Unique uniform polytopes in dimensions 9 and higher

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Unique uniform polytopes in dimensions 9 and higher

Postby pentagonalpolytope747 » Mon Jun 21, 2021 11:10 am

In this topic, unique polytope is a polytope that doesn't derive from simplex, cube or orthoplex.

For example, 421 in 8 dimensions.

I wonder if in dimensions starting 9 there are unique polytopes, from any family is enough.
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Re: Unique uniform polytopes in dimensions 9 and higher

Postby wendy » Mon Jul 26, 2021 10:57 am

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.
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Re: Unique uniform polytopes in dimensions 9 and higher

Postby Mecejide » Mon Jul 26, 2021 12:19 pm

Yes.

In 24 dimensions, it is possible to blend troops of the rectified orthoplex and expanded simplex to get uniforms in a Leech subregiment.
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Re: Unique uniform polytopes in dimensions 9 and higher

Postby Mecejide » Tue Jul 27, 2021 12:28 pm

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.

Are you only considering convex polytopes here?
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