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9-polytope
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope Ridge (geometry), ridge being shared by exactly two 8-polytope Facet (mathematics), facets. A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope Facet (geometry), facets. Regular 9-polytopes Regular 9-polytopes can be represented by the Schläfli symbol , with w 8-polytope Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 9-polytopes: # - 9-simplex # - 9-cube # - 9-orthoplex There are no nonconvex regular 9-polytopes. Euler characteristic The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic used t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 9-simplex
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a Rectification (geometry), rectification of the regular 9-simplex. These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry. There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedron, tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex. Rectified 9-simplex The rectified 9-simplex is the vertex figure of the 10-demicube. Alternate names * Rectified decayotton (reday) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''rectified 9-simplex'' can be most simply positioned in 10-space as permutations of (0,0,0,0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-orthoplex
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells ''4-faces'', 5376 5-simplex ''5-faces'', 4608 6-simplex ''6-faces'', 2304 7-simplex ''7-faces'', and 512 8-simplex ''8-faces''. It has two constructed forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or Coxeter symbol 611. It is one of an infinite family of polytopes, called cross-polytopes or ''orthoplexes''. The dual polytope is the 9-hypercube or enneract. Alternate names * Enneacross, derived from combining the family name ''cross polytope'' with ''ennea'' for nine (dimensions) in Greek * Pentacosidodecayotton as a 512- facetted 9-polytope (polyyotton) Construction There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or ,37symmetry group, and a lower symmetry with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified Enneacross
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 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 ,37Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or 6,1,1Coxeter group. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-demicube
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM9 for a 9-dimensional ''half measure'' polytope. Coxeter named this polytope as 161 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract: : (±1,±1,±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Images References * H.S.M. Coxeter: ** Coxeter, ''Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-simplex T06
In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°. It can also be called a decayotton, or deca-9-tope, as a 10- facetted polytope in 9-dimensions.. The name ''decayotton'' is derived from ''deca'' for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and ''-on''. Coordinates The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are: :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 9-cube
In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube. There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of ei ... centers of the 9-cube. These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry. Rectified 9-cube Alternate names * Rectified enneract (Acrony ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-simplex T03
In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°. It can also be called a decayotton, or deca-9-tope, as a 10- facetted polytope in 9-dimensions.. The name ''decayotton'' is derived from ''deca'' for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and ''-on''. Coordinates The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are: :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right) :\left(\sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
