Isospectral Riemannian manifolds
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the title of an article by Mark Kac in the American Mathematical Monthly in 1966, but the phrasing of the title is due to Lipman Bers. These questions can be traced back all the way to Hermann Weyl.
For the 1966 paper that made the question famous, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968.
The frequencies at which a drumhead can vibrate depends on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known. No other shape than a square vibrates at the same frequencies as a square. Kac did not know whether it was possible for two different shapes to yield the same set of frequencies.
More formally, the drum is conceived as an elastic membrane whose boundary is clamped. It is represented as a domain D in the plane. Denote by λn the Dirichlet eigenvalues for D: that is, the eigenvalues of the Dirichlet problem for the Laplacian:
Two domains are said to be isospectral (or homophonic) if they have the same eigenvalues. 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.
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