9-orthoplexRectified 9-orthoplexBirectified 9-orthoplexTrirectified 9-orthoplexQuadrirectified 9-cubeTrirectified 9-cubeBirectified 9-cubeRectified 9-cube9-cubeOrthogonal projections in A9 Coxeter plane In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.
There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.
These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.
Rectified 9-orthoplex Rectified 9-orthoplexTypeuniform 9-polytopeSchläfli symbol t1{37,4}Coxeter-Dynkin diagrams7-faces6-faces5-faces4-facesCellsFacesEdges2016Vertices144Vertex figure7-orthoplex prismPetrie polygonoctakaidecagonCoxeter groupsC9, [4,37]D9, [36,1,1]Propertiesconvex
The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb. or
Alternate names Rectified enneacross (Acronym: riv) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0,0,0,0)
Root vectors Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.
Images
Birectified 9-orthoplex Alternate names Rectified 9-demicube Birectified enneacross (Acronym: brav) (Jonathan Bowers)
Images
Trirectified 9-orthoplex Alternate names Trirectified enneacross (Acronym: tarv) (Jonathan Bowers)
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) o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv
External links Polytopes of Various Dimensions Multi-dimensional Glossary