  Concerning the Laplace operator with homogeneous Dirichlet boundary
conditions, the classical notion of isospectrality assumes that two domains are
related when they give rise to the same spectrum. In two dimensions, non
isometric, isospectral domains exist. It is not known however if all the
eigenvalues relative to a specific domain can be preserved under suitable
continuous deformation of its geometry. We show that this is possible when the
2D Laplacian is replaced by a finite dimensional version and the geometry is
modified by respecting certain constraints. The analysis is carried out in a
very small finite dimensional space, but it can be extended to more accurate
finite-dimensional representations of the 2D Laplacian, with an increase of
computational complexity. The aim of this paper is to introduce the preliminary
steps in view of more serious generalizations.
