Given an n-dimensional electrically neutral, solid metal ball (a point for n=0; a rod, n=1; a disc, n=2; a solid ball, n=3; ...), place N=(n+1)! identical ions on the ball. As one of my favorite physics professors used to say, forget the mathematics, intuitively I expect the ions to equilibrate to the vertices of an n-D [permutohedron](http://en.wikipedia.org/wiki/Permutohedron) inscribed in the n-ball (e.g., a hexagon inscribed in the circumference of a disc with 6 ions and a [truncated octahedron](http://en.wikipedia.org/wiki/Truncated_octahedron) for n=3 ). Similarly with N= [2(n+1)]! /[(n+1)! (n+2)!] (the [Catalan numbers](http://oeis.org/A000108)), I expect to see an [associahedron](http://en.wikipedia.org/wiki/Associahedron) (a Stasheff polytope, e.g., a pentagon for the disc with 5 ions).


Is my intuition correct? Has anyone seen this worked out mathematically, as an extremum problem? An electrostatics simulation for n=3 would be interesting also. (See OEIS [A019538](https://oeis.org/A049019) for refined f-vectors of permutohedra / permutahedra and [A133437](http://oeis.org/A133437) for associahedra.)


After reviewing the very interesting, proposed answers, I think I need to clarify and qualify my question:
I initially was thinking of a real physical situation, a conductive metal ball in 3 dimensions or disc in 2 with Coulomb's inverse-square law applying. Then, if I remember my physics correctly, free charges must always reside on the suface with the potential vanishing in the interior of the ball and with the gradient of the potential at the surface being normal to the surface in the equilibrium state. The number of ions for the 3-D ball is constrained as stated above to either 4!=24 for the 3-D permutahedron or 8!/(4! 5!) = 14 for the 3-D associahedron. Given either number, how will the ions distribute themselves on the surface? Whether one should or can sensibly modify the repulsion for higher dimensions n and whether analogous surface (n-1) physics and equilibrium topology would occur are the next logical considerations; however, the number of "ions" would still be restricted to the number of vertices of either the n-D permutohedron or associahedron.



Equilibrium configurations of ions on n-Dim balls.
It turns out to pretty hard to guess the answers to these kinds of extremal problems. The permutohedron is almost certainly not the answer in $\mathbb{R}^n$ for $n \ge 3$, for any reasonable potential function (for example, an inverse power law). Specifically, the truncated octahedron is not good at minimizing things, because of the square and hexagonal facets, and it seems that you can always do better with the same number of particles, although I haven't carefully proved this for all potential functions. In $\mathbb{R}^4$ the situation is even worse, since the 120 vertices of a regular 600-cell always lead to lower energy than those of a permutohedron. I'm confident that the permutohedron will be suboptimal also for $n \ge 5$, and that the associahedron behaves similarly.


More generally, it's far from clear when one should expect symmetry in ground states of physical systems like this (see [Link](https://web.archive.org/web/20220121031512/http://arxiv.org/abs/1003.3053) for an expository article on this issue). For small systems you often get symmetry, but in large systems defects can actually lower energy in surprising ways.


It's natural to guess that regular polytopes should lead to energy minima, and the ones with simplicial facets actually do. (By contrast, the cube does not, because one can lower energy by rotating one facet relative to the opposite one, to get a square antiprism.) These polytopes actually minimize a very wide class of energies, including all inverse power laws; specifically, they are optimal whenever the potential function is a completely monotonic function of squared Euclidean distance. (Recall that a completely monotonic function is a nonnegative $C^\infty$ function whose derivatives alternate in sign.) Abhinav Kumar and I called this phenomenon universal optimality.


In three dimensions there is no universal optimum larger than an icosahedron, but in higher dimensions there are more examples, such as the $E\_8$ root system or the minimal vectors in the Leech lattice. Another interesting example is the Schläfli configuration of 27 points on a sphere in six dimensions, which correspond to the 27 lines on a cubic surface. See [Link](https://web.archive.org/web/20220121033941/http://arxiv.org/abs/math/0607446) for details and proofs.


Universal optimality seems to be quite rare, and in most cases the optimal configuration will depend on the potential function being used. See [Link](https://web.archive.org/web/20210506222437/https://arxiv.org/abs/math/0611451) for some large-scale numerical experiments, as well as references to many other papers in the literature.


