Abstract: Lord Rayleigh asserted in 1877, and Faber and Krahn proved fifty years later, that “If the area of a membrane be given, there must evidently be some form of boundary for which the pitch (of the principal tone) is the gravest possible, and this form can be no other than the circle.” In modern terminology, Rayleigh was claiming that among all planar domains of given area, the one that minimizes the first eigenvalue of the Laplacian (under Dirichlet boundary conditions) is the disk. In terms of heat flow in 3 dimensions, that means a room of given volume whose boundary is maintained at temperature zero will cool off slowest when it is a ball. What about a room whose boundary is perfectly insulated (Neumann boundary conditions)? Szego and Weinberger discovered in the 1950s that the room’s temperature will equilibrate fastest when it is spherical - which is admittedly an unlikely shape for a room, except for fans of Futuro Flying Saucer homes. Insulation is never perfect, as every homeowner knows, and so we are led to the Robin boundary condition for partial insulation, and to recent progress and open problems on “isoperimetric type” eigenvalue inequalities that extend the work of Rayleigh-Faber-Krahn and Szego-Weinberger.