We have developed a new method for finding and representing dividing surfaces which can, for example, be used to identify "atoms" in molecules or condensed phases based on Bader's definition. Given the total electron density of the system, the dividing surface is taken to be the zero-flux surface, i.e., the surface on which the normal component of the gradient vanishes. Our method for finding this surface involves creating an "elastic sheet" represented by a swarm of fictitious particles which interact with each other so as to give a nearly uniform distribution of points on the sheet. Two kinds of forces act on the particles: (1) the component of the gradient of the density normal to the elastic sheet, and (2) an interparticle force which only acts in the local tangent plane of the sheet. Starting with a spherical surface and applying an optimization algorithm that minimizes the forces leads to convergence of the particles to the zero-flux surface. The elastic sheet tends to round off regions where the zero-flux surface has sharp cusps or points, but this appears not to be a serious problem in cases we have studied. The elastic sheet method is robust and can converge in situations where currently used methods fail. We demonstrate the method with a study of water clusters and a Si interstitial in a Si crystal.