I present three papers written on the theme of the interaction between polyhedra and Hamil- tonian mechanics. In the first, I extend Delzant’s classification of toric symplectic mani- folds to a classification of toric symplectic stacks. These are singular objects whose moment polytopes may be irrational. In the second, with Jeremy Lane we construct a completely integrable system on the dual of the Lie algebra of a compact group. This generalizes the celebrated Gelfand-Zeitlin system. The third concerns a construction called partial tropi- calization, which was motivated by considering the limits of families of Poisson structures on certain Poisson-Lie groups. Together with Anton Alekseev, Jeremy Lane, and Yanpeng Li, we develop basic results about partial tropicalizations and use them to build symplectic embeddings into multiplicity-free spaces.