8-cubeRectified 8-cubeBirectified 8-cubeTrirectified 8-cubeTrirectified 8-orthoplexBirectified 8-orthoplexRectified 8-orthoplex8-orthoplexOrthogonal projections in B8 Coxeter plane In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.
There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.
Rectified 8-cube Rectified 8-cubeTypeuniform 8-polytopeSchläfli symbol t1{4,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces 256 + 166-faces 2048 + 1125-faces 7168 + 4484-faces 14336 + 1120Cells 17920 + 1792Faces 4336 + 1792Edges 7168Vertices1024Vertex figure6-simplex prism{3,3,3,3,3}×{}Coxeter groupsB8, [36,4]D8, [35,1,1]Propertiesconvex
Alternate names Rectified octeract Acronym: recto (Jonathan Bowers)
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Birectified 8-cube Birectified 8-cubeTypeuniform 8-polytopeCoxeter symbol 0511Schläfli symbol t2{4,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces 256 + 166-faces 1024 + 2048 + 112 5-faces 7168 + 7168 + 4484-faces 21504 + 14336 + 1120Cells 35840 + 17920 + 1792Faces 35840 + 14336Edges21504 Vertices1792Vertex figure{3,3,3,3}x{4}Coxeter groupsB8, [36,4]D8, [35,1,1]Propertiesconvex
Alternate names Birectified octeract Rectified 8-demicube Acronym: bro (Jonathan Bowers)
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Trirectified 8-cube Triectified 8-cubeTypeuniform 8-polytopeSchläfli symbol t3{4,3,3,3,3,3,3}Coxeter diagrams7-faces 16+2566-faces 1024 + 2048 + 112 5-faces 1792 + 7168 + 7168 + 4484-faces 1792 + 10752 + 21504 +14336 Cells8960 + 26880 + 35840Faces17920+35840 Edges17920Vertices1152Vertex figure{3,3,3}x{3,4}Coxeter groupsB8, [36,4]D8, [35,1,1]Propertiesconvex
Alternate names Trirectified octeract Acronym: tro (Jonathan Bowers)
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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. o3o3o3o3o3o3x4o - recto, o3o3o3o3o3x3o4o - bro, o3o3o3o3x3o3o4o - tro
External links Polytopes of Various Dimensions Multi-dimensional Glossary