Here's a self-intersecting line I occasionally imagine when I'm trying to take a nap:
I don't know the proper name for a shape like this. I tend to think of it as a spiral, but it's technically not a spiral, nor is it some other kind of whorl. If there is a proper term and you know it, please email me.
We can make a similar design out of elliptical arcs:
This shape feels more likely to have a proper name, though maybe not from math but from some other field, such as graphic design. It reminds me a little of retro line art.