Hit a drum and it rings at a fixed set of frequencies — its pure tones, fixed entirely by the shape of the drumhead. Run that backward: if you knew every frequency, could you reconstruct the shape? For 26 years nobody knew. The answer is no — and here are the two drums that prove it.
A drumhead is a membrane clamped at its rim. Strike it and it vibrates, but not at just any frequency: only at a discrete ladder of them, the same way a guitar string sounds a fundamental and its overtones. Mathematically those frequencies are the square roots of the eigenvalues of the Laplacian on the drum's region, with the boundary held fixed (the rim can't move). Bigger drum, lower tones; the full ladder of frequencies is the drum's spectrum — everything you could possibly hear from it.
In 1966 the mathematician Mark Kac wrote a famous paper with a one-line title: “Can One Hear the Shape of a Drum?” He meant it precisely. Two drums of different shapes — do they always have at least one different tone, so your ear (given perfect pitch and infinite patience) could tell them apart? Or could two genuinely different shapes ring identically?
Some of the answer was already known and is genuinely surprising on its own: from the spectrum you can recover the drum's area (Hermann Weyl proved this in 1911), and later its perimeter, and even the number of holes in it. So two drums that sound alike must have the same area and the same perimeter. That feels like almost enough to pin down the shape. It isn't.
In 1992, Carolyn Gordon, David Webb, and Scott Wolpert built two flat drums — concave eight-sided polygons — that are not the same shape and cannot be moved or flipped onto each other, yet have exactly the same spectrum. Every frequency matches. The ear cannot tell them apart. Here they are. Nothing below is asserted on my word; the page computes it in front of you.
Below are the two drums. They are each built from seven identical triangular tiles, folded together two different ways. Press strike and the page does, live, what no closed formula can: it lays a fine triangular mesh over each drum (a finite-element model), assembles the stiffness and mass matrices of the membrane, and solves for the lowest vibration modes by inverse iteration, then turns the frequencies it finds into sound. The two drums produce the same chord. Not approximately: each triangle gets the same mesh, laid down in mirror image across every shared edge, so the two computed frequency sets agree to rounding (about twelve digits) at every mesh size. What the finite mesh does change is how close both land to the true frequencies, and that gap shrinks as you refine.
Added 28 September 2026: the strike buttons play each drum's eight lowest frequencies with one invented loudness rule, the same for both drums, so what they let you hear is the list of frequencies, which is exactly what Kac asked about. A real strike also sets how loud each note is, and that depends on the shape. The last button, strike the whole head, uses the loudness each drum really gives each note when every point of it is hit at once, computed live from the same solve. Those loudnesses are not the same: see section IV.
The spectrum is not silent about the shape — it just stops short. Three things are recoverable from the list of frequencies alone, by reading how fast the frequencies pile up:
· the area, from Weyl's law: the number of frequencies below a bound ω grows like (area / 4π) times ω² (in terms of the eigenvalues, which are the squared frequencies: area / 4π times the eigenvalue bound). Count the tones, and you've measured the drumhead's area.
· the perimeter, from the next, finer term in the same expansion (the heat trace).
· the number of holes (here, none), from the term after that.
So our two drums are forced to share area and perimeter — and they do, both built from seven tiles of total area 7⁄2. The lesson of 1992 is that area + perimeter + holes is necessary but nowhere near sufficient: you can match all three and still be a different shape. The map from shape to spectrum loses exactly the rest.
Equal area. Equal perimeter. Equal every overtone. Still not the same drum.
Why are the spectra exactly equal: not by coincidence, not to twelve digits, but provably? Because of a beautiful piece of bookkeeping called transplantation (Bérard; Buser–Conway–Doyle–Semmler 1994; popularized by Chapman 1995). Take any way the first drum can vibrate (any eigenfunction). Restrict it to each of the seven triangles. Now build a function on the second drum where each of its seven triangles is a specific signed sum of three of those pieces. The recipe is fixed (worked out for these two drums from how their triangles meet); here it is. Hover a triangle of Drum B to see which three pieces of Drum A build it.
