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Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally int

Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.

Someone 10y ago
And even if you can find a numerical solution to arbitrary precision, at best you only get arbitrarily close to the mathematically optimal solution (at best because a purely numerical approach cannot rule out that it misses a very small, but high peak in the function to be optimized) As an extreme example, for n=2 and using binary floating point, you probably will find the mathematically correct solution, but there's no way to tell numerically whether your answer is off by a fraction of your floating point's epsilon value. That's probably not important to physicists who want to know an answer, but it is for mathematicians. What I find very surprising is (from https://en.m.wikipedia.org/wiki/Thomson_problem https://en.m.wikipedia.org/wiki/Thomson_problem): "Numerical solutions for N=8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively." I would like to see the visualizations of the better solutions for n=20 (wikipedia links to the one for n=8) and/or hear a heuristic argument as to how that can happen.
- grkvlt 10y ago
  For N=8 the solution listed on the Wikipedia page is a square antiprism (https://en.m.wikipedia.org/wiki/Square_antiprism https://en.m.wikipedia.org/wiki/Square_antiprism) which seems intuitively reasonable. But, I don't see anything listed for N=20. Look at the table in S.5 of the article (https://en.m.wikipedia.org/wiki/Thomson_problem#Configurations_of_smallest_known_energy https://en.m.wikipedia.org/wiki/Thomson_problem#Configuratio...) for more links to visualisations in the rightmost column, if one exists.
  - deleted 10y ago
    [deleted]
- tamana 10y ago
  The function here has a simple continuity, so there is no risk of narrow peaks beyond some computable threshold. The challenge is the high dimensionality.

Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.

Someone 10y ago

And even if you can find a numerical solution to arbitrary precision, at best you only get arbitrarily close to the mathematically optimal solution (at best because a purely numerical approach cannot rule out that it misses a very small, but high peak in the function to be optimized) As an extreme example, for n=2 and using binary floating point, you probably will find the mathematically correct solution, but there's no way to tell numerically whether your answer is off by a fraction of your floating point's epsilon value. That's probably not important to physicists who want to know an answer, but it is for mathematicians. What I find very surprising is (from https://en.m.wikipedia.org/wiki/Thomson_problem https://en.m.wikipedia.org/wiki/Thomson_problem): "Numerical solutions for N=8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively." I would like to see the visualizations of the better solutions for n=20 (wikipedia links to the one for n=8) and/or hear a heuristic argument as to how that can happen.

grkvlt 10y ago
For N=8 the solution listed on the Wikipedia page is a square antiprism (https://en.m.wikipedia.org/wiki/Square_antiprism https://en.m.wikipedia.org/wiki/Square_antiprism) which seems intuitively reasonable. But, I don't see anything listed for N=20. Look at the table in S.5 of the article (https://en.m.wikipedia.org/wiki/Thomson_problem#Configurations_of_smallest_known_energy https://en.m.wikipedia.org/wiki/Thomson_problem#Configuratio...) for more links to visualisations in the rightmost column, if one exists.
- deleted 10y ago
  [deleted]
tamana 10y ago
The function here has a simple continuity, so there is no risk of narrow peaks beyond some computable threshold. The challenge is the high dimensionality.