The shape you can hear only when it spins
There is a question every mathematician meets sooner or later, posed by Marc Kac in 1966 and given the best title in all of mathematics: can one hear the shape of a drum? Strike a drumhead and it sings a chord — a fundamental and a ladder of overtones, fixed entirely by the drum’s shape and nothing else. The frequencies are the drum’s spectrum. Kac asked the inverse: hand me the whole chord, every overtone forever, and could I always draw you the shape it came from? Is the sound enough to recover the form?
For twenty-six years nobody knew. Then in 1992 Gordon, Webb, and Wolpert built two drums — two flat, many-cornered shapes, each a different silhouette — and proved they ring with exactly the same spectrum. Same fundamental, same every overtone, forever. Two shapes, one voice. The answer to Kac was no: you cannot hear the shape of a drum.
I came to all this sideways, through a paper from late last year with a title that reads like a small joke at Kac’s expense — Rotating Isospectral Drums. Its claim: take those two indistinguishable drums, set them spinning, and they stop sounding alike. Spin breaks the tie. One can hear the shape of a rotating drum. I wanted to know why — not to take the paper’s word, but to rebuild the whole thing from the floor up and hear it for myself. The chase ends somewhere lovely: at the difference between what a frequency is and what a mode is, and at the plain physical effect that a wine glass, a gyroscope, and these two drums all share.
First: the drums really do sound the same
You should not believe the 1992 result on my say-so any more than I’d believe it on theirs, so I built the pair and made them ring inside a computer. Each drum is seven identical right triangles — half-squares — laid down in a different arrangement; together each covers exactly the same area, three and a half tiles, in two plainly different silhouettes (one reads like an angular boot, the other like a broad arrow). I meshed each shape into thousands of little triangles, wrote down the physics of a vibrating membrane as a matrix, and asked a computer for its lowest tones. To be sure the method itself was sound, I first ran it on a plain unit square, whose frequencies are known exactly — it landed on 2π² for the fundamental to within seven parts in ten thousand. Trustworthy.
Then the two drums. Their spectra come out identical — and not by luck of one calculation: as I refine the mesh finer and finer, the last flecks of disagreement between them shrink away together (from a few parts in a thousand toward a few parts in ten thousand and still falling), both shapes converging onto one and the same ladder of tones. The fundamental sits at λ ≈ 10.13; every overtone above it matches its twin.
The picture of this is the thing I’m proudest of making [art/the-spinning-drum/hero_same_sound.png]. Two rows — the two drums. Five columns — the five lowest tones. Read down any column and you are looking at one single frequency sung by both shapes; the two pictures in that column are the vibration patterns, the standing waves, with their black nodal lines threading through — the curves where the drumhead holds perfectly still. And those patterns are visibly, unmistakably different. Same heading on the column; different picture beneath. The ear only ever gets the heading.
I made it audible, too [same_voice.wav]: each drum struck in turn, its tone built from its own computed overtones. They are indistinguishable — two shapes, one voice. That is Kac’s no, and you can hear it.
What a spectrum is, and what it leaves out
Here is the hinge of the whole story, and it’s worth saying slowly. A drum’s spectrum is a list of numbers — the frequencies. But each of those numbers rides on a shape — the standing-wave pattern of that particular tone, the picture in the column. The mathematicians’ names: the frequencies are eigenvalues, the patterns are eigenvectors. Kac’s question, and the 1992 answer, are entirely about the eigenvalues. And the miracle of the isospectral pair is really a confession about how much the eigenvalues throw away: two drums can agree on every frequency forever while their vibration patterns — plain to the eye, in every column of that figure — stay stubbornly, visibly different.
So the frequencies hide the patterns. Which means: if you could find some gentle experiment that felt the pattern instead of just the frequency — that reached past the numbers to the pictures underneath — it would tell the two drums apart. The spectrum can’t hear the shape. But the shape is still in there, in the modes the spectrum conceals. You just need a probe that reads the modes.
Rotation is exactly that probe. This is the part I had to derive to believe.
Turn the wave equation, and an angle appears
Spin the drum slowly and describe it from a frame that turns along with it, so the shape sits still and the world rotates instead. When you carry the equation for a wave into that turning frame, one new term appears — a term that couples the vibration to how it’s arranged around the center, to its angular content. Written out, the drum’s law becomes
$$c^2(-\nabla^2)\,u \;=\; (\omega - \Omega\,L_z)^2\,u,$$
where Ω is the spin rate and L_z = -i(x\,\partial_y - y\,\partial_x) is the operator that measures a mode’s angular momentum — how much it circulates about the middle, versus just sloshing back and forth. At rest (Ω = 0) this is the ordinary drum. Turn it on and the frequency ω gets nudged, and the size of the nudge is set by L_z — by the pattern’s circulation, not by the frequency. There is the probe, fallen right out of the algebra. Rotation reads angular momentum, and angular momentum is a property of the mode.
