# Most Regularity of a Polygon

Conseider $n$ electrons in an empty sphere. What structure do they make?

This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of sphere has so much positive electronic charge; (ii) if electrons can occur in everywhere of sphere.

It seems that if there is a Platonic solid (see for example http://en.wikipedia.org/wiki/Platonic_solid) with $n$ vertices, then the electrons make this solid in case (ii) of question.But we know there is Platonic solids just for $n = 4, 6, 8, 12, 20$. What is the structure of electrons, when for example $n = 7$ or $11$ and other numbers?

As a generalization, I ask the similar question for spheres in higher dimensions.

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What is the definition of an electron here? – Emil Jeřábek Jul 25 '14 at 21:03
I mean $n$ points that want to have most possible distance from together. You can define an interesting metric for "distance between two electron". Also one can consider the problem of maximising the minimum distance between two electrons. – Arash Ahadi Jul 25 '14 at 21:17
According to the Wikipedia article, the corners of the cube and dodecahedron do not give optimal configurations. So it's not just a matter of having the 'most regularity' in how they are arranged. – Colin Reid Jul 25 '14 at 23:51
I don't understand the title, "Most regularity of a polygon", since I don't know what "regularity" means and I don't see any polygons in the question. – Gerry Myerson Jul 26 '14 at 0:29

The on-the-surface instance of your problem is sometimes known as the Thomson problem, after J.J. who posed the problem over a century ago. There is a detailed Wikipedia page on the topic. The $n=5$ case was only settled in 2010—"a rigorous computer-assisted proof" established that the triangular bipyramid is the unique minimizer of the Coulomb potential.

Richard Evan Schwartz. "The 5 Electron Case of Thomson's Problem." 2010. arXiv link

"In the [triangular bipyramid], two points are antipodal points on [the sphere] and the remaining 3 points form an equilateral triangle on the equator midway between the two antipodal points":

(Image from brakke @this link)
Many other candidates for minimal energy configurations have been identified but not rigorously proven. For example, for $n=14$, it seems that the gyroelongated hexagonal dipyramid is the minimal energy configuration (combinatorially).

For a deeper look:

Cohn, Henry, and Abhinav Kumar. "Universally optimal distribution of points on spheres." Journal of the American Mathematical Society. 20.1 (2007): 99-148.

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It is not clear that OP is asking about Coulomb potential. (It is not clear what OP is asking.) – Gerry Myerson Jul 26 '14 at 0:27
@GerryMyerson: It is true that I extrapolated from the Arash's use of the words "electron" and "charge" and "structure" to a specific interpretation of his question. – Joseph O'Rourke Jul 26 '14 at 0:49
Gerry and Joseph, Joseph's answer is so relevant. In mathoverflow.net/questions/177107/…, I present a formal case of the question. – Arash Ahadi Jul 26 '14 at 7:09