In this topic, unique polytope is a polytope that doesn't derive from simplex, cube or orthoplex.
For example, 4_{21} in 8 dimensions.
I wonder if in dimensions starting 9 there are unique polytopes, from any family is enough.
wendy wrote:There are no 'unique' forms (outside the A, B, C, D groups) for nine or higher dimensions. Instead, some interest is to be had filling in the large holes that sphere-packing leaves. There is a lot of empty space in sphere-packing. 4_21 has spheres that cover only 1/3 of the volume, and the Leech lattice is 1/512 of space is filled with spheres.
The most efficient packing of spheres comprises of two quarter-cubics, but these can be set separately, the result rattles.
pentagonalpolytope747 wrote:I wonder if in dimensions starting 9 there are unique polytopes, from any family is enough.
mr_e_man wrote:[...]
The convex uniform polytopes in 5 dimensions (never mind 9 dimensions) are still unknown.
[...]
The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.
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