Isospectral shapes with Neumann and alternating boundary conditions

The isospectrality of a well-known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the Neumann boundary condition is verified numerically to high precision. Equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating Dirichlet and Neumann boundary conditions along successive edges. Good agreement with theory is obtained for the corresponding spectral staircase functions. Strong numerical evidence is also presented for isospectrality in an example of a different pair of shapes whose basic building-block triangle is not isosceles. Some possible confirmatory experiments involving fluids are suggested.