J. Brossard and R. A. Carmona, who then suggested that one should replace the Hausdorff dimension with the upper box dimension. In the plane, this was proved if the boundary has dimension 1 (1993), but mostly disproved for higher dimensions (1996); both results are by Lapidus and Pomerance. See also Gassmann triple Isospectral Spectral geometry Vibrations of a circular membrane an extension to iterated function system fractals Notes References (In Russian). . (Revised and enlarged second edition to appear in 2005.) External links Simulation showing solutions of the wave equation in two isospectral drums Isospectral Drums by Toby Driscoll at the University of Delaware Some planar isospectral domains by Peter Buser, John Horton Conway, Peter Doyle, and Klaus-Dieter Semmler 3D rendering of the Buser-Conway-Doyle-Semmler homophonic drums Drums That Sound Alike by Ivars Peterson at the Mathematical Association of America web site Partial differential equations Spectral theory Drumming Mathematics papers