We survey the recent joint work with V. Ovsienko and R. Schwartz on the Pentagram Map, a projectively natural transformation of polygons. The main result is that this map is completely integrable. The Pentagram Map is a geometrically natural discretization of the Boussinesq equation, one of the most popular completely integrable PDEs. We show that the integrability mechanism that works for the Pentagram Map fails for similar projectively natural iterations.

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