First, the curve-shortening process is quite beautiful in its own right:
[![Spiral][1]][1]
[YouTube video](https://www.youtube.com/watch?v=8Ez0QoJ3XG8) showing self-intersection is avoided.
Second, curve-shortening is a method for proving the existence of simple closed geodesics on a topological $2$-sphere. See [The theorem of the three geodesics](https://en.wikipedia.org/wiki/Theorem_of_the_three_geodesics).