Comment by gukoff
1 day ago
Thanks a lot for the detailed perspective!
The potential fields are a curious phenomenon. I feel like - and I haven't verified this in any way yet - the smoothness is dictated by the fact that all values are within a not-so-broad range. Then as you "peel" the hexagon from the outer layer using those 6-rings, building the potential field, each next layer inwards shouldn't changed too much.
I had also explored a case of n->inf where the sums turn into integrals, and consecutiveness turns into the uniform distribution. Then it's not so hard to find a solution. It's not in the article, because it turned out to be a dead end for solving the discrete case, and I didn't want to make the article too heavy - but if I remember correctly, there I also observed the smoothness.
Can you clarify what the “potential field” actually is? The text is not really precise, and the little interactive tool has the very curious property that I can set all the potentials to 0 and I don’t get all zeros in the magic hexagon.
I would guess that the smooth mountain-looking structure come from a very simple observation: a gadget consisting, in potential space, of a 2 surrounded by a ring of 6 1’s fully cancels at the center and in the ring immediately around the center and leaves a nice pattern of +1 and -1 residuals in the ring around it. I suspect that, fairly generally, as you try to build out small numbers around the outside of the magic hexagon, you end up with a large pile of things like this in the center, and a sum, even a very noisy one, of things that even vaguely Gaussians, tends to produce Gaussians. (That’s the central limit theorem.)
Apologies for the confusion - the playground had a typo (7 instead of 8) which is fixed now. Now it works as you would expect, and zeroed potentials correspond to a zeroed hexagon.
The "wide ring" gadget is an interesting idea. I think it will be linear pyramids and not gaussian. A set of these gadgets can also form a basis, and in such a different basis the potential fields may look very different - almost flat, perhaps?..