The full Dirichlet spectrum of the Laplacian determines a Euclidean triangle up to congruence, but the known proof reads the triangle off from coefficients in the short-time expansion of the heat trace and therefore consumes the entire (infinite) spectrum. Since the space of triangles is three-dimensional — two shape parameters plus scale — it is natural to ask whether three eigenvalues already suffice. Numerical evidence of Antunes and Freitas suggests that the first three Dirichlet eigenvalues λ1,λ2,λ3\lambda_1,\lambda_2,\lambda_3 determine a triangle uniquely among triangles, while the analogous statement for λ1,λ2,λ4\lambda_1,\lambda_2,\lambda_4 is false; the failure for (λ1,λ2,λ4)(\lambda_1,\lambda_2,\lambda_4) is now a computer-assisted theorem of Gómez-Serrano and Orriols, who constructed non-isometric triangles sharing the first, second and fourth eigenvalues. This problem — Open Problem 1 of Grieser–Maronna (2013) and Conjecture 3 of Lu–Rowlett (2015) — asks for a proof or disproof: must two triangles whose first three Dirichlet eigenvalues coincide be congruent? The problem remains open as of August 2026.

Background

In 1966 Mark Kac popularized the question “Can one hear the shape of a drum?” (Kac 1966): is a bounded planar domain determined, up to rigid motion, by the sequence of eigenvalues of the Laplacian on that domain with Dirichlet boundary condition, i.e. with eigenfunctions required to vanish on the boundary? These eigenvalues are the idealized resonant frequencies of a drumhead with the domain as its face, and two domains with the same eigenvalue sequence are called isospectral. The general answer is negative: Gordon, Webb and Wolpert constructed planar isospectral domains that are not congruent (Gordon–Webb–Wolpert 1992), and all known planar counterexamples are non-convex polygons.

Triangles, by contrast, are audible. Durso proved in her 1988 thesis that a triangle is determined, among all triangles, by its full Dirichlet spectrum (Durso 1988), and Grieser–Maronna (2013) gives a short proof. The heat trace h(t)=∑ke−λkth(t)=\sum_k e^{-\lambda_k t} admits a short-time asymptotic expansion h(t)∼a0t−1+a1/2t−1/2+a1+O(e−c/t)h(t)\sim a_0 t^{-1}+a_{1/2} t^{-1/2}+a_1+O(e^{-c/t}) whose coefficients are explicit universal expressions in the area AA, the perimeter PP, and the angles α1,α2,α3\alpha_1,\alpha_2,\alpha_3 of the triangle; in particular a0=A/(4π)a_0=A/(4\pi), a1/2=−P/(8π)a_{1/2}=-P/(8\sqrt{\pi}) and a1=(1/24)∑i(π/αi−αi/π)a_1=(1/24)\sum_i(\pi/\alpha_i-\alpha_i/\pi), so the spectrum determines (A,P,∑i1/αi)(A,P,\sum_i 1/\alpha_i). The central theorem of the article is that these three numbers determine a triangle uniquely, proved by convexity arguments and the partial fraction expansion of sin⁡−2x\sin^{-2}x. Note that this argument uses asymptotic information about the whole spectrum.

Because the known proof consumes infinitely many eigenvalues, the article asks whether finitely many — ideally three — already suffice. The space of triangles up to congruence is three-dimensional (for instance, it is parametrized by the three side lengths), and since eigenvalues rescale by the factor c−2c^{-2} under a dilation by cc, the eigenvalue data carry scale as well as shape information: the counting matches exactly. Chang and DeTurck proved a finitary analogue with a caveat: for every ε>0\varepsilon>0 there is a number NN such that the first NN Dirichlet eigenvalues determine any triangle uniquely among triangles whose angles are all at least ε\varepsilon, but NN depends on ε\varepsilon and no uniform bound over all triangles is known (Chang–DeTurck 1989).

