First, the curve-shortening process is quite beautiful in its own right:
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![Spiral][1]][1]
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[YouTube video](https://www.youtube.com/watch?v=8Ez0QoJ3XG8) showing self-intersection is avoided.
<hr />
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).


  [1]: https://i.sstatic.net/V60z5.jpg