In 1954, Nash imagined an explicit way to build isometric embeddings in codimension 1 of regularity only $C^1$. In the 1970s, Gromov generalized this technique to one which allows to solve a large family of non linear PDE, and giving a geometrical understanding of the construction. Even if the consctruction is explicit, it's done taking the limit of an induction, and each step involves a lot of parameters. We finally have a first complete explicit construction in 2012 by the Hevea team. What was the contribution of the explicit constructions that followed? And which questions arose during the construction process?