ⓘ Geometric Folding Algorithms
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 ORourke. It was published in 2007 by Cambridge University Press. A Japaneselanguage translation by Ryuhei Uehara was published in 2009 by the Modern Science Company.
1. Audience
Although aimed at computer science and mathematics students, much of the book is accessible to a broader audience of mathematicallysophisticated readers with some background in highschool level geometry. Mathematical origami expert Tom Hull has called it "a mustread for anyone interested in the field of computational origami". It is a monograph rather than a textbook, and in particular does not include sets of exercises.
The Basic Library List Committee of the Mathematical Association of America has recommended this book for inclusion in undergraduate mathematics libraries.
2. Topics and organization
The book is organized into three sections, on linkages, origami, and polyhedra.
Topics in the section on linkages include the Peaucellier–Lipkin linkage for converting rotary motion into linear motion,Kempes universality theorem that any algebraic curve can be traced out by a linkage, the existence of linkages for angle trisection, and the carpenters rule problem on straightening twodimensional polygonal chains. This part of the book also includes applications to motion planning for robotic arms, and to protein folding.
The second section of the book concerns the mathematics of paper folding, and mathematical origami. It includes the NPcompleteness of testing flat foldability, the problem of map folding determining whether a pattern of mountain and valley folds forming a square grid can be folded flat, the work of Robert J. Lang using tree structures and circle packing to automate the design of origami folding patterns, the foldandcut theorem according to which any polygon can be constructed by folding a piece of paper and then making a single straight cut, origamibased angle trisection,rigid origami, and the work of David A. Huffman on curved folds.
In the third section, on polyhedra, the topics include polyhedral nets and Durers conjecture on their existence for convex polyhedra, the sets of polyhedra that have a given polygon as their net, Steinitzs theorem characterizing the graphs of polyhedra, Cauchys theorem that every polyhedron, considered as a linkage of flat polygons, is rigid, and Alexandrovs uniqueness theorem stating that the threedimensional shape of a convex polyhedron is uniquely determined by the metric space of geodesics on its surface.
The book concludes with a more speculative chapter on higherdimensional generalizations of the problems it discusses.
A History of Folding in Mathematics 

Geometric Exercises in Paper Folding 

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