within the interior of a polygonal region , subject to Dirichlet boundary conditions . This equation can be applied to the mechanical vibration of plates with clamped boundaries, in which case . Here, is an allowed eigenfrequency of vibration, with representing the speed of waves in the vibrating medium. In quantum mechanics, the Helmholtz equation in the form

represents a particle in an infinite two-dimensional potential well.

The Helmholtz equation in a unit square has the simple solutions

,

with .

A linear combination of degenerate solutions

, ,

has the additional property of vanishing along the diagonal . This suggests an additional solution of the Helmholtz equation for isosceles right triangles, and by stacking a set of such triangles, one can build some irregular polygons with the same solutions. We consider here the cases of a parallelogram, a trapezoid and a (nonregular) hexagon. These all contain 45° or 135° vertex angles.

For each of the four irregular polygons, you can choose the quantum numbers and to produce a contour plot of . The corresponding vibration eigenvalue is also shown.