Some planar isospectral domains

The authors give a number of examples of two noncongruent isospectral domains in the plane and a particularly simple method of proof. One of their examples is a pair of domains that are not only isospectral but homophonic, i.e. each domain has a distinguished point such that the corresponding normalized Dirichlet eigenfunctions take equal values at the distinguished points. They interprete this to mean that if the corresponding drums'' are struck at these special points, then they sound the same'' in the very strong sense that every frequency will be excited to the same intensity for each. This shows that one really cannot hear the shape of a drum.