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.
Famous quotes containing the words hearing the, drum, hearing and/or shape:
“I used to be angry all the time and Id sit there weaving my anger. Now Im not angry. I sit there hearing the sounds outside, the sounds in the room, the sounds of the treadles and heddlesa music of my own making.”
—Bhakti Ziek (b. c. 1946)
“Not a drum was heard, not a funeral note,
As his corse to the rampart we hurried;”
—Charles Wolfe (17911823)
“Therefore I have uttered what I did not understand, things too wonderful for me, which I did not know. ... I had heard of you by the hearing of the ear, but now my eye sees you; therefore I despise myself, and repent in dust and ashes.”
—Bible: Hebrew, Job 42:3-6.
Job to God.
“And yonder in the gymnasts garden thrives
The self-sown, self-begotten shape that gives
Athenian intellect its mastery,
Even the grey-leaved olive-tree
Miracle-bred out of the living stone....”
—William Butler Yeats (18651939)