Timeline for Equilibrium configurations of ions on n-Dim balls.
Current License: CC BY-SA 3.0
16 events
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| Sep 1, 2022 at 21:39 | comment | added | Tom Copeland | From "Realizing the associahedron: Mysteries and questions" by Ceballos and Ziegler (arxiv.org/abs/1110.4059): The associahedron constructed as the secondary polytope of n + 3 equally-spaced points on a quadratic planar curve turns out to have all its vertices on an ellipsoid. This phenomenon extends to the permuto-associahedron and to the cyclohedron | |
| Dec 2, 2011 at 22:23 | comment | added | j.c. | There is an applet by Cris Cecka, Mark Bowick and Alan Middleton at this webpage thomson.phy.syr.edu which allows you to visualize the minimum energy configurations for $N$ particles interacting by Coulomb repulsion on a sphere in 3 dimensions. This case is called the "Thomson problem", by the way en.wikipedia.org/wiki/Thomson_problem | |
| Oct 23, 2011 at 18:45 | vote | accept | Tom Copeland | ||
| Oct 23, 2011 at 18:18 | comment | added | Henry Cohn | Actually, that calculation is a little unfair, since I used the standard embedding of the permutohedron (permutations of $(1,2,\dots,n)$). It's not surprising that that's not in equilibrium, but it will evolve to an equilibrium invariant under the same group. However, I am confident that this does not lead to a stable equilibrium that's a polytope combinatorially equivalent to the permutohedron. | |
| Oct 23, 2011 at 18:11 | comment | added | Henry Cohn | Actually, that also settles it: the truncated octahedron is not even in equilibrium at all (stable or unstable) for Coulomb energy, and the same is true for the permutohedra in $\mathbb{R}^4$ and $\mathbb{R}^5$ with the analogue of Coulomb energy. I haven't done the calculation, but I am sure it is true in all higher dimensions, and the same is probably true for the associahedron as well. | |
| Oct 23, 2011 at 17:43 | comment | added | Tom Copeland | A stable equilibrium should be another qualification. A ring of ions on a great circle of a 3-D ball would be an unstable equilibrium configuration. | |
| Oct 23, 2011 at 17:00 | comment | added | Tom Copeland | Sorry, I was thinking of analytic geometry and geometric algebra. ag tag has been deleted. | |
| Oct 23, 2011 at 16:54 | history | edited | Tom Copeland | CC BY-SA 3.0 |
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| Oct 23, 2011 at 14:58 | answer | added | Henry Cohn | timeline score: 10 | |
| Oct 23, 2011 at 13:32 | answer | added | José Figueroa-O'Farrill | timeline score: 2 | |
| Oct 23, 2011 at 11:53 | comment | added | Mattia Talpo | algebraic geometry? | |
| Oct 23, 2011 at 11:10 | answer | added | Igor Rivin | timeline score: 1 | |
| Oct 23, 2011 at 10:30 | comment | added | Tom Copeland | That's part of the problem to be fleshed out from the math. Obviously, for n=0 it's inconsequential what the potential is. For n=1 and 2, the genuine 3-D electrostatic potential is sufficient, but even for n=3 I don't know if this is true, so the generalization to higher dimensions is certainly an open question to me. | |
| Oct 23, 2011 at 10:18 | comment | added | Pietro Majer | The potential to minimize is $\sum_{i < j } | x_i - x_j |^{2-n}$ in dimension $n$, right? | |
| Oct 23, 2011 at 9:36 | history | edited | Tom Copeland | CC BY-SA 3.0 |
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| Oct 23, 2011 at 8:32 | history | asked | Tom Copeland | CC BY-SA 3.0 |