Finite geometries and diffractive orbits in isospectral billiards

Several examples of pairs of isospectral planar domains have been produced in the two-dimensional Euclidean space by various methods. We show that all these examples rely on-the symmetry between points and blocks in finite projective spaces; from the properties of these spaces, one can derive a relation between Green's functions as well as a relation between diffractive orbits in isospectral billiards.