Artificial Wasteland · Instruments · hearable mathematics

Can You Hear the Shape?

In 1966 Mark Kac asked whether a drum's sound determines its shape. Strike these two — they have the same answer. Then prove your ears still work on a circle.

Two drums, cut to two different non-convex octagons. Tap each to strike it. They ring the identical set of overtones — the same spectrum, to twelve digits — so at a shared pitch they are the same sound. Your eyes say two shapes; your ears say one drum.

Audio starts on your first tap — phones stay silent until then. Each strike is the drum's eigenvalue spectrum; nothing is recorded.

A round plays you two drums with their shapes hidden. Strike both, then judge: same sound, or different? Sometimes it is the two octagons above (same). Sometimes one of them is swapped for a circle or a square — a genuinely different shape (different). Catch the impostor by ear.

Press New round to begin.

Strike both pads, then call it.
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Here is what your ear was reading: the overtone fingerprint of each shape — the ratios √(λₙ/λ₁) of its eigenvalues, drawn as the partials of a struck tone and normalised so the fundamental matches. Pick a drum, then tap a bar to hear that one partial, or strike the whole tone. The two octagons share one ladder; the circle and square stand apart.

octagon (both)
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The bar heights are exact: octagon from Driscoll's 12-digit eigenvalues (1997), circle from the Bessel zeros, square from π²(p²+q²). The fundamental, the number of partials and the mallet are choices — held equal across every drum, so a difference you hear is a difference of shape.

What you are hearing

A drumhead, pinned at its rim, can only vibrate at a discrete set of pure frequencies — its spectrum. Mathematically these are the square roots of the Dirichlet eigenvalues of the Laplacian on the drum's shape: solve −Δφ = λφ with φ = 0 on the boundary, and the natural frequencies are fₙ ∝ √λₙ. Strike the drum and you hear those frequencies at once, decaying — a chord fixed entirely by the shape.

In 1966 Mark Kac asked the question this instrument is named for: can one hear the shape of a drum? — does the spectrum determine the shape? For twenty-six years it was open. In 1992 Carolyn Gordon, David Webb and Scott Wolpert answered no: they built two non-congruent planar drums with exactly the same spectrum. The pair at the top of this page is theirs — two eight-sided drums, each seven right-isosceles triangles, that ring an identical ladder of tones.

The shape sets the spectrum. But the spectrum does not set the shape — and that is something you can hear, by failing to.

The fingerprint, and the trick of the pitch

Two drums of the same shape but different size are easy to tell apart: the smaller is higher (λ scales as 1/area). That is hearing the size, not the shape. So this instrument normalises every drum to one fundamental and leaves only the pattern of overtones above it — the ratios rₙ = √(λₙ/λ₁), what we've called the shape's fingerprint. A circle's fingerprint is wide and bright; the octagon's is compressed and clustered; the square's sits between. At a common pitch those are audibly different instruments — which is exactly why you can catch the circle.

The two Gordon–Webb–Wolpert octagons have the same fingerprint, mode for mode. Not similar — identical, to every digit anyone has computed. That is why the "same or different" round is, for them, genuinely undecidable by ear: there is no acoustic difference to find. When you call them same and the shapes are revealed, you have heard the theorem.

What you can hear, and what you can't

Equal spectra are not nothing — they force a great deal to agree. By Weyl's law the two drums must have the same area (both exactly 7/2); the next term of the heat trace forces the same perimeter; the corners contribute the same again. So "same area and perimeter" is necessary but not sufficient — which is what makes the counterexample hard, and why it took until 1992.

The honest fine print. What the theorem equates is the spectrum — the frequencies. The drums' eigenfunctions (the shapes of each vibration) genuinely differ, so where you strike a real drumhead changes the loudness balance of its overtones, and that balance can differ between the two. This instrument holds the strike envelope equal for every drum on purpose, to isolate the frequencies — the thing Kac's question is about. The companion stratum carries that nuance, and the elementary transplantation proof (each piece of one drum's vibration is a signed sum of three of the other's), in full.

The mapping, and what is true

Every overtone you hear is the spectrum, not an imitation of it. The bar heights and the partial frequencies are rₙ = √(λₙ/λ₁) taken from primary data: the octagon from Tobin Driscoll's published 12-digit eigenvalues (SIAM Review, 1997), the circle from the zeros of the Bessel functions, the square from π²(p²+q²). Our own from-scratch finite-element solver reproduces the octagon spectrum and confirms the two drums agree mode-for-mode (the residual is pure mesh error and shrinks as the mesh refines). The numbers, the equality check, and the arrays this page embeds are all in research/isospectral-drums/.

What is a choice, not a fact: the common fundamental pitch, the number of partials, and the mallet (how fast the higher overtones fade). Real drumheads carry many more modes, with their own slightly-inharmonic detail and a richer decay; this is an honest sketch of a membrane's spectrum, held identical across all three drums so that any difference you hear is a difference of shape and nothing else. The pitch, partial count and mallet live in the URL, so Share this tuning hands someone the exact sound you made.

Sources

An exhibit in the Wasteland's instrument room — hearable mathematics, where every sound is the object. It is the playable companion to the stratum You Can't Hear the Shape of a Drum, which solves the eigenvalues live with a finite-element method and sets out the proof in full.