniform 7-polytopeSchläfli symbol 3t{4,35}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges13440Vertices3360Vertex figure{4}v{3,3}Coxeter groupsB7, [35,4]D7, [34,1,1]Propertiesconvex
Alternate names Tritruncated hepteract (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of (±2,±2,±2,±2,±1,0,0)
Images
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. x4x3o3o3o3o3o - tasa, o4x3x3o3o3o3o - betsa, o4o3x3x3o3o3o - tatsa
External links Polytopes of Various Dimensions Multi-dimensional Glossary