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It is known that there are five Platonic solids:

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If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will eventually move particles to form a tetrahedron.

I have two questions here:

1) If number of particles are 8, 6, 20, and 12, would resulting solid (in similar experiment) be hexahedron, octahedron, dodecahedron, and icosahedron, respectively?

2) What would happen if number of particles is not 4, 8, 6, 20, or 12? Would an equilibrium be reached ever? Would the equilibrium shape be unique or not? Are those solids (resulting from such experiments) widely known? What would be their other properties?

I attempted some simulations, and researche the internet but didn't reach any sound conclusion.

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This is known as the Thomson problem. For $4$, $6$, and $12$ particles, the minimum energy configuration is the corresponding Platonic solid, but not for $8$ or $20$ particles. In particular, for $8$ particles a square antiprism has lower energy than a cube. Minimum energy configurations of course exist for other $n$, but finding them is a difficult numerical problem; as with most packing problems, there appears to be no general pattern for large $n$.

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  • $\begingroup$ Thanks for answer, @Rahul! But is there any formal proof that any distribution of particles would eventually lead to the one with minimum energy? In particular, if particles were initially at vertices of a cube, would they move to vertices of an antiprism? $\endgroup$ – VividD Apr 30 '14 at 14:14
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    $\begingroup$ No, in general if you start from an arbitrary distribution of particles and perform, say, gradient descent, you will probably only land on a local minimum, not the global minimum. This is why the problem is hard for large $n$ if you care about global minima. // Starting from a cube, actually, you may not end up anywhere because the cube is a critical point with zero gradient -- sort of like a stationary bicycle that's perfectly balanced vertically. But if you perturb the particles slightly away from the cube randomly, you should end up at the antiprism. $\endgroup$ – Rahul Apr 30 '14 at 15:25
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    $\begingroup$ Is it known that the energy function on the space of particle distributions is a Morse function? If so, then if you start with not any distribution but with almost any distribution (with respect to Lebesgue measure on the sphere) then gradient descent will converge to a global minimum. $\endgroup$ – Lee Mosher May 1 '14 at 12:51
  • $\begingroup$ @Lee: Can't Morse functions have multiple local minima? $\endgroup$ – Rahul May 1 '14 at 15:19
  • $\begingroup$ @LeeMosher: Min-Energy Configurations of Electrons On A Sphere says that N = 16 electrons has 2 local minima, but only one of them is the unique global minimum. $\endgroup$ – David Cary May 2 '14 at 4:04

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