8-orthoplexTruncated 8-orthoplexBitruncated 8-orthoplexTritruncated 8-orthoplexQuadritruncated 8-cubeTritruncated 8-cubeBitruncated 8-cubeTruncated 8-cube8-cubeOrthogonal projections in B8 Coxeter plane In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.
Truncated 8-orthoplex Truncated 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t0,1{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges1456Vertices224Vertex figure( )v{3,3,3,4}Coxeter groupsB8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex
Alternate names Truncated octacross (acronym: tek) (Jonthan Bowers)
Construction There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.
Coordinates Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of (±2,±1,0,0,0,0,0,0)
Images
Bitruncated 8-orthoplex Bitruncated 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t1,2{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{ }v{3,3,3,4}Coxeter groupsB8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex
Alternate names Bitruncated octacross (acronym: batek) (Jonthan Bowers)Klitizing, (o3x3x3o3o3o3o4o - batek)
Coordinates Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of (±2,±2,±1,0,0,0,0,0)
Images
Tritruncated 8-orthoplex Tritruncated 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t2,3{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{3}v{3,3,4}Coxeter groupsB8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex
Alternate names Tritruncated octacross (acronym: tatek) (Jonthan Bowers)Klitizing, (o3o3x3x3o3o3o4o - tatek)
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,±1,0,0,0,0)
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. (1966) x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek
External links Polytopes of Various Dimensions Multi-dimensional Glossary