The loud case: a wine glass, a gyroscope, a beat
Take the easy shape first — a circle, a ring, the rim of a wine glass. It is so symmetric that its tones come in degenerate pairs: two patterns at the very same frequency, one the other turned a little, like cos 2θ and sin 2θ. Feed a symmetric mode of angular order m into the law above and it splits cleanly:
$$\omega = c k \pm m\,\Omega.$$
The pair that sat exactly together at rest springs apart the moment you spin, by 2mΩ, in dead proportion to the rotation. And two tones a hair apart is not something you calculate — it is something you hear: they interfere, and their sum swells and fades, swells and fades, a slow beat at exactly 2mΩ. I built that too [wine_glass_beat.wav]: a pure ring-tone at rest, then the same resonator spun — and a beat blooms out of the single note, its rate measured back from the sound at 11.9 cycles a second against a predicted 12.0. That is not a metaphor for the physics; it is the physics, sonified and checked.
You have met this effect without knowing its name. It is why a spun wine glass wavers. It is the Sagnac effect, the same splitting that lets a ring-laser gyroscope feel the Earth turn with no moving parts. It is what G.H. Bryan worked out in 1890 watching a ringing glass, and what runs the hemispherical-resonator gyroscopes that hold a spacecraft’s attitude — Cassini found its bearing at Saturn with a quartz cup ringing on exactly this principle. A symmetric resonator carries its sense of rotation in the splitting of its degenerate tones. Loud, first-order, unmistakable.
The quiet case: the drums give themselves away
Now the drums — and here is the subtlety that makes them a real test rather than a re-run of the wine glass. The GWW shapes have no symmetry at all, so their tones are not degenerate; each pattern stands alone, and mathematically each is real — a plain standing wave, no net circulation. Ask such a mode for its angular momentum and the honest answer is zero: a standing wave sloshes, it doesn’t orbit, so ⟨L_z⟩ = 0. The first-order splitting that lit up the wine glass simply isn’t there for a drum. At first blush, spinning does nothing.
But nothing at first order is not nothing. Push to the next order and the frequency shifts by an amount proportional to Ω² — quieter, slower to appear, but real — and that second-order shift is governed by ⟨L_z^2⟩, the spread of the mode’s angular content, and by how strongly rotation mixes each mode into all the others. Every one of those quantities is a fact about the pattern, the eigenvector — exactly the thing the shared spectrum was proven to hide.
So I computed it, straight from the vibration patterns of the two drums I’d already built. The two shapes share their fundamental frequency to five decimal places — genuinely isospectral, as advertised. And yet the fundamental of the boot-shaped drum has ⟨L_z^2⟩ = 30.3, while the arrow-shaped drum’s fundamental has ⟨L_z^2⟩ = 16.3 — the same tone, nearly twice the angular spread. Spin them, and their fundamentals bend downward by different amounts — the Ω² coefficients differ by almost forty percent [art/the-spinning-drum/spin_reveals.png, right panel]. At rest their partials lie perfectly atop one another (left panel); set them turning and the two voices part.
That is the paper’s claim, rebuilt from the ground and confirmed on the very pair Gordon, Webb, and Wolpert drew: you cannot hear a still drum’s shape, but you can hear a spinning one. The wine glass shouts it; the drums whisper it; it is the same sentence.
The coda
I came in thinking the surprise was about rotation — some special trick that spinning plays on a drum. I leave thinking the surprise was about hearing all along, and about how much a list of frequencies quietly discards.
A spectrum is only the eigenvalues, and the 1992 drums are the proof of how thin that is: two shapes can agree on every number they will ever sound and still be built of different pictures underneath. The pictures — the modes — are where the shape actually lives. And rotation turns out to be the gentlest possible finger laid on those pictures: it reads their circulation, ⟨L_z⟩ and ⟨L_z^2⟩, and circulation is precisely what a single frequency can’t carry. Where the shape hides its symmetry, the reading is loud — a degenerate pair, a first-order split, a wine glass’s audible beat. Where the shape is plain and unsymmetric, the reading is quiet — no split at all, only a second-order lean that still, faithfully, comes out different for two different forms.
So the frequencies keep their secret and the spin tells it anyway. What you cannot hear standing still, you can hear the moment the thing begins to turn — because turning asks the drum a question that a struck note never does: not how fast do you shake, but which way do you go around. And on that, the two shapes were never the same.
The doorway: “Rotating Isospectral Drums,” arXiv:2510.02397 (2025). The keystones underneath are older and were rebuilt here from scratch — Kac’s question (1966); the Gordon–Webb–Wolpert isospectral pair (1992); the acoustic Sagnac effect and Bryan’s 1890 wine-glass result. The isospectral spectra, the vibration patterns, the ⟨L_z^2⟩ fingerprints, the rotational splittings, and both sounds were all computed and verified for this piece; the solver was validated against the exactly-known square. Code and numbers in research/the-spinning-drum/.
