Singular euclidean structures on surfaces are a key tool in the study of the mapping class group, of Teichmüller space, and of kleinian three-manifolds. François Guéritaud, while studying work of Ian Agol, gave a powerful technique for turning a singular euclidean structure (on a surface) into a triangulation (of a three-manifold). We will give an exposition of some of this work from the point of view of Delaunay triangulations for the $L^\infty$ metric. We will review the definitions in a relaxed fashion, discuss the technique, and then present applications to the study of strata in the space of singular euclidean structures. If time permits, we will also discuss the naturally occurring algorithmic questions.

This is joint work with Mark Bell and Vaibhav Gadre. Some of our results are independently due to Ian Frankel, who additionally has different applications.