---
tag: X3VH8H1XV3
kind: problem
name: Hearing the shape of a drum
msc: [58J53, 35P05, 58J50]
uses: [0E0B6RA8K0, EETVA6QYX4, 0BQAA2BK3Q]
about: [0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum
same_as:
  wikidata: Q5691670
  wikipedia: Hearing the shape of a drum
history:
  posed_by: "Mark Kac"
  first_stated: "1966"
---

## Problem

Let $\Omega_1$ and $\Omega_2$ be bounded domains in $\R^2$.
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]].
Suppose that $-\Delta$ with the [[Dirichlet boundary condition|EETVA6QYX4]] has the same [[eigenvalues|0BQAA2BK3Q]] on $\Omega_1$ and on $\Omega_2$, with the same multiplicities.
Must $\Omega_1$ and $\Omega_2$ be congruent?

## Remarks

Mark Kac popularised the question in his 1966 article "Can one hear the shape of a drum?".
The answer is no.
In 1964 John Milnor had already found two flat tori of dimension $16$ that have the same spectrum but are not isometric.
In 1992 Carolyn Gordon, David Webb and Scott Wolpert constructed two plane domains, both polygons, that have the same Dirichlet spectrum but are not congruent.
Some features can be heard: by [[Weyl's law|2JK9YMZV2B]] the spectrum determines the area of the domain, and for domains with smooth boundary it also determines the perimeter.
Among plane domains with smooth boundary, the disc is therefore determined by its spectrum, by the equality case of the isoperimetric inequality.

## Explanations

### L1 human

A drumhead of a given shape can play certain pitches, which are the eigenvalues of the Laplacian.
Kac asked whether a listener who knows all these pitches can tell the shape of the drum.
The answer is no: there are two differently shaped drums that sound exactly alike, though some features, such as the area, can be heard.