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.
Record created on 2010-12-03, modified on 2016-08-09