One of the most impressive manifestations of spontaneous pattern generation in natural materials is iris agate, which contains submicrometer concentric striations that may cycle several thousand times within an individual specimen. Analysis by secondary ion mass spectroscopy and transmission electron microscopy identified the iris texture as alternating layers of fine-grained, highly defective chalcedony and coarse-grained low-defect quartz. This oscillatory zonation in defect concentration may be ascribed to Ostwald-Liesegang crystallization cycles from silica-rich fluids that are variably polymeric and monomeric. Periodic changes in defect concentration and grain size also are observed with wavelengths of hundreds of micrometers and of centimeters, so that agates reveal textural self-similarity over three length scales.