I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the famous video 'Outside In' by Bill Thurston and collaborators, and book (Amazon, free pdf) by Scott Carter.

However, we already know when such eversions can be constructed, as the embeddings of $k$-spheres up to regular homotopy in $\mathbb{R}^n$ have been characterized by Smale's theorem. So here is my question:

**What is the mathematical value, aside from the aesthetic quality of the proofs, of finding explicit regular homotopies?**

Let me be clear: these proofs are extraordinarily beautiful, and exhibiting beauty is (in my opinion) absolutely valid as a primary goal of mathematics. But, just for the purposes of this question, I would like to put this aside.

To put my question in a different way: suppose I exhibited in an ingenious way some new explicit regular homotopy $M \sim M'$ of immersed manifolds, such that $M$ and $M'$ were already known in a nonconstructive sense to be regular homotopic. Other than aesthetic appreciation, why would a homotopy theorist be interested in studying my result? Are there, for example, important open conjectures about the minimal length of regular homotopies?