Modal logic of planar polygons

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.