8-cubeTruncated 8-cubeBitruncated 8-cubeTritruncated 8-cubeQuadritruncated 8-cubeTritruncated 8-orthoplexBitruncated 8-orthoplexTruncated 8-orthoplex8-orthoplexOrthogonal projections in B8 Coxeter plane In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.
There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.
Truncated 8-cube Truncated 8-cubeTypeuniform 8-polytopeSchläfli symbol t{4,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure( )v{3,3,3,3,3}Coxeter groupsB8, [3,3,3,3,3,3,4]Propertiesconvex
Alternate names Truncated octeract (acronym: tocto) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of (±2,±2,±2,±2,±2,±2,±1,0)
Images
Related polytopes The truncated 8-cube, is seventh in a sequence of truncated hypercubes:
Bitruncated 8-cube Bitruncated 8-cubeTypeuniform 8-polytopeSchläfli symbol 2t{4,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{ }v{3,3,3,3}Coxeter groupsB8, [3,3,3,3,3,3,4]Propertiesconvex
Alternate names Bitruncated octeract (acronym: bato) (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of (±2,±2,±2,±2,±2,±1,0,0)
Images
Related polytopes The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:
Tritruncated 8-cube Tritruncated 8-cubeTypeuniform 8-polytopeSchläfli symbol 3t{4,3,3,3,3,3,3}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{4}v{3,3,3}Coxeter groupsB8, [3,3,3,3,3,3,4]Propertiesconvex
Alternate names Tritruncated octeract (acronym: tato) (Jonathan Bowers)Klitizing, (o3o3o3o3x3x3o4o – tato)
Coordinates Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of (±2,±2,±2,±2,±1,0,0,0)
Images
Quadritruncated 8-cube Quadritruncated 8-cubeTypeuniform 8-polytopeSchläfli symbol 4t{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{3,4}v{3,3}Coxeter groupsB8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex
Alternate names Quadritruncated octeract (acronym: oke) (Jonathan Bowers)Klitizing, (o3o3o3x3x3o3o4o – oke)
Coordinates Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of (±2,±2,±2,±2,±1,0,0,0)
Images
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. o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke
External links Polytopes of Various Dimensions Multi-dimensional Glossary