CAN YOU HEAR THE SHAPE OF A DRUM?

Tap a drum. Where you tap changes the mixture, not the notes.

WHAT THIS IS

In 1966 Mark Kac asked whether the set of notes a drum can make determines its shape. A drum skin has a spectrum, the eigenvalues of the Laplacian on its shape, and every note you can hear is one of them. The round drum is the one you know. The two seven-sided drums are Gordon, Webb and Wolpert's 1991 answer: different shapes, identical spectra, every note, forever. You cannot hear the shape; you can hear where the mallet lands.

The last drum is the one I wanted to hear. It is a regular octagon with each side glued to the one opposite, which closes it into a surface with two holes (genus 2) that is curved like a saddle at every point. That is the Bolza surface, from 1887. No such shape fits in our space at any size, so no one can stretch a skin over it, but the wave equation does not care. Drawn in the Poincaré disk, every octagon in the picture is the same drum seen again, so a wave leaving one side comes straight back in at the opposite one. Its lowest note, 98 Hz, comes three times over, as three different shapes at one pitch, because the surface has so much symmetry.

Every sound here is additive synthesis from the drums' own computed modes (300 for the flat drums, 500 for the octagon), struck with a 1.5 ms mallet, heard at one fixed point on the skin, with the same damping as the film. The film's sounds are the full time-domain wave equation on meshes of up to 267,000 points, which is what the page cannot do live. The octagon's computed spectrum agrees with Strohmaier and Uski's published values (3.8389, 5.3536, 8.2496, with multiplicities 3, 4, 2) to a tenth of a percent.

THE FILM

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