Hearing the shape of a triangle

TL;DR

This paper discusses the inverse spectral problem for triangles, providing a new proof that the shape of a triangle can be uniquely determined by its spectral data, specifically area, perimeter, and angle reciprocals.

Contribution

It offers a novel proof that the shape of a triangle is uniquely determined by spectral data, using convexity and partial fraction techniques.

Findings

A triangle's shape is uniquely determined by its area, perimeter, and angle reciprocals.

Convexity arguments and partial fraction expansion are effective in inverse spectral problems.

The new proof simplifies understanding of spectral determination of triangles.

Abstract

In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general inverse spectral problem we will give a new proof of this fact. The central point of the argument is to show that area, perimeter and the sum of the reciprocals of the angles determine a triangle uniquely. This is proved using convexity arguments and the partial fraction expansion of $\sin^{-2}x$.