Hearing the shape of a drum

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.

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?" 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. Similar questions can be traced back all the way to physicist Arthur Schuster in 1882.[1] For his paper, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968.[2]

The frequencies at which a drumhead can vibrate depend 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; for example, whether a Reuleaux triangle can be recognized in this way.[3] Kac admitted that he did not know whether it was possible for two different shapes to yield the same set of frequencies. 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.

  1. ^ Crowell, Rachel (2022-06-28), "Mathematicians Are Trying to 'Hear' Shapes—And Reach Higher Dimensions", Scientific American, retrieved 2022-11-15
  2. ^ "Can One Hear the Shape of a Drum? | Mathematical Association of America"
  3. ^ Kac, Mark (April 1966), "Can One Hear the Shape of a Drum?" (PDF), American Mathematical Monthly, 73 (4, part 2): 16, doi:10.2307/2313748, JSTOR 2313748