A Counter-Example to the Theorem of Hiemer and Snurnikov

A planar polygonal billiard $\mathcal{P}$ is said to have the finite blocking property if for every pair (O, A) of points in $\mathcal{P}$ there exists a finite number of “blocking” points B ₁,...,B _n such that every billiard trajectory from O to A meets one of the B _i 's. As a counter-example to a theorem of Hiemer and Snurnikov, we construct a family of rational billiards that lack the finite blocking property.