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You Can't Always Hear the Shape of a Drum

You Can't Always Hear the Shape of a Drum

You Can't Always Hear the Shape of a Drum

In 1966, Marc Kac posed the question "Can you hear the shape of a drum?"More precisely, can you deduce the shape of a plane region by knowing thefrequencies at which it resonates (where, as in a physical drum, the boundaryis assumed to be held fixed)?

Long before Kac posed this question, mathematicians had beeninvestigating the analogous questions in higher dimensions: Is a Riemannianmanifold (possibly with boundary) determined by its spectrum?

The problem was first settled, in the negative, in higher dimensions. In 1964,John Milnor found two distinct 16-dimensional manifolds with the same spectrum. But the problem for plane regions remained open until 1991, when CarolynGordon, David Webb, and Scott Wolpert found examples of distinct plane "drums"which "sound" the same. See the illustrations below.

The story of the problem and its solution can be found in the article YouCan't Always Hear the Shape of a Drum by Barry Cipra, which appeared inVolume 1 of What's Happening in the MathematicalSciences.

David Webb and Carolyn Gordon, former faculty at Washington
University in St. Louis, with paper models of a pair of
"sound-alike" drums.
(Photo courtesy of Washington UniversityPhotographic Services).

View an animation of the top two drums in Figure 1 beating.
(Reproduced with permission of the Cornell TheoryCenter)

(Note: This animation is a large file (1.7 megabytes). It is in MPEG format, so you musthave an MPEG player to view this file. If you need to locate aplayer, here is a list of some of the MPEG resourceson the web.)

- Steven Weintraub

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