The sound of symmetry

TL;DR

This paper explores inverse spectral problems, demonstrating that certain geometric shapes like parallelograms and regular polygons can be uniquely identified by their spectral data, illustrating the 'sound of symmetry.'

Contribution

It provides an accessible introduction to inverse isospectral problems and proves that specific polygons' shapes can be determined from their eigenvalues.

Findings

Shapes like parallelograms and regular polygons are uniquely determined by their spectra.

The regular n-gon can be distinguished among convex n-gons by a finite set of eigenvalues.

The paper offers an interactive approach to understanding inverse spectral theory.

Abstract

This note begins with an introduction to the inverse isospectral problem popularized by M. Kac's 1966 article in the American Mathematical Monthly, "Can one hear the shape of a drum?" Although the answer has been known for some twenty years now, many open problems remain. Intended for general audiences, readers are challenged to complete exercises throughout this interactive introduction to inverse spectral theory. Following the introduction, the main techniques used in inverse isospectral problems are collected and discussed. These are then used to prove that one can hear the shape of: parallelograms, acute trapezoids, and the regular n-gon. Finally, we show that one can realistically hear the shape of the regular n-gon amongst all convex n-gons because it is uniquely determined by a finite number of eigenvalues; the sound of symmetry can really be heard!