  The pentagram map is a projectively natural iteration defined on polygons,
and also on objects we call twisted polygons (a twisted polygon is a map from Z
into the projective plane that is periodic modulo a projective transformation).
We find a Poisson structure on the space of twisted polygons and show that the
pentagram map relative to this Poisson structure is completely integrable in
the sense of Arnold-Liouville. For certain families of twisted polygons, such
as those we call universally convex, we translate the integrability into a
statement about the quasi-periodic motion for the dynamics of the pentagram
map. We also explain how the pentagram map, in the continuous limit,
corresponds to the classical Boussinesq equation. The Poisson structure we
attach to the pentagram map is a discrete version of the first Poisson
structure associated with the Boussinesq equation.
