An intriguing question about drums kicked off decades of inquiry
More than 50 years ago Polish-American mathematician Mark Kac popularized a zany but mathematically deep question in his 1966 paper “Can One Hear the Shape of a Drum?” In other words, if you hear someone beat a drum, and you know the frequencies of the sounds it makes, can you work backward to figure out the shape of the drum that created those sounds? Or can more than one drum shape create the exact same set of...
The researchers’ findings began to crystallize in a new way while one of the mathematicians—Carolyn Gordon, now an emeritus professor at Dartmouth College—was on a short visit to Europe. She had traveled to Germany’s Mathematical Research Institute of Oberwolfach, nestled in the Black Forest.
Effectively, their work had answered a question that earlier researchers considered intractable. In 1882 Arthur Schuster, a German-born British physicist, wrote, “To find out the different tunes sent out by a vibrating system is a problem which may or may not be solvable in certain special cases, but it would baffle the most skilful [sic] mathematician to solve the inverse problem and to find out the shape of a bell by means of the sounds which it is capable of sending out.
Yet among all the individual results about hearing shapes, a different team of researchers pointed out a gaping unsettled idea: it remains to be seen whether it is generally true that you will be able to discern the outline of a given type of shape or surface from its sounds. “We’re thinking about [flat] tori,” Rowlett says. In one dimension, a torus “is just a circle,” she notes. In three dimensions, mathematicians often describe tori as having the shape of a glazed doughnut, though they’re usually only referring to the surface of the sugary delight, not its doughy innards.
Rowlett’s recent preprint paper, which she co-authored with two researchers who were then her students, was motivated by her desire to discover “the tipping point” between when you can and can’t hear the shape of a flat torus. “Three is the magic number,” meaning one can’t hear the shape of tori in four or more dimensions, she says.