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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Re: Unique uniform polytopes in dimensions 9 and higher

Postby AliceAusten » Sun Mar 06, 2022 4:15 pm

How come there is no Coxeter diagram for the 24-dimensional Leech polytope?
How come there is no "exceptional family name" designated for the leech polytope?
Isn't the Leech polytope an exceptional figure?
How come the Leech polytope is never discussed as being made from lower dimensional polytopes (for example 23-dimensional, simplexes or orthoplexes)
For example, I understand that the families A, B=C, D are the simplex, measure polytope (cube)/orthoplex, and demi-cubes and they go on forever, the families E6,E7,E8 come to an end in eight dimensions,
F4 24-cell exists only in four dimensions, and the non-crystallographic H family (pentagonal family icosahedron, dodecahedron, 600-cell, 120-cell) ends in four dimension.
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Re: Unique uniform polytopes in dimensions 9 and higher

Postby mr_e_man » Mon Aug 08, 2022 3:37 pm

pentagonalpolytope747 wrote:I wonder if in dimensions starting 9 there are unique polytopes, from any family is enough.

The convex uniform polytopes in 5 dimensions (never mind 9 dimensions) are still unknown.

So we cannot answer this yet.
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