8-orthoplexRectified 8-orthoplexBirectified 8-orthoplexTrirectified 8-orthoplexTrirectified 8-cubeBirectified 8-cubeRectified 8-cube8-cubeOrthogonal projections in A8 Coxeter plane In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.
Rectified 8-orthoplex Rectified 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t1{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces2726-faces30725-faces89604-faces12544Cells10080Faces4928Edges1344Vertices112Vertex figure6-orthoplex prismPetrie polygonhexakaidecagonCoxeter groupsC8, [4,36]D8, [35,1,1]Propertiesconvex
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.
Related polytopes The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb. or
Alternate names Rectified octacross Rectified diacosipentacontahexazetton (acronym: rek) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0,0,0)
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Birectified 8-orthoplex Birectified 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t2{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces2726-faces31845-faces161284-faces34048Cells36960Faces22400Edges6720Vertices448Vertex figure{3,3,3,4}x{3}Coxeter groupsC8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex
Alternate names Birectified octacross Birectified diacosipentacontahexazetton (acronym: bark) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,0,0,0,0,0)
Images
Trirectified 8-orthoplex Trirectified 8-orthoplexTypeuniform 8-polytopeSchläfli symbol t3{3,3,3,3,3,3,4}Coxeter-Dynkin diagrams7-faces 16+2566-faces 1024 + 2048 + 1125-faces 1792 + 7168 + 7168 + 4484-faces 1792 + 10752 + 21504 + 14336 Cells 8960 + 126880 + 35840Faces 17920 + 35840Edges17920Vertices1120Vertex figure{3,3,4}x{3,3}Coxeter groupsC8, [3,3,3,3,3,3,4]D8, [35,1,1]Propertiesconvex The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Alternate names Trirectified octacross Trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,0,0,0,0)
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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. o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark
External links Polytopes of Various Dimensions Multi-dimensional Glossary