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A8 Lattice
In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb. A8 lattice This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the _8 Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle. _8 contains _8 as a subgroup of index 5760. Both _8 and _8 can be seen as affine extensions of A_8 from different nodes: The A lattice is the union of three A8 lattices, and also identical to the E8 la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Uniform 8-honeycomb
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] |