Isospectral Domains in Euclidean 3-Space

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 , where is subject to Dirichlet boundary conditions. This paper presents methods to expand the already existing two dimensional transplantation proof into Euclidean 3space and, through these means, provides a number of three dimensional isospectral domains.