Numerics strongly support the sharpest possible statement. Antunes and Freitas computed the map from triangles to the first few eigenvalues and found that (λ1,λ2,λ3)(\lambda_1,\lambda_2,\lambda_3) appears to determine the triangle uniquely; they conjectured this, and also observed numerically that the triple (λ1,λ2,λ4)(\lambda_1,\lambda_2,\lambda_4) fails, because near a saddle point of the quotient λ4/λ1\lambda_4/\lambda_1 the quotient λ2/λ1\lambda_2/\lambda_1 is regular, which produces pairs of distinct triangles (Antunes–Freitas 2011). Gómez-Serrano and Orriols turned that observation into a computer-assisted theorem, constructing non-isometric triangles whose first, second and fourth Dirichlet eigenvalues coincide — indeed an open set in the moduli space of triangles consists of such pairs (Gómez-Serrano–Orriols 2021). Thus, among low-lying triples of eigenvalues, the first three appear to be special; whether they determine the triangle remains unknown. The same question is recorded as Conjecture 3 in Lu–Rowlett (2015).

Problem Statement

Let TT be a non-degenerate Euclidean triangle and let 0<λ1(T)≤λ2(T)≤λ3(T)≤…0<\lambda_1(T)\le\lambda_2(T)\le\lambda_3(T)\le\dots be the Dirichlet eigenvalues of the Laplacian on TT, i.e. the numbers for which the equation −Δu=λu-\Delta u=\lambda u in TT has a nonzero solution uu vanishing on the boundary of TT. Question (Open Problem 1 of Grieser–Maronna 2013; Conjecture 3 of Lu–Rowlett 2015): if TT and T′T' are triangles with λi(T)=λi(T′)\lambda_i(T)=\lambda_i(T') for i=1,2,3i=1,2,3, must TT and T′T' be congruent? Equivalently, is the map T↦(λ1(T),λ2(T),λ3(T))T\mapsto(\lambda_1(T),\lambda_2(T),\lambda_3(T)) from the three-dimensional space of triangles modulo congruence to R3\mathbb{R}^3 injective? The problem is to prove that this map is injective — the conjectured affirmative answer, consistent with the numerical evidence of Antunes–Freitas — or to exhibit a pair of non-congruent triangles whose first three Dirichlet eigenvalues coincide.

References

  1. Daniel Grieser, Svenja Maronna, Hearing the shape of a triangle, Notices of the American Mathematical Society 60(11):1440–1447, 2013. DOI: 10.1090/noti1063; arXiv:1208.3163. https://doi.org/10.1090/noti1063
  2. Catherine Durso, On the inverse spectral problem for polygonal domains, PhD thesis, Massachusetts Institute of Technology, 1988. https://dspace.mit.edu/handle/1721.1/84196
  3. Mark Kac, Can one hear the shape of a drum?, The American Mathematical Monthly 73(4, part II):1–23, 1966. DOI: 10.2307/2313748. https://doi.org/10.2307/2313748
  4. Carolyn Gordon, David L. Webb, Scott Wolpert, One cannot hear the shape of a drum, Bulletin of the American Mathematical Society 27(1):134–138, 1992. DOI: 10.1090/S0273-0979-1992-00289-6. https://doi.org/10.1090/S0273-0979-1992-00289-6
  5. Pei-Kun Chang, Dennis DeTurck, On hearing the shape of a triangle, Proceedings of the American Mathematical Society 105(4):1033–1038, 1989. DOI: 10.1090/S0002-9939-1989-0953738-7. https://doi.org/10.1090/S0002-9939-1989-0953738-7
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  7. Javier Gómez-Serrano, Gerard Orriols, Any three eigenvalues do not determine a triangle, Journal of Differential Equations 275:920–938, 2021. DOI: 10.1016/j.jde.2020.11.002; arXiv:1911.06758. https://doi.org/10.1016/j.jde.2020.11.002
  8. Zhiqin Lu, Julie Rowlett, The sound of symmetry, The American Mathematical Monthly 122(9):815–835, 2015. DOI: 10.4169/amer.math.monthly.122.9.815. https://doi.org/10.4169/amer.math.monthly.122.9.815