  We study the critical points of Laplace eigenfunctions on polygonal domains
with a focus on the second Neumann eigenfunction. We show that if each convex
quadrilaterals has no second Neumann eigenfunction with an interior critical
point, then there exists a convex quadrilateral with an unstable critical
point. We also show that each critical point of a second-Neumann eigenfunction
on a Lip-1 polygon with no orthogonal sides is an acute vertex.
