Search-based texture synthesis algorithms are sensitive to the order in which texture samples are generated; different synthesis orders yield different textures. Unfortunately, most polygon rasterizers and ray tracers do not guarantee the order with which surfaces are sampled. To circumvent this problem, textures are synthesized beforehand at some maximum resolution and rendered using texture mapping. We describe a search-based texture synthesis algorithm in which samples can be generated in arbitrary order, yet the resulting texture remains identical. The key to our algorithm is a pyramidal representation in which each texture sample depends only on a fixed number of neighboring samples at each level of the pyramid. The bottom (coarsest) level of the pyramid consists of a noise image, which is small and predetermined. When a sample is requested by the renderer, all samples on which it depends are generated at once. Using this approach, samples can be generated in any order. To make the algorithm efficient, we propose storing texture samples and their dependents in a pyramidal cache. Although the first few samples are expensive to generate, there is substantial reuse, so subsequent samples cost less. Fortunately, most rendering algorithms exhibit good coherence, so cache reuse is high.