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Solving the wave equation
Strike the skin, then change its shape. Every pitch here is solved from the wave equation, live, on your device.
The first twelve modes of this drum, drawn the way Chladni did it: sand settles where the skin stays still. A strike is a mix of all of them. Tap one to hear it alone.
All four are glued from seven identical triangles, with the same area, the same perimeter and the same corners. Two of them also share every frequency. Ring plays a drum's frequencies at equal strength, so only pitch matters. Listen, then pick the pair.
In 1911 Hermann Weyl showed that a drum's frequencies give away its area. Count the modes below a frequency f: as f grows, the count approaches π·area·f²/c², where c is the wave speed of the skin. A second, smaller term gives away the perimeter, and for a polygon a third gives away its corners.
The panel above does exactly this. It sees only the list of frequencies, fits Weyl's formula to them, and reads off the area. For the current drum that gives · against a measured ·.
So when Mark Kac asked in 1966 whether one can hear the shape of a drum, the real question was whether the frequencies reveal anything beyond these few numbers.
Carolyn Gordon, David Webb and Scott Wolpert answered Kac in 1992, and the answer was no. Their two drums are each glued from seven copies of one triangle. Take any way one drum can vibrate and cut it along the triangles. Added together in the right combinations, some of them flipped upside down, the pieces rebuild a vibration of the other drum at exactly the same frequency. Mathematicians call this transplantation. It works in both directions, so the two drums share every frequency.
This page checks it numerically. On your device, drums 2 and 4 agree across all 48 computed frequencies to within ·, which is the rounding limit of 64-bit arithmetic. The grid lines up with the triangles, so even the discretised drums are exact twins.
The other two drums came from a search. Of the 316 shapes with a simple outline that can be glued from seven of these triangles, exactly four have the twins' area, perimeter and corner angles, and the lineup is all four. Drum 3 matches the twins to within a quarter of a percent for its first seven frequencies, then drifts by up to 2%. Drum 1 starts 4% flat.
Each drum becomes a grid of points, and the wave equation becomes a matrix problem: find the patterns in which every point accelerates in proportion to its own displacement. Each pattern is a mode, and its rate of oscillation is a frequency.
The solver is written from scratch and runs in a background thread: a sparse Cholesky factorisation of the grid Laplacian, a block Lanczos iteration on its inverse with full reorthogonalisation so that repeated frequencies are not missed, then a Householder and QL eigensolver on the small projected matrix. The current drum: ·.
The sound is additive synthesis of those modes. A strike at a point excites each mode in proportion to how much the skin moves there, and each mode then decays at its own rate. The picture is the same sum, slowed down about 150 times.
The skin is an ideal membrane on a 60 cm frame with a wave speed of 100 m/s, roughly a medium-tuned tom. Real drums add air loading and stiffness, which shift the overtones by a few percent.