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Those are the best known configurations, we don't have proofs that they are the lowest possible.
by Jabbles 10y ago
Those are the best known configurations, we don't have proofs that they are the lowest possible.
- paulpauper 10y agoIsn't this an optimization problem that can be solved with computer to solve a high degree equation?
- jessriedel 10y agoNot 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 agoAnd 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 agoFor 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 agoThe function here has a simple continuity, so there is no risk of narrow peaks beyond some computable threshold. The challenge is the high dimensionality.
- contravariant 10y agoThe unsolved problem referred to in the title is finding a configuration close to the global minimum. Most optimization algorithms need such a configuration as a starting point.