This is a ring made from a hexagonal mesh torus. It's currently sized for a ring around size 10 or 10.5 (my left ring finger is size 10.5 and this doesn't quite fit, but it fits on my other ring finger). I made it by using this mathematica code:
http://mathematica.stackexchange.com/questions/39879/create-a-torus-with-a-hexagonal-mesh-for-3d-printing
and then I dropped it in Tinkercad and bored a cylinder hole out of it.

I'm going to keep iteratively tinkering with this design because (a) I'm not entirely satisfied with it yet and (b) I think I will get to learn lots of interesting things about (i) conformal mappings on the torus and (ii) stl files.
In particular, I don't love how it's divoted in on each hexagonal facet, and I think this is an artifact of how Mathematica exported the stl file. I suspect that fixing this problem will also fix the problem of the jaggies on the top/bottom of the ring.
Also it is kinda bulky right now.