Strike the drum below — anywhere. The displacement resolves into a superposition of normal modes whose frequencies are the zeros of Bessel functions — inharmonic, which is why drums sound like drums and not like bells:
Strike near the center to excite only radial modes m=0. Strike off-center to wake asymmetric modes m≥1. You can hear the difference — the timbre shifts as the excited eigenfunctions change.
The drum size slider (top-right) scales R, shifting all frequencies simultaneously as ω ∝ 1/R.
Each eigenfrequency ωₙ = jₙ c/R is a rotating phasor. Toggle modes to add or remove their rotation — and hear the resulting tone shift in real time. The tones are the exact physical frequencies of the drum.
With all four modes active, the waveform is not a pure sine — it is the inharmonic superposition that gives drums their characteristic sound.
↑ each toggle starts / stops its sine tone
Each card is a single eigenmode φₘₙ = Jₘ(jₘₙ r/R)cos(mθ) oscillating at its pure frequency. Click any card to hear and see that tone alone. Higher modes have higher, shorter-lived frequencies — the drum's voice decays from high to low as modes ring out.
When two modes have nearby eigenvalues, their superposition beats — the amplitude oscillates slowly at rate |ω₁−ω₂|/2. You can hear this clearly: select a pair and press Play.
With three modes the superposition φ_A cos(ω_A t) + φ_B cos(ω_B t) + φ_C cos(ω_C t) generates richer interference. When no two frequency differences are commensurate, the pattern is quasi-periodic — never repeating exactly, yet bounded. The beating structure becomes a tangle of three rhythms at once.
Choose any three modes and hear the combination. Watch how the spatial pattern morphs as the three standing waves drift in and out of phase.
Weyl proved in 1911 that the density of eigenvalues recovers the area. Let N(λ) count eigenvalues below λ:
For the unit disk (Area = π), this gives N(λ) ∼ λ/4. Each step of the staircase is a new eigenfrequency clicking into place — as the Weyl animation runs, you hear each new mode added as a brief pure tone.
In 1992, Carolyn Gordon, David Webb, and Scott Wolpert gave the definitive answer to Kac's question: No. They exhibited two planar domains — each assembled from seven right triangles — that share every eigenvalue of −Δ with Dirichlet boundary conditions, yet are not congruent. The drums below are identical to the ear. Click either to hear the shared voice.
Let M be a compact Riemannian manifold and G a finite group acting on M by isometries. Let H₁, H₂ ≤ G be subgroups that are almost conjugate — meaning: for every conjugacy class [g] ⊂ G, we have |H₁ ∩ [g]| = |H₂ ∩ [g]|. Then the quotient manifolds H₁\M and H₂\M are isospectral.
The almost-conjugacy condition ensures that the representation multiplicities of G on L²(H₁\M) and L²(H₂\M) agree, forcing equal spectra. Note: almost-conjugate does not mean conjugate — that would force H₁\M ≅ H₂\M isometrically. The gap between almost-conjugacy and conjugacy is exactly the room where isospectral non-congruent pairs live.
Fix a right triangle T with legs of length 1 and 2 (the specific shape matters for the geometry, not for the spectral argument). Seven congruent copies of T will tile each domain, assembled by gluing along their hypotenuses and legs.
The isosceles right triangle with equal legs also works; Gordon–Webb–Wolpert used a right triangle whose smaller angle is arctan(1/2). What matters is that reflections across every edge of T generate a finite group action on a larger base surface.
The ambient group is G = GL(3, 𝔽₂), the group of invertible 3×3 matrices over the two-element field — a group of order 168, isomorphic to PSL(2, 7). Gordon and Webb found two subgroups H₁, H₂ ≤ G of order 24 that are almost conjugate but not conjugate in G.
Almost-conjugacy was verified by computing, for each of the 6 conjugacy classes of G, that |H₁ ∩ [g]| = |H₂ ∩ [g]| — a finite check. The two subgroups are not conjugate, so the quotient domains they produce are genuinely non-isometric.
Construct a base surface M by reflecting the triangle T across its sides repeatedly, tracking the group element applied at each step, until 168 copies tile a larger polygon (a fundamental domain for G acting on the plane by reflections).
The orbit of each copy under G is labeled by elements of G. The quotient H_i\M is then a union of exactly |G|/|H_i| = 168/24 = 7 triangles — which is why both domains consist of exactly 7 copies of T.
Both domains Ω₁, Ω₂ automatically inherit the same Weyl invariants. They have equal area (7 copies of T each), equal perimeter (the boundary lengths agree because the gluing patterns are balanced), and the same number of corners with the same angles. Weyl's law and its boundary corrections are therefore consistent — the spectra could in principle be equal, and by Sunada's theorem they are.
