**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?" was the witty title of an article by Mark Kac in the American Mathematical Monthly 1966 (see the references below), but these questions can be traced back all the way to Hermann Weyl.

The frequencies at which a drumhead can vibrate depend on its shape. The Helmholtz equation tells us the frequencies if we know the shape. These frequencies are the eigenvalues of the Laplacian in the region. A central question is: can they tell us the shape if we know the frequencies? No other shape than a square vibrates at the same frequencies as a square. Is it possible for two different shapes to yield the same set of frequencies? Kac did not know the answer to that question.

Read more about Hearing The Shape Of A Drum: Formal Statement, The Answer, Weyl's Formula, The Weylâ€“Berry Conjecture

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“When, at rare intervals, some thought visits one, as perchance he is walking on a railroad, then, indeed, the cars go by without his hearing them. But soon, by some inexorable law, our life goes by and the cars return.”

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“If all would lead their lives in love like me,

Then bloody swords and armor should not be;

No *drum* nor trumpet peaceful sleeps should move,

Unless alarm came from the camp of love.”

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“We can say that the sound is the primary object of the act of *hearing*, and that the act of *hearing* itself is the secondary object.”

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As with the might of waters; an apt type

This label seemed of the utmost we can know,

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His steadfast face and sightless eyes, I gazed,

As if admonished from another world.”

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