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.
pentagonalpolytope747
Mononian
 
Posts: 7
Joined: Wed May 22, 2019 6:10 pm

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2011
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

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.
Mecejide
Trionian
 
Posts: 86
Joined: Sun Mar 10, 2019 1:58 am
Location: Nethlekempt Farrows

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?
Mecejide
Trionian
 
Posts: 86
Joined: Sun Mar 10, 2019 1:58 am
Location: Nethlekempt Farrows

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.
AliceAusten
Nullonian
 
Posts: 1
Joined: Sun Mar 06, 2022 12:12 pm

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.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian
 
Posts: 448
Joined: Tue Sep 18, 2018 4:10 am

Re: Unique uniform polytopes in dimensions 9 and higher

Postby quickfur » Wed Aug 24, 2022 5:22 pm

mr_e_man wrote:[...]
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?
quickfur
Pentonian
 
Posts: 2884
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Unique uniform polytopes in dimensions 9 and higher

Postby mr_e_man » Thu Aug 25, 2022 5:59 pm

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. :)
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian
 
Posts: 448
Joined: Tue Sep 18, 2018 4:10 am

Re: Unique uniform polytopes in dimensions 9 and higher

Postby quickfur » Thu Aug 25, 2022 8:21 pm

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.
quickfur
Pentonian
 
Posts: 2884
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 0 guests

cron