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The precipitation of barium or strontium carbonates in alkaline silica-rich environments leads to crystalline aggregates that have been named silica/carbonate biomorphs because their morphology resembles that of primitive organisms. These aggregates are self-assembled materials of purely inorganic origin, with an amorphous phase of silica intimately intertwined with a carbonate nanocrystalline phase. We propose a mechanism that explains all the morphologies described for biomorphs. Chemically coupled coprecipitation of carbonate and silica leads to fibrillation of the growing front and to laminar structures that experience curling at their growing rim. These curls propagate in a surflike way along the rim of the laminae. We show that all observed morphologies with smoothly varying positive or negative Gaussian curvatures can be explained by the combined growth of counterpropagating curls and growing laminae.