Hearing the shape of a drum
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 Gordon, Webb and Wolpert.
Gassmann triple
Isospectral
Spectral geometry
Vibrations of a circular membrane
Brossard, Jean; Carmona, René (1986), , Comm. Math. Phys., 104 (1): 103–122, Bibcode:1986CMaPh.104..103B, doi:10.1007/BF01210795, S2CID 121173871
"Can one hear the dimension of a fractal?"
; Conway, John; Doyle, Peter; Semmler, Klaus-Dieter (1994), "Some planar isospectral domains", International Mathematics Research Notices, 1994 (9): 391–400, doi:10.1155/S1073792894000437
Buser, Peter
Chapman, S.J. (1995), "Drums that sound the same", , 102 (February): 124–138, doi:10.2307/2975346, JSTOR 2975346
American Mathematical Monthly
Giraud, Olivier; (2010), "Hearing shapes of drums – mathematical and physical aspects of isospectrality", Reviews of Modern Physics, 82 (3): 2213–2255, arXiv:1101.1239, Bibcode:2010RvMP...82.2213G, doi:10.1103/RevModPhys.82.2213, S2CID 119289493
Thas, Koen
; Webb, David (1996), "You can't hear the shape of a drum", American Scientist, 84 (January–February): 46–55, Bibcode:1996AmSci..84...46G
Gordon, Carolyn
; Webb, D.; Wolpert, S. (1992), "Isospectral plane domains and surfaces via Riemannian orbifolds", Inventiones Mathematicae, 110 (1): 1–22, Bibcode:1992InMat.110....1G, doi:10.1007/BF01231320, S2CID 122258115
Gordon, C.
Ivrii, V. Ja. (1980), "The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary", Funktsional. Anal. I Prilozhen, 14 (2): 25–34, :10.1007/BF01086550, S2CID 123935462 (In Russian).
doi
(April 1966), "Can One Hear the Shape of a Drum?" (PDF), American Mathematical Monthly, 73 (4, part 2): 1–23, doi:10.2307/2313748, JSTOR 2313748
Kac, Mark
Lapidus, Michel L. (1991), "Can One Hear the Shape of a Fractal Drum? Partial Resolution of the Weyl-Berry Conjecture", Geometric Analysis and Computer Graphics, Math. Sci. Res. Inst. Publ., vol. 17, New York: Springer, pp. 119–126, :10.1007/978-1-4613-9711-3_13, ISBN 978-1-4613-9713-7
doi
Lapidus, Michel L. (1993), "Vibrations of fractal drums, the , waves in fractal media, and the Weyl–Berry conjecture", in B. D. Sleeman; R. J. Jarvis (eds.), Ordinary and Partial Differential Equations, Vol IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, London: Longman and Technical, pp. 126–209
Riemann hypothesis
Lapidus, M. L.; van Frankenhuysen, M. (2000), Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions, Boston: Birkhauser. (Revised and enlarged second edition to appear in 2005.)
Lapidus, Michel L.; Pomerance, Carl (1993), , Proc. London Math. Soc., Series 3, 66 (1): 41–69, CiteSeerX 10.1.1.526.854, doi:10.1112/plms/s3-66.1.41
"The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums"
Lapidus, Michel L.; Pomerance, Carl (1996), "Counterexamples to the modified Weyl–Berry conjecture on fractal drums", Math. Proc. Cambridge Philos. Soc., 119 (1): 167–178, :1996MPCPS.119..167L, doi:10.1017/S0305004100074053, S2CID 33567484
Bibcode
(1964), "Eigenvalues of the Laplace operator on certain manifolds", Proceedings of the National Academy of Sciences of the United States of America, 51 (4): 542ff, Bibcode:1964PNAS...51..542M, doi:10.1073/pnas.51.4.542, PMC 300113, PMID 16591156
Milnor, John
(1985), "Riemannian coverings and isospectral manifolds", Ann. of Math., 2, 121 (1): 169–186, doi:10.2307/1971195, JSTOR 1971195
Sunada, T.
Zelditch, S. (2000), "Spectral determination of analytic bi-axisymmetric plane domains", Geometric and Functional Analysis, 10 (3): 628–677, :math/9901005, doi:10.1007/PL00001633, S2CID 16324240
arXiv
showing solutions of the wave equation in two isospectral drums
Simulation
by Toby Driscoll at the University of Delaware
Isospectral Drums
Some planar isospectral domains
3D rendering of the Buser-Conway-Doyle-Semmler homophonic drums
by Ivars Peterson at the Mathematical Association of America web site