One of the basic mechanisms responsible for the formation of three-dimensional organs during embryogenesis relies on the regulated folding of two-dimensional sheets of cells. This process is driven by the spatially nonuniform and dynamic distribution of multiple chemical components (products of gene expression). Some of the key questions in this class of problems are related to the number of involved genes and to the diversity and dynamics of their expression patterns. I will present the results of our experimental and computational work that explores these questions in fruit fly development. I will discuss the mathematical and computational issues associated with the combinatorial construction of complex patterns from a small number of building blocks and the dynamics of piecewise linear models of patterning. Finally, I will compare and contrast models of biological pattern formation with models of purely chemical systems.