One cannot hear the shape of a drum

TL;DR

This paper constructs a pair of different-shaped but isospectral planar domains, demonstrating that one cannot determine a drum's shape solely from its sound, by extending Sunada's theorem.

Contribution

It introduces a novel method to create simply connected isospectral domains in the plane, answering Kac's question negatively.

Findings

Constructed nonisometric isospectral simply connected domains

Extended Sunada's theorem for planar domains

Provided counterexamples to hearing the shape of a drum

Abstract

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold.