Comment by amluto
20 hours ago
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?..