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Let $P_i$ be a set of points in the plane and $P_{i+1}$ the corners of the Voronoi diagram of $P_i$. Start with some $P_1$ and iterate away. What happens? Can you choose a $P_1$ so the iteration goes into a stable loop? (If you play the game on a torus, $P_1=${some point} works, on a sphere $P_1=...