,

,

with the Dirichlet boundary condition (on the boundary of the polyhedron). Analytic solutions are known for , and (equilateral triangle, square and disk, respectively). Accurate numerical solutions can be obtained for all by the finite element method. This entails discretizing the -gon and applying the Mathematica algorithm NDEigensystem. Comparing with the known analytic solutions shows that the first eight computed eigenvalues are accurate to better than .01%.

Consider the cases to (equilateral triangle to regular octagon) and (disk). The size of each -gon is adjusted so that it has an area of 1. The lowest eigenvalue represents the ground state, which is labeled . The second and third eigenvalues are equal. The degenerate first excited states are labeled as and . Continuing, we find , , , . Only for the pentagon () and hexagon () do the levels deviate from this pattern.

You can display a contour plot for any of the eight eigenstates of any -gon. You can also choose to display multiple plots for all eigenstates of a selected -gon or the states of all -gons for a given quantum number.

The energy-level diagram shows the energy spectrum for all -gons. As increases, the spectrum approaches that for the disk ().