Twists, Tilings, and Tessellations

Robert J. Lang has been an avid student of origami for over fifty years and is now recognized as one of the world’s leading masters of the art. He is noted for designs of great detail and realism, and his repertoire includes some of the most complex origami designs ever created. His work combines aspects of the Western school of mathematical origami design with the Eastern emphasis upon line and form to yield designs that are at once distinctive, elegant, and challenging to fold. They have been shown in exhibitions in New York (Museum of Modern Art), Paris (Carrousel du Louvre), Salem (Peabody Essex Museum), San Diego (Mingei Museum of World Folk Art), and Kaga, Japan (Nippon Museum of Origami), among others. He is one of the pioneers of computational origami techniques, and has published widely on the theory and mathematics of folding. Dr. Lang was born in Ohio and raised in Atlanta, Georgia. Along the way to his current career as a full-time origami artist and consultant, he worked as a physicist, engineer, and R&D manager, during which time he authored or co-authored over 80 technical publications and 50 patents on semiconductor lasers, optics, and integrated optoelectronics. He was elected a Fellow of the Optical Society of America and served as Editor-in-Chief of the IEEE Journal of Quantum Electronics from 2007–2010. After switching his primary focus to origami, he authored or co-authored numerous technical articles on the mathematics of folding and on design techniques for folding in technological applications. In 2009, he received Caltech’s Distinguished Alumni Award for his origami work, and in 2013 was elected a Fellow of the American Mathematical Society. Dr. Lang resides in Alamo, California.

Introduction. Vertices. Periodicity. Simple Twists. Twist Tiles. Tilings. Primal-Dual Tessellations. Rigid Foldability. Spherical Vertices. 3D Vectors. Rotational Solids. Afterword