7-orthoplexRectified 7-orthoplexBirectified 7-orthoplexTrirectified 7-orthoplexBirectified 7-cubeRectified 7-cube7-cubeOrthogonal projections in B7 Coxeter plane In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.
Rectified 7-orthoplex Rectified 7-orthoplexTypeuniform 7-polytopeSchläfli symbol r{3,3,3,3,3,4}Coxeter-Dynkin diagrams6-faces1425-faces13444-faces3360Cells3920Faces2520Edges840Vertices84Vertex figure5-orthoplex prismCoxeter groupsB7, [3,3,3,3,3,4]D7, [34,1,1]Propertiesconvex
The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. or
Alternate names rectified heptacross rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon
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Construction There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [34,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0,0)
Root vectors Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.
Birectified 7-orthoplex Birectified 7-orthoplexTypeuniform 7-polytopeSchläfli symbol 2r{3,3,3,3,3,4}Coxeter-Dynkin diagrams6-faces1425-faces14284-faces6048Cells10640Faces8960Edges3360Vertices280Vertex figure{3}×{3,3,4}Coxeter groupsB7, [3,3,3,3,3,4]D7, [34,1,1]Propertiesconvex
Alternate names Birectified heptacross Birectified hecatonicosaoctaexon (Acronym: barz) (Jonathan Bowers) - birectified 128-faceted polyexon
Images
Cartesian coordinates Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,0,0,0,0)
Trirectified 7-orthoplex A trirectified 7-orthoplex is the same as a trirectified 7-cube.
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. o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz
External links Polytopes of Various Dimensions Multi-dimensional Glossary