Hearing the Shape: Geometric Information in Spectra

• Geometric information is extracted from Laplace-type operators’ spectra, allowing retrieval of invariants like area, perimeter, and curvature via heat and wave trace methods.
• Computational techniques, including neural inversion and isospectralization, effectively reconstruct shape features from spectral data despite inherent non-uniqueness.
• The research highlights that while global traits such as volume and boundary length are audible, finer geometric details may remain inaudible, challenging unique reconstruction.

Geometric information that can be extracted from the spectrum of Laplace-type operators is one of the central questions in inverse spectral geometry, encapsulated since Kac's question: “Can one hear the shape of a drum?” For a broad class of geometric and combinatorial objects—planar domains, polygons, higher-dimensional manifolds, and discrete graphs—the spectrum encodes certain invariants, while others can remain inaudible. The interplay between audible and inaudible geometric data underlies a vast body of theoretical, computational, and applied research across mathematics, physics, and engineering.

1. The Inverse Isospectral Problem and Classical Spectral Invariants

Let Ω ⊂ ℝ² be a bounded domain with piecewise smooth boundary. The Dirichlet Laplacian Δ defines the eigenvalue problem

$Δu + λu = 0,\quad u|_{∂Ω} = 0,$

which yields a discrete spectrum $0 < λ_1 < λ_2 \leq \cdots \to \infty$. The "inverse isospectral problem" asks: if two domains Ω and Ω′ have the same spectrum $\{λ_k\}$, must they be congruent (i.e., isometric under a rigid motion)?

Spectral invariants arise via trace expansions:

• Heat trace expansion: $$H(t) = \operatorname{Tr}(e^{-tΔ}) = \sum_k e^{-λ_k t}.$$ For planar polygonal domains (e.g., convex $n$-gons), as $t \to 0^+$: $$H(t) \sim \frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + a_0 + O(t),$$ where

$$a_0 = \sum_{i=1}^n \frac{\pi^2 - \alpha_i^2}{24\pi\alpha_i},$$

with $\alpha_i$ the interior angles.

• Wave trace: $W(t) = \sum_{k=1}^\infty e^{i \sqrt{λ_k} t},$ whose singular support set is the length spectrum of closed billiard trajectories; that is, the "sound" hears periodic orbits (Lu et al., 2020).

From the heat trace, the coefficients of the expansions explicitly encode:

• $t^{-1}$: the area $|\Omega|$.
• $t^{-1/2}$: the perimeter $|\partial\Omega|$.
• $t^0$ (constant term): a function of the angle multiset.

These invariants are universal for polygonal domains; for $C^\infty$ domains, the constant term depends on boundary curvature rather than angles (Lu et al., 2020).

2. Rigidity and Uniqueness for Special Domain Classes

While the general inverse spectral problem has negative answers (there exist non-congruent isospectral domains), for several distinguished classes, the geometric information is uniquely determined:

• Triangles: The Dirichlet spectrum encodes area $A$, perimeter $P$, and $S = \sum_i 1/\alpha_i$; this triplet determines the triangle up to congruence, by a convexity argument related to the partial fraction expansion of $\csc^2 x$ (Grieser et al., 2012).
• Parallelograms: The heat trace determines area, perimeter, and a monotonic function of angle; once an angle is fixed, geometry is rigid (Lu et al., 2020).
• Acute trapezoids: Combined spectral and wave trace data recover height, area, perimeter, and a system of equations in the angles that has at most one solution, forcing congruence (Lu et al., 2020).
• Regular $n$-gons: Among all convex $n$-gons, the regular $n$-gon uniquely maximizes $f(\Omega)=|\Omega|/|\partial\Omega|^2$. There exists $N=N(n)$ such that matching the first $N$ eigenvalues with those of the regular $n$-gon determines it uniquely up to congruence. Hence, “the sound of symmetry can be heard” (Lu et al., 2020).

3. Limits of Geometric Determination: Isospectrality and Inaudible Features

Isospectral but non-isometric domains exist, constructed by combinatorial gluing and transplantation methods. For example, 17 infinite families of planar isospectral drums (Buser–Conway–Doyle–Semmler) and their analogues in 3D show the limitations: the Laplace spectrum encodes global invariants (volume, boundary area, length spectrum of periodic rays), but not the finer geometrical details of how tiles are arranged (Liu et al., 2017). In higher dimensions, Milnor identified isospectral but non-isometric 16-dimensional tori (Mårdby et al., 26 Jun 2024).

Table: Audible and Inaudible Geometric Data

Spectrally Audible	Not Always Audible
Dimension, area/volume	Exact boundary shape (general)
Perimeter/boundary length	Arrangement of isospectral tiles
Euler characteristic (2D)	Interior angles (except in special cases)
Heat-trace invariants	Finer domain topology in general
Corners vs. smooth boundary	Full metric for generic manifolds

(Lu et al., 2020, Mårdby et al., 26 Jun 2024, Liu et al., 2017)

The presence or absence of corners is itself spectrally determined for planar domains: the constant term in the heat trace is strictly larger for domains with corners than for any smooth-boundary domain of equal genus, so one can “hear corners” (Lu et al., 2020).

