| Abstract: | We prove that if a neighborhood of a polygonal region admits a two-dimensional eigenfunction of the Laplacian, having a nonzero eigenvalue and such that its normal derivative vanishes on all the bounding edges, then the polygonal region must be a union of complete pieces of a tiling of the plane by congruent rectangles, or by congruent (45°, 45°, 90°) or (30°, 60°, 90°) triangles. Hydrodynamically, this means that during critical convection in a horizontal fluid layer uniformally heated from below, the mere occurrence of one arbitrary closed vertical polygonal fluid surface across which there is no transportation of fluid automatically guarantees the presence of one of the usual special convection patterns. In addition it shows that linear convection theory seldom predicts a regular fluid pattern: e.g., for the case of a triangular container having angles substantially different from (45°, 45°, 90°), (30°, 60°, 90°), (60°, 60°, 60°) or (30°, 30°, 120°), it predicts that the convection cells not touching the boundary, if any, should be noticeably nonpolygonal. We also consider a nonlinear generalization and the noneuclidean analogues of such polygons. |
