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Given a discrete set of points in a metric space, called nuclei, one associates to each such nucleus its Voronoi cell, which consists of all points closer to it than to other nuclei. Voronoi diagrams are now widely used in mathematics, science, and engineering; they are used even in baking. In Euclidean space, one commonly uses a homogeneous Poisson point process to assign the locations of the nuclei. As the intensity of the point process tends to 0, the nuclei spread out and disappear in the limit, with each pair of space points eventually belonging to the same cell. Surprisingly, this does not happen in other settings such as hyperbolic space, which is the third-most important geometry after Euclidean and spherical. We will describe properties of such a limiting tessellation, as well as analogous behavior on Cayley graphs of finitely generated groups. We will illustrate results with many pictures and videos.   The talk is based on work of Sandeep Bhupatiraju and joint work with Matteo d'Achille, Nicolas Curien, Nathanael Enriquez, and Meltem Unel. We will not assume knowledge of Poisson point processes or of hyperbolic space. The talk will be accessible to graduate students.