Cardiac fibrillation (spontaneous, asynchronous contractions of cardiac muscle fibres) is the leading cause of death in the industrialized world *RF 1*, yet it is not clear how it occurs.It has been debated whether or not fibrillation is a random phenomenon. There is some determinism during fibrillation [2,3], perhaps resulting from rotating waves of electrical activity [4-6]. Here we present a new algorithm that markedly reduces the amount of data required to depict the complex spatiotemporal patterns of fibrillation. We use a potentiometric dye  and video imaging [8,9] to record the dynamics of transmembrane potentials at many sites during fibrillation. Transmembrane signals at each site exhibit a strong periodic component centred near 8 Hz. This periodicity is seem as an attractor in two-dimensional-phase space and each site can be represented by its phase around the attractor. Spatial phase maps at each instant reveal the 'sources' of fibrillation in the form of topological defects, or phase singularities , at a few sites. Using our method of identifying phase singularities, we can elucidate the mechanisms for the formation and termination of these singularities, and represent an episode of fibrillation by locating singularities. Our results indicate an unprecedented amount of temporal and spatial organization during cardiac fibrillation.