A Universal Triangulation for Flat Tori

construction due to Burago and Zalgaller (Vestnik Leningrad Univ 15:66–80, 1960; St Petersburg Math J 7(3):369–385, 1995) shows that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear embedding into Euclidean space 𝔼^3 . A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. The embeddings constructed according to Burago and Zalgaller may have a huge number of vertices, moreover distinct for every flat torus. Based on another construction of Zalgaller (J Math Sci 100(3):2228–2238, 2000. https://doi.org/10.1007/s10958-000-0007-3 ) and on recent works by Arnoux et al. (2021, in preparation), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.