As is well-known from the smooth setting, Laplace operators provide one way for studying Riemannian manifolds. In the discrete setting, Laplace operators are ubiquitous in applications spanning numerical analysis, geometric modeling, and physical simulation. Many applications require discrete operators that retain key structural properties inherent to the smooth case -- such as symmetry, locality of definition, positive semi-definitness, and a maximum principle.

Building on the smooth setting, we present a set of natural properties for Laplace operators on discrete surface meshes. We point out an important theoretical limitation: discrete Laplacians cannot satisfy all of these properties on general unstructured triangle meshes; retroactively, this explains the diversity of existing discrete Laplacians found in the literature. Furthermore, building on insights that date back to James Clerk Maxwell, we provide a characterization of those triangle meshes that do allow for "perfect" Laplacians. Finally, we present a principled construction that extends discrete Laplacians from triangle meshes to arbitrary polygonal surface meshes.