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Pólya's Shire theorem from 1927 says that the zero sets $Z(f^{(n)})$ of the iterated derivatives $f^{(n)}$ of a meromorphic function $f$ with set of poles $S$ accumulate along (the boundaries of) the Voronoi diagram associated with $S$. But it leaves open the question of the asymptotic distribution of the zero-set. We will show that the answer is interesting, leading to a probability measure on Voronoi diagrams, by describing what is known for some classes of functions, and also discuss possible extensions to higher dimensions.