It is indeed fascinating that the completely different physics of these systems produce such similar shapes of affect. A question that occurs to me is that, while magnetic fields are measurable with Hall effect sensors, I am unaware (as a layman) of any means of testing electron orbital shape, so is there any way of demonstrating this mathematical treatment does actually represent the physics in operation?

The idea that fractal structure results in such similar shapes at different scales is familiar to me from the similarity between solar systems and simple atoms, which is so cliche the movie 'Animal House' presented it as a sophomoric experience of marijuana smokers. Are there other similar forms at macro scales, say in galactic lobes and nodes, oceanic currents, or intertidal zones where water and land intersect?

Thanks!