
Researchers chip away at Smale's 7th unsolved problem in mathematics - dnetesn
http://phys.org/news/2016-07-chip-smale-7th-unsolved-problem.html
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ddumas
Correction: The article states that Thomson's problem has only been solved for
2, 3, 4, 6, and 12 charges. However, in 2013 the 5-electron case was solved by
Richard Schwartz. Here's the paper:

[http://www.tandfonline.com/doi/abs/10.1080/10586458.2013.766...](http://www.tandfonline.com/doi/abs/10.1080/10586458.2013.766570)

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jacobolus
Needless to say, there has been a _lot_ of work on this subject from a number
of different directions, and any particular new paper is unlikely a priori to
be a revolutionary breakthrough. Look at Google Scholar or a similar citation
graph and you can find thousands of papers about it. (Some key words:
“spherical cap discrepancy”, “Fekete points”, “Riesz energy”, “logarithmic
potential”, obviously also “Thomson’s problem”, etc.)

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copenja
I actually was asked this problem in a programming interview. This is first
I've read that it is an unsolved problem, which definitely makes me chuckle.

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jbpetersen
Reminds me of reading about at least one case where a problem was solved in a
situation like that. Anyone recall what it was?

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jordigh
Dantzig solved an open problem in statistics which he thought was being
causally assigned as homework:

[https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st...](https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_statistics)

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jbn
I think you mean "casually", not "causally" (even though it's kind the
interesting Freudian slip, IMHO).

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paulpauper
but Wikipedia shows a huge table of values.it's just that we don't have a name
or certain constructions for values above 12

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Jabbles
Those are the best known configurations, we don't have proofs that they are
the lowest possible.

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paulpauper
Isn't this an optimization problem that can be solved with computer to solve a
high degree equation?

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jessriedel
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.

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Someone
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.

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grkvlt
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#Configuratio...](https://en.m.wikipedia.org/wiki/Thomson_problem#Configurations_of_smallest_known_energy))
for more links to visualisations in the rightmost column, if one exists.

