Turing patterns (TPs) underlie many fundamental developmental processes, but they operate over narrow parameter ranges, raising the conundrum of how evolution can ever discover them. Here we explore TP design space to address this question and to distill design rules. We exhaustively analyze 2- and 3-node biological candidate Turing systems, amounting to 7,625 networks and more than 3 × 10^11 analyzed scenarios. We find that network structure alone neither implies nor guarantees emergent TPs. A large fraction (>61%) of network design space can produce TPs, but these are sensitive to even subtle changes in parameters, network structure, and regulatory mechanisms. This implies that TP networks are more common than previously thought, and evolution might regularly encounter prototypic solutions. We deduce compositional rules for TP systems that are almost necessary and sufficient (96% of TP networks contain them, and 92% of networks implementing them produce TPs). This comprehensive network atlas provides the blueprints for identifying natural TPs and for engineering synthetic systems.