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?..