The core of the proof is an explicit linear map T: L²(Ω₁) → L²(Ω₂) that intertwines the Laplacians. Label the 7 triangles in Ω₁ as A₁…A₇ and in Ω₂ as B₁…B₇. Given an eigenfunction φ of −Δ on Ω₁ with eigenvalue λ, the transplant is:
where the matrix (cᵢⱼ) ∈ {−1, 0, 1}^{7×7} encodes the almost-conjugacy relation between H₁ and H₂ in G. Three properties must be verified:
All three properties follow from the almost-conjugacy condition on H₁, H₂. Smoothness across interior edges uses the fact that reflections across each edge correspond to elements of G, and the matrix (cᵢⱼ) is precisely calibrated to cancel discontinuities. The argument is purely algebraic — no analysis of the actual eigenvalues is needed.
All known planar isospectral counterexamples have corners. Whether a generic smooth planar domain is spectrally rigid — determined by its spectrum among smooth domains — is unknown.
No isospectral pair of convex planar domains is known. It is conjectured that convex domains are spectrally determined, but this remains unproved.
The GWW domains are isospectral for both Dirichlet and Neumann boundary conditions simultaneously — a stronger result. Gordon–Webb showed this also follows from transplantation.
The nodal set of an eigenfunction φ is the zero locus {φ=0} — the curves where the membrane neither rises nor falls. The connected components of Ω ∖ {φ=0} are the nodal domains: regions of pure positive or pure negative displacement, alternating like a checkerboard across the dividing lines.
On the circular drum, the nodal set of φₘₙ = Jₘ(jₘₙr/R)cos(mθ) consists of n−1 interior concentric circles (where Jₘ vanishes) and 2m radial lines (where cos(mθ)=0). These divide the disk into exactly 2mn nodal domains for m≥1, and n for m=0. Click any mode to see its nodal architecture.
The n-th eigenfunction (counting with multiplicity) has at most n nodal domains. The bound is sharp for n=1,2 but becomes increasingly slack for large n. Equality holds, for instance, for the n-th mode of a 1D interval.
Pleijel proved that the Courant bound is not asymptotically sharp in 2D. The Pleijel constant is γ = (j₀,₁/π)² = (2.4048/π)² ≈ 0.691. For any sequence of eigenfunctions, the ratio of nodal domains to eigenvalue index is bounded above by a constant strictly less than 1.
For surfaces with ergodic geodesic flow, Jung and Zelditch proved that quantum ergodic sequences of eigenfunctions — those equidistributing in phase space — have the number of nodal domains tending to infinity. The equidistribution of eigenfunctions forces their zero sets to become increasingly complex.
Jang and Jung study the precise threshold at which quantum ergodicity forces nodal complexity. Their results sharpen the relationship between L²-equidistribution of eigenfunctions and the combinatorial topology of their zero sets — showing that even partial equidistribution in phase space has detectable consequences for the nodal domain count.
The theory of nodal domains sits at the meeting point of spectral theory, dynamical systems, probability, and topology. The following directions represent some of the most active and enticing open problems in the field — questions where a single idea could unlock a decade of mathematics.
Yau conjectured (1982) that the (n-1)-dimensional measure of the nodal set satisfies c√λ ≤ H^{n-1}({φ=0}) ≤ C√λ. Donnelly–Fefferman proved the upper bound; Logunov proved the lower bound in full generality in 2018. But the precise constant — and its dependence on the geometry — remains unknown.
Berry (1977) conjectured that high-energy eigenfunctions on ergodic surfaces locally resemble random monochromatic waves. Under this model, nodal domains form a percolation pattern with a precise critical density. Proving that real eigenfunctions exhibit this percolation structure — even on a torus — is wide open.
Can two isospectral drums — like the GWW pair — have different nodal domain counts for corresponding eigenfunctions? The transplantation map is an isometry of L² but may not preserve the sign of eigenfunctions or the topology of their zero sets. This offers a potential geometric invariant beyond the spectrum.
Pleijel's theorem extends to d-dimensional manifolds with constant γ_d = (j_{d/2−1,1} / ω_d^{1/d})^2 where ω_d is the volume of the unit ball. The sharp limsup — achieving the Pleijel constant along some sequence — is not known in dimension d ≥ 3, nor is it known which geometric features drive the maximum.
What topological types can nodal curves realize? On a surface of genus g, the nodal set of the n-th eigenfunction is a union of smooth embedded curves. Their collective topology — how many connected components, their linking, their genus — is essentially unconstrained by current theory. Jung's recent work begins to probe this frontier.
On arithmetic hyperbolic surfaces, Hecke–Maass eigenfunctions are highly constrained by number-theoretic symmetries. Jang–Jung results on quantum ergodicity apply, but the rate of nodal growth, and connections to the Riemann Hypothesis via L-function zeros, hint at deep arithmetic content encoded in the nodal structure of these extraordinary functions.