Preface: Special issue on advances in the mathematical study of pattern formation

Natural physical mechanisms are capable of forming an incredible array of self-organized and regular spatio-temporal structure. Often referred to as patterns, such structures have elicited interest by scientists, engineers, and mathematicians alike. Inspired by pioneers like Liesegang, Haeckel, Wentworth Thompson, and Turing, researchers seek to understand how the combination of simple mechanisms, such as diffusion, reaction, convection, or transport, combine to create and mediate regular periodic patterns. Classic examples (for which many interesting questions are still unanswered) arise in the formation of roll states in convective fluids, stripes and spots in animal coats, ordered cell differentiation in biological morphogenesis, and spiral arrangements of primordia in flowers. As such diverse systems often produce qualitatively similar forms, it is of interest to distill any universal properties of patterns across these systems.