Mathematicians 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.
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