Select contributions from the Fourth International Conference on Origami in Science, Mathematics, and Education (4OSME), held September 8-10, 2006, in Pasadena, CA (sponsored by OrigamiUSA, in collaboration with the California Institute of Technology). The conference has been held approximately once every five years (since the first one in 1989), and it focuses on the mathematics of origami and applications of origami in the sciences.