7-orthoplexTruncated 7-orthoplexBitruncated 7-orthoplexTritruncated 7-orthoplex7-cubeTruncated 7-cubeBitruncated 7-cubeTritruncated 7-cubeOrthogonal projections in B7 Coxeter plane In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.
Truncated 7-orthoplex Truncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol t{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCells3920Faces2520Edges924Vertices168Vertex figure( )v{3,3,4}Coxeter groupsB7, [35,4]D7, [34,1,1]Propertiesconvex
Alternate names Truncated heptacross Truncated hecatonicosaoctaexon (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of (±2,±1,0,0,0,0,0)
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Construction There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.
Bitruncated 7-orthoplex Bitruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol 2t{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges4200Vertices840Vertex figure{ }v{3,3,4}Coxeter groupsB7, [35,4]D7, [34,1,1]Propertiesconvex
Alternate names Bitruncated heptacross Bitruncated hecatonicosaoctaexon (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of (±2,±2,±1,0,0,0,0)
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Tritruncated 7-orthoplex The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.
Tritruncated 7-orthoplexTypeuniform 7-polytopeSchläfli symbol 3t{35,4}Coxeter-Dynkin diagrams6-faces5-faces4-facesCellsFacesEdges10080Vertices2240Vertex figure{3}v{3,4}Coxeter groupsB7, [35,4]D7, [34,1,1]Propertiesconvex
Alternate names Tritruncated heptacross Tritruncated hecatonicosaoctaexon (Jonathan Bowers)
Coordinates Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of (±2,±2,±2,±1,0,0,0)
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. x3x3o3o3o3o4o - taz, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz
External links Polytopes of Various Dimensions Multi-dimensional Glossary