Unique uniform polytopes in dimensions 9 and higher

[pentagonalpolytope747]

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.

[wendy]

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.

[Mecejide]

Yes.

In 24 dimensions, it is possible to blend troops of the rectified orthoplex and expanded simplex to get uniforms in a Leech subregiment.

[Mecejide]

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?

[AliceAusten]

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.

[mr_e_man]

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.

[quickfur]

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

Really?? I thought the families of uniform polytopes derived from the simplex, cross/hypercube, demihypercubes are well-established for all dimensions. Or do you mean the full set of convex uniform polytopes is not yet known for dimension ≥5? I kinda doubt that... I think we know at least up to dimension 9 with the conclusion of the 4_21 series of uniform polytopes. Perhaps an argument could be made for dimensions 10 and above, but last I checked, I was pretty sure we already know that all higher dimensions no longer has uniform polytopes outside the simplex/cube/demicube families.

Unless I misunderstood, and the aforementioned families are only the ones we know about, but there may be more out there?

[mr_e_man]

Yes, I mean the full set.
See https://en.wikipedia.org/wiki/Uniform_5-polytope

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.

And following the links leads to this page http://www.openproblemgarden.org/op/con ... _polytopes

Those sites are not "official". However, if we can't find anything saying that the full set is known, then it is unknown, by default.

[quickfur]

Interesting! I would daresay all the Wythoffian polytopes have been found. But there remains a small chance for non-Wythoffian uniform polytopes that may further add to the set. Like the grand antiprism in 4D, which was found by computer.

Wonder if there's a way to brute-force search for any extra uniform polytera by computer. Since vertices have to be transitive, you're limited to just considering facets surrounding a single vertex, which is a much smaller set of combinations than, say, CRFs; and then checking closure.