Revisions to Can solutions to Thomson's problem have pentagons?

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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edit approved Oct 10, 2022 at 5:54

Community Bot

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Thomson's problem asks for the minimum energy-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large N$N$, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large N$N$, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

Thomson's problem asks for the minimum energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large N, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large N, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

Thomson's problem asks for the minimum-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large $N$, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large $N$, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

singular polyhedron, oops :)

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edited Oct 9, 2022 at 22:07

Alex Meiburg

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Thomson's problem asks for the minimum energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedrapolyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large N, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large N, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

Thomson's problem asks for the minimum energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedra, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large N, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large N, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

Thomson's problem asks for the minimum energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large N, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large N, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.