You can now mathematically design a crazy shape that 'rings' with any specific musical notes you want.
March 24, 2026
Original Paper
An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature
arXiv · 2603.21240
The Takeaway
A famous math puzzle asks if the shape of an object determines the 'notes' it plays. This paper proves the inverse: you can mathematically construct a higher-dimensional shape to match any specific list of frequencies, effectively allowing you to 'prescribe' the sound of a geometric object.
From the abstract
A classical theorem of Colin de Verdière shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including multiplicities, subject to the usual topological constraints) by choosing a sufficiently heterogeneous smooth metric. In this paper, we study the same inverse problem under the rigid geometric constraint of \emph{constant negative sectional curvature}. Allowing the topological complexity to vary, we prove that any fin