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 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 Alternate names Bitruncated octacross (acronym batek) (Jonthan Bowers) 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 Alternate names Tritruncated octacross (acronym tatek) (Jonthan Bowers) 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 Ivic Weiss, Wiley-Interscience Publication, 1995, (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 8-polytopes