  We study the modal logic of the closure algebra $P_2$, generated by the set
of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic
is finitely axiomatizable, is complete with respect to the class of frames we
call "crown" frames, is not first order definable, does not have the Craig
interpolation property, and its validity problem is PSPACE-complete.
