Hearing the shape of a drum

is the title of a 1966 article by Mark Kac in the American Mathematical Monthly which made the question famous, though this particular phrasing originates with Lipman Bers.

The question of whether the frequencies determine the shape was finally answered in the negative in the early 1990s by Carolyn S. Gordon, David Webb and Scott A. Wolpert.

The term "homophonic" is justified because the Dirichlet eigenvalues are precisely the fundamental tones that the drum is capable of producing: they appear naturally as Fourier coefficients in the solution wave equation with clamped boundary.

In 1964, John Milnor observed that a theorem on lattices due to Ernst Witt implied the existence of a pair of 16-dimensional flat tori that have the same eigenvalues but different shapes.

However, the problem in two dimensions remained open until 1992, when Carolyn Gordon, David Webb, and Scott Wolpert constructed, based on the Sunada method, a pair of regions in the plane that have different shapes but identical eigenvalues.

This idea has been generalized by Buser, Conway, Doyle, and Semmler[4] who constructed numerous similar examples.

On the other hand, Steve Zelditch proved that the answer to Kac's question is positive if one imposes restrictions to certain convex planar regions with analytic boundary.

It is also known, by a result of Osgood, Phillips, and Sarnak that the moduli space of Riemann surfaces of a given genus does not admit a continuous isospectral flow through any point, and is compact in the Fréchet–Schwartz topology.

Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.
One-parameter family of isospectral drums
Eigenmodes and corresponding eigenvalues of the Laplace operator on the GWW domains