Added 28 September 2026. Kac's question is about the list of frequencies, and for these two drums the list is the same. But a drum carries more than its list. Hit every point of the head at once, as a flat beater would, and each note sounds with a loudness set by how much of the head moves together in it: the square of the integral of that vibration, (∫φ)². The same numbers decide something with no sound in it at all. Warm a drumhead to temperature 1, hold its rim at 0, and let it cool: the heat left at time t is Σ (∫φ)² e−λt, the heat content.
For these two drums those numbers differ, and not by a little. The second note weighs about 0.58 in Drum A and 0.13 in Drum B; the third about 0.06 in A and 0.39 in B. So struck across the whole head, A sounds mostly its first two notes, a little over a minor third apart, and B mostly its first and third, near a tritone. The button above plays them, with the weights computed live.
And the heat is different, provably. The total heat over all time is the drum's torsional rigidity. We bounded it from both sides for each drum, with trial functions rounded to exact fractions and every integral done in exact integer arithmetic, so that neither rounding nor a sloppy solver can fake the result: Drum A's is at most 0.2211958 and Drum B's is at least 0.2219662 (unit legs, area 7⁄2). An independent audit with its own quadratic elements and finite differences found 0.2209138 and 0.2222505. At late times only the lowest note is left, and B keeps about 11% more heat than A. At early times the difference is too small for our meshes to say which drum is warmer, and we don't claim it.
Why doesn't the proof above carry the heat across as it carries the notes? Heat content would match if the recipe carried a constant temperature to a constant, which for a norm-preserving recipe means every column summing to 1 (the criterion in Herbrich 2014, §6.5). This recipe's columns sum to −1, −1, 1, 1, 3, 1, 1.
Same notes. Different heat.
As far as we could find, nobody had shown this before for these drums. One refereed paper says it cannot happen: “It is a result of Gilkey that heat content cannot distinguish isospectral planar domains constructed via the Sunada method” (Colladay, Kaganovskiy and McDonald, arXiv:1607.05752, published in J. Phys. A 50, 2017; we quote the arXiv text, as the journal's is paywalled), and a 2021 preprint by Comer and McDonald repeats it. Gilkey's theorem (2009) is about finite coverings of manifolds with smooth boundary. As we read Gordon, Webb and Wolpert's own construction, the drums' clamped-rim vibrations are the anti-invariant ones on a double cover, and Gilkey's argument, which lifts the constant temperature 1, reaches only the invariant ones. That reading is ours and deserves checking by someone who knows these constructions from the inside. If you know of prior work, tell us.
The proof is one file, no dependencies, about twelve seconds: verify-heat.mjs (node verify-heat.mjs in an empty folder). The full notebook, with the literature quoted and the audit's own codes, is here.
The 1992 drums are concave: they have inward dents. The most natural follow-up question is still unanswered: can two convex drums (no dents, like an ellipse versus an egg) sound identical? Nobody knows. And the news isn't all negative: if a drum's boundary is perfectly smooth (analytic) and has a symmetry, you often can hear its shape (Zelditch); and an ellipse of small eccentricity (a nearly round one) is pinned down by its spectrum (Hezari and Zelditch, Annals 2022). So the honest summary is narrow and exact: among general shapes, the spectrum does not determine the drum; among nice enough ones, sometimes it does.
This wasn't the first time geometry hid from its own spectrum. John Milnor found two different 16-dimensional doughnuts that sound alike back in 1964 (the first isospectral pair of non-isometric manifolds); Toshikazu Sunada turned the trick into a machine in 1985 (the method that built these drums, using two subgroups of a 168-element group that are almost but not quite interchangeable). Gordon, Webb, and Wolpert's achievement was to bring it all the way down to a shape you can draw on paper.
None of the above is taken on faith. The drums' frequencies were recomputed in your browser; the engine was checked first against shapes whose tones are known exactly, and against a published 12-digit table; and the transplantation recipe is applied live to the computed vibrations. The apparatus says precisely what is solid and what is left open.