Polyhedral Realizations of Flat Tori
A polyhedral surface is obtained by gluing a family of Euclidean polygons along their edges under the condition that glued edges have the same length.
A flat torus is a polyhedral surface without singularity obtained by gluing the opposite sides of a parallelogram.
While it is easy to give a polyhedral realization of a right cylinder, it seems much more difficult to think of such a realization for a given flat torus.
In this gallery, we give PL isometric embeddings of various flat tori by making effective a general method due to Burago and Zalgaller to embed any polyhedral surface. The building block of the construction is a procedure which allows to pleat in a PL and isometric way a big acute triangle above a smaller acute one. It remains to compute an acute triangulation and a short, smooth and conformal embedding of the flat torus we want to embed in order to apply the previous building block.
The pictures were computed thanks to the CGAL library.
A PL isometric embedding of the square flat torus
A PL isometric embedding of the square flat torus with 170,040 triangles. Each big triangle is pleated inside the initial short embedding to stretch the metric of the initial short embedding.
