Isospectral Domains in Euclidean 3-Space

Abstract

The question as to whether the shape of a drum can be heard has existed for around fifty years. The simple answer is ‘no’ as shown through the construction of isospectral domains. Isospectral domains are non-isometric domains that display the same spectra of frequencies of sound. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a domain omega , where alpha-omega is subject to Dirichlet boundary conditions. This paper presents methods to expand the already existing two dimensional transplantation proof into Euclidean 3-space and, through these means, provides a number of three dimensional isospectral domains.

We can't hear the shape of drum: revisited in 3D case

“…Actually, they are a simplified version of the pair 7 3 . Previously based on the reflection rule of 7 3 , Cox [40] constructed three dimensional iso-spectral domains. In addition, 45 shapes assembled by unit cube, square-based prism and right-angled wedge were depicted to show non-isospectrality in Moorhead's Ph.D. thesis [39].…”