A fruitful way to frame the study of symplectic geometry is as the search for the boundary between symplectic flexibility (when symplectic objects behave like topological objects) and rigidity (when symplectic behavior is more restrictive). I will introduce this central theme through a visually appealing problem involving geometric properties of special cylinders in R^4, and will then describe a technique (capacities derived from generating families) for solving this problem. This is joint work with Lisa Traynor.