It's fun to puzzle through why optical reflections, active radar, and things like that are inverse R^4. (It's just two inverse-square laws composed together).

Dipoles, like magnetic fields and planetary tidal forces, decay as an inverse R^3. That's less intuitive.

Totally unrelated, but was quite thrilled to see this map of the sphere

https://substackcdn.com/image/fetch/$s_!8JEP!,f_auto,q_auto:...

because I had been entertaining myself by making toy globes out of paper and it seems this polyconic map was the one I had used (an interrupted version of this).

The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip. What I was working with were more like flowers, one for each hemisphere, with the pole at the center.

Now, on a more related note (related to the actual post, not my tangent), things get complicated and interesting in 2d flatland.

There the field has to decay as 1/r because the circumference of the boundary scales as O(r). But that's the field. To get to the potential you need to integrate and then you get a function that is logarithmic.

A consequence of that is the potential does not drop to zero as you go away further. Unlike what is the case for the 3d case.

If you have heard that a random walking bird returns infinitely often but a random flying bird one doesn't, that's sorta related.

While we're on this... it is arguably even more befuddling to recognize that the information content potential of space apparently correlates with the surface area rather than volume https://en.wikipedia.org/wiki/Holographic_principle