STROMATOLITES are laminated, accretionary structures, which are commonly regarded to have formed by the sediment-binding or precipitating activities of ancient microbial mats or biofilms (composed mainly of cyanobacteria), possibly supplemented by abiotic surface precipitation1-4. Stromatolites are thus considered to be a proxy for early life on Earth, as the record of these structures extends back to 3.5 Gyr ago5. But as stromatolites only rarely contain fossil microbes, their biogenicity is tacitly assumed on the basis of morphological comparisons with modern, demonstrably biological, structures6. Little is known about the physical, chemical and biological processes that controlled the growth of ancient stromatolites4 and, with pioneering exceptions7-9, the analysis of the inherent geometric characteristics of the structures has not been pursued. Here we present a morphological characterization of ancient stromatolites that have growth surfaces with self-afime fractal geometry. We deduce, from both the microscopic textures and the fractal dimension, a purely abiotic dynamical model of stromatolite surface growth that combines chemical precipitation on the growing interface, fallout and diffusive rearrangement of suspended sediment, and uncorrelated random noise. This result calls into question the assumption that organisms-even if present-necessarily played an essential role in determining stromatolite morphology during times when precipitation at the sea floor was common, such as the earlier Precambrian.