  In this paper we consider eigenfunctions of the Laplacian on a planar domain
with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions.
The main result is a quantitative estimate on the $L^2$ mass of eigenfunctions
near a point in terms of the distance to the nearest non-adjacent boundary
face. In particular, eigenfunctions cannot concentrate completely at any one
single point. The technique of proof is to use the commutator ideas from the
recent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.
