Re: n - charged particles

quasi <quasi@xxxxxxxx> writes:
On Sat, 31 Mar 2007 22:24:49 +0100, David Hartley <me9@xxxxxxxxxxx>
wrote:

Try n=21. I'm fairly sure it has no rotational symmetry, but haven't
ruled out reflections yet.

No, there _is_ symmetry for n=21. In fact, there are 3 non-trivial
symmetries -- a rotation and 2 reflections, just like the symmetries
for n=13.

symmetry #1 (180 degree rotation about point 20):

map 20 -> 20, 8 -> 10, 11 -> 14

symmetry #2 (reflection about the plane through the points
4,5,13,15,19,20,21):

map 20 -> 20, 8 -> 11, 14 -> 10

symmetry #3 (reflection about the plane through the points
2,3,7,9,12,17,20):

map 20 -> 20, 8 -> 14, 11 -> 10)

Each of the planes above has 7 points on the plane and 7 points on
each side. This suggests an analogous symmetrical construction for any
multiple of 3. In fact, I believe I can prove that for any composite n
which is not a power of 2, there is at least one stable equilibrium
with a non-trivial symmetry.

As to the question of whether every stable equilibrium has a
non-trivial symmetry, you must admit that the evidence for the truth
of that claim is looking stronger. I even have some tentative ideas as
to how that might be proved but they're too sketchy to discuss at this
point.

I'm prepared do hallucinate that there exists one symmetry.
I think my seeding logic will fail when one tries to find
a minimal counter example, so none exists.

Phil
--
"Home taping is killing big business profits. We left this side blank
so you can help." -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
.

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