4. Extensions to Discrete, Quantum, and Noncommutative Settings

The notion of “hearing the shape” generalizes to discrete (graph-theoretic), quantum, and noncommutative frameworks.

• Discrete graphs: Ising model spectra provide graph invariants; the full spectrum of the quantum transverse-field Ising Hamiltonian is a strictly stronger invariant than the classical spectrum or the partition function, resolving graph pairs undistinguished by classical invariants (Vinci et al., 2013).
• Noncommutative geometry: In the NCG framework of Connes and collaborators, the spectrum of a generalized Dirac operator (on a spectral triple) determines not only the Riemannian metric but also internal/gauge structure; the spectral action encodes geometry and physics, including all Standard Model fields and gravitational terms (Chamseddine, 8 Nov 2025).
• Quantum boundary conditions: On quantum graphs or systems with boundary junctions, the entire family of self-adjoint realizations can be classified spectrally. For a ring with a junction, only parity-symmetric boundary conditions are uniquely determined by the spectrum; generic parity-broken cases are isospectral within a continuous family (Angelone et al., 2022).

5. Computational and Data-Driven Approaches

Recent advances leverage machine learning and numerical optimization to reconstruct geometry from spectral data:

• Neural inversion: Encoder–decoder networks trained on many random polygon spectra can reliably reconstruct the area, perimeter, and angle-sum functional of unseen convex polygons and produce high-fidelity boundary images on a grid (Zhao et al., 2022).
• Isospectralization: Differentiable optimization frameworks deform an initial mesh to match a target Laplacian spectrum, subject to geometric constraints. The result is practical geometric reconstruction (2D or 3D) and enhanced deformable correspondence—even though theoretical non-uniqueness exists, in practice the spectrum serves as a strong prior (Cosmo et al., 2018).
• Audio-based inference: CNNs trained on time-invariant scattering features encoded from drum waveforms can regress physically meaningful shape and damping parameters, generalizing the notion of “hearing the shape” into the regime of supervised learning and generative models (Han et al., 2020).

6. "Hearing the Shape" Beyond Classical Domains: Topology, Robotics, and Physical Inference

Spectral geometry extends into topological data analysis, robotics, and applied physical inference:

• Topological data analysis: Persistent Laplacians track the evolution of geometric and topological features through a filtration (e.g., in sequences of musical bells or instrument design), enabling spectral signatures and a precise mathematical foundation for “musical shape” (Wei, 2023).
• Robotic sensing: Autonomous robots/robotic swarms recover room or arena geometry from acoustic impulse responses (image–source method) or from distributed Laplacian-based diffusion protocols, achieving high-fidelity geometric reconstruction from auditory or local diffusion data (Nguyen et al., 2019, Cazenille et al., 25 Mar 2024, Deleforge et al., 10 Sep 2025).
• Curvature from quantum spectra: For magnetic Laplacians with geometric discontinuities (magnetic steps), semiclassical expansions show that low-lying eigenvalue asymptotics encode the maximum and second derivative of curvature at the edge—a direct link between spectral gaps and local geometric features (Assaad et al., 2021). For Dirac-type operators on surfaces with quantum Hall states, the spectral scaling exponents distinguish shape-blind and geometry-dependent branches; reciprocal surfaces possess dual quantum spectra, giving a spectral fingerprint of curvature (Dusa et al., 21 Mar 2025).

7. Open Problems, Conjectures, and Outlook

Despite extensive progress, key questions remain open. Notable conjectures include:

• Pólya–Szegő conjecture: The regular $n$-gon minimizes the first Dirichlet eigenvalue among all $n$-gons of fixed area (proven for some cases; open in general) (Lu et al., 2020).
• Bat conjecture: For each $n$, a finite set of closed billiard trajectory lengths suffices to determine a convex $n$-gon (Lu et al., 2020).
• Further generalizations: Whether the length spectrum is determined by the Laplacian spectrum for smooth manifolds; the completeness of spectral invariants from the quantum transverse-field Ising model for graph isomorphism; and the possibility of reconstructing arbitrary convex domains from a finite set of eigenvalues (Mårdby et al., 26 Jun 2024, Vinci et al., 2013).

A significant outcome is the recognition that while some geometric and topological data are spectrally rigid—volume, perimeter, Euler characteristic, corner structure—other features can remain silent to the spectrum, and reconstruction of arbitrary geometry from spectral data alone is impossible in general. Nonetheless, both classical trace expansions and modern computational, quantum, and data-driven approaches continue to push the boundary of what can be "heard" from spectral data.

References:

(Lu et al., 2020, Lu et al., 2020, Grieser et al., 2012, Vinci et al., 2013, Liu et al., 2017, Cosmo et al., 2018, Nguyen et al., 2019, Han et al., 2020, Assaad et al., 2021, Zhao et al., 2022, Angelone et al., 2022, Wei, 2023, Cazenille et al., 25 Mar 2024, Mårdby et al., 26 Jun 2024, Dusa et al., 21 Mar 2025, Deleforge et al., 10 Sep 2025, Chamseddine, 8 Nov 2025)