  A result due to Burago and Zalgaller states that every orientable polyhedral
surface, one that is obtained by gluing Euclidean polygons, has an isometric
piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A flat
torus, resulting from the identification of the opposite sides of a Euclidean
parallelogram, is a simple example of polyhedral surface. In a first part, we
adapt the proof of Burago and Zalgaller, which is partially constructive, to
produce PL isometric embeddings of flat tori. In practice, the resulting
embeddings have a huge number of vertices, moreover distinct for every flat
torus. In a second part, based on another construction of Zalgaller and on
recent works by Arnoux et al., we exhibit a universal triangulation with 5974
triangles which can be embedded linearly on each triangle in order to realize
the metric of any flat torus.
