Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press ( ISBN 978-0-521-85757-4). [1] [2] [3] [4] A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company ( ISBN 978-4-7649-0377-7). [5]
Although aimed at computer science and mathematics students, [3] [4] much of the book is accessible to a broader audience of mathematically-sophisticated readers with some background in high-school level geometry. [2] [4] Mathematical origami expert Tom Hull has called it "a must-read for anyone interested in the field of computational origami". [6] It is a monograph rather than a textbook, and in particular does not include sets of exercises. [4]
The Basic Library List Committee of the Mathematical Association of America has recommended this book for inclusion in undergraduate mathematics libraries. [1]
The book is organized into three sections, on linkages, origami, and polyhedra. [1] [2]
Topics in the section on linkages include the Peaucellier–Lipkin linkage for converting rotary motion into linear motion, [4] Kempe's universality theorem that any algebraic curve can be traced out by a linkage, [1] [4] the existence of linkages for angle trisection, [1] and the carpenter's rule problem on straightening two-dimensional polygonal chains. [4] This part of the book also includes applications to motion planning for robotic arms, and to protein folding. [1] [2]
The second section of the book concerns the mathematics of paper folding, and mathematical origami. It includes the NP-completeness of testing flat foldability, [2] the problem of map folding (determining whether a pattern of mountain and valley folds forming a square grid can be folded flat), [2] [4] the work of Robert J. Lang using tree structures and circle packing to automate the design of origami folding patterns, [2] [4] the fold-and-cut theorem according to which any polygon can be constructed by folding a piece of paper and then making a single straight cut, [2] [4] origami-based angle trisection, [4] rigid origami, [2] and the work of David A. Huffman on curved folds. [4]
In the third section, on polyhedra, the topics include polyhedral nets and Dürer's conjecture on their existence for convex polyhedra, the sets of polyhedra that have a given polygon as their net, Steinitz's theorem characterizing the graphs of polyhedra, Cauchy's theorem that every polyhedron, considered as a linkage of flat polygons, is rigid, and Alexandrov's uniqueness theorem stating that the three-dimensional shape of a convex polyhedron is uniquely determined by the metric space of geodesics on its surface. [4]
The book concludes with a more speculative chapter on higher-dimensional generalizations of the problems it discusses. [4]