Timeline for Can solutions to Thomson's problem have pentagons?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| S Oct 10, 2022 at 5:54 | history | suggested | CommunityBot | CC BY-SA 4.0 |
The question is not "Among all energy configurations, which is the minimum?" but "Among all configurations, which has the minimum energy?" Hence this hyphen.
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| Oct 10, 2022 at 1:59 | review | Suggested edits | |||
| S Oct 10, 2022 at 5:54 | |||||
| Oct 9, 2022 at 22:30 | comment | added | Oscar Lanzi | We do have the modified dodecahedron where each pentagon loops five triangular faces (base of a pyramid, $n=32$). In my answer the presence of the pentagons is implied by Wikipedia listing the symmetry as $I_h$. | |
| Oct 9, 2022 at 22:25 | comment | added | Alex Meiburg | Haha, good point @GerryMyerson. :) My dad would be disappointed, as would my younger self who made paper models of all the Archimedean solids. Among other vertex-transitive polyhedra with pentagons, the Icosidodecahedron and Snub dodecahedron are clearly bad candidates (large "gaps"), but the Rhombicosidodecahedron -- as well as soccer balls -- do look like natural candidates. Surprising that neither is optimal. | |
| Oct 9, 2022 at 22:07 | history | edited | Alex Meiburg | CC BY-SA 4.0 |
singular polyhedron, oops :)
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| Oct 9, 2022 at 2:44 | comment | added | Gerry Myerson | @Alex, concerning "Dodecahedra are the only places that pentagons belong!" you must not be a soccer fan. en.wikipedia.org/wiki/File:Soccerball.svg | |
| Sep 5, 2022 at 18:10 | comment | added | Oscar Lanzi | @AlexMeiburg I would guess that there are infinitely many solutions with squares. But the squares tend to become an ever smaller fraction of the faces as we add more points. | |
| Sep 5, 2022 at 17:12 | answer | added | Oscar Lanzi | timeline score: 5 | |
| Feb 8, 2021 at 0:32 | comment | added | Alex Meiburg | I guess a related conjecture: among all the optimal solutions to Thomson's problem, only a finite number of them have squares. The same idea that, eventually, everything becomes triangle packings. | |
| Feb 6, 2021 at 17:03 | comment | added | Will Jagy | apparently the correct comparison is with circle packing in the plane; each circle center joins to six near neighbors to make equilateral triangles | |
| Feb 6, 2021 at 10:09 | comment | added | Denis Serre | @BrianHopkins. Did you notice the following sentence of the introduction ? A particularly interesting application of polyhedra in biology is provided by the structure of spherical shells, such as HIV which is built around a trivalent polyhedron with icosahedral symmetry. A few years later, the authors would have changed HIV into Coronavirus. | |
| Feb 6, 2021 at 7:29 | comment | added | Alex Meiburg | @M.Winter I find it disappointingly unasthetic too! I can't help feeling that, if we can't even get a pentagon on a dodecahedron, then surely they can't show up anywhere else. Dodecahedra are the only places that pentagons belong! :) | |
| S Feb 6, 2021 at 7:26 | history | suggested | gmvh |
Added closest top-level tags (this is both about optimization and about geometry)
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| Feb 6, 2021 at 7:19 | review | Suggested edits | |||
| S Feb 6, 2021 at 7:26 | |||||
| Feb 6, 2021 at 5:50 | comment | added | Brian Hopkins | @M.Winter Surprisingly, it seems that the minimal energy for 20 points is not related to the dodecahedron. Nor are the vertices of the cube the optimal arrangement for 8 points. A 2003 survey article by Atiyah & Sutcliffe includes citations and nice illustrations of the polyhedra (arxiv.org/abs/math-ph/0303071). | |
| Feb 6, 2021 at 2:50 | comment | added | M. Winter | Wait, the dodecahedron is not a minimum energy configuration? | |
| Feb 6, 2021 at 2:21 | history | asked | Alex Meiburg | CC BY-SA 4.0 |
