  The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental
invariant counting lattice points in integer dilates of $P$. The quasi-period
of this quasi-polynomial divides the denominator of $P$ but is not always equal
to it: this situation is called quasi-period collapse. Polytopes experiencing
quasi-period collapse appear widely across algebra and geometry, and yet the
phenomenon remains largely mysterious. Using techniques from algebraic geometry
- specifically the $\mathbb{Q}$-Gorenstein deformation theory of orbifold del
Pezzo surfaces - we explain quasi-period collapse for rational polygons dual to
Fano polygons and describe explicitly the discrepancy between the quasi-period
and the denominator.
