have a need to express every aspect of nature in terms of an equation.
That's a good thing... if not a bit obsessive. The March 2013 edition
of SciAm has an article about "overcurved
rings" such as those in a flat spiral spring; e.g., a Slinky. If
you cut a full rotation of a Slinky (360°)
and join the ends, you find that it does not lay flat due to
overcurvature, but instead it assumes a
saddle shape. Another familiar example of an overcurved ring is
found in a pop-up tent. Interestingly, the author describes a method
for folding an overcurved ring into a set of three concentric rings
that will lay flat. I immediately recognized it as the method used to
package large bandsaw blades, fan belts, etc. It can take a bit of noodling
to figure out how to get the ring into that configuration if you don't
have instructions. The video below is one I made a while back demonstrating
how to fold a bandsaw blade.