Tanessi Quintanar

A surface $(S,g)$ is called polyedric if all point has a neighbourhood isometric to a disk or to a neighbourhood of the  (non flat) vertex of an euclidean cone.

In 1996, Burago et Zalgaller prooved that every oriented  polyedric surface admits a piecewise linear embedding (PL), isometric in $\mathbb{R}^3$. The proof is not constructive and finding the explicit isometric embeddings is a very delicate work. Thanks to a smart folding allowing to curve a cilynder in a $PL$ way, Zalghaller shows explicites constructions of isometric embeddings of "large" rectangular tori. In this exposition, I will present you a construction of "short" rectangular tori trough origamic foldings. This construction is made with 40 vertices.

If we remove the isometric condition, it has been prooved that 7 vertices are sufficient to embed  the torus in $\mathbb{R}^3$. Could it exist an origamic flat tourus having only 7 vertices?