''Geometric Folding Algorithms: Linkages, Origami, Polyhedra'' is a

monograph A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject. In library cataloging, ''monograph ...

on the mathematics and

computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (). A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company ().

Audience

Although aimed at computer science and mathematics students, much of the book is accessible to a broader audience of mathematically-sophisticated readers with some background in high-school level geometry. Mathematical origami expert Tom Hull has called it "a must-read 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 The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has recommended this book for inclusion in undergraduate mathematics libraries.

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,

Kempe's universality theorem In 1876 Alfred B. Kempe published his article ''On a General Method of describing Plane Curves of the nth degree by Linkwork,'' which showed that for an arbitrary algebraic plane curve a linkage can be constructed that draws the curve. This direct ...

that any algebraic curve can be traced out by a linkage, the existence of linkages for

angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge an ...

, and the

carpenter's rule problem The carpenter's rule problem is a discrete geometry problem, which can be stated in the following manner: ''Can a simple planar polygon be moved continuously to a position where all its vertices are in convex position, so that the edge lengths and ...

on straightening two-dimensional 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 NP-completeness 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 Robert J. Lang (born May 4, 1961) is an American physicist who is also one of the foremost origami artists and theorists in the world. He is known for his complex and elegant designs, most notably of insects and animals. He has studied the mathe ...

using tree structures and circle packing to automate the design of origami folding patterns, the

fold-and-cut theorem The fold-and-cut theorem states that any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include polygons, which may be concave, shapes with hol ...

according to which any polygon can be constructed by folding a piece of paper and then making a single straight cut, origami-based angle trisection,

rigid origami Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain ...

, and the work of David A. Huffman on curved folds. 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 In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar grap ...

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

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s on its surface. The book concludes with a more speculative chapter on higher-dimensional generalizations of the problems it discusses.

References

External links

Authors' web site for ''Geometric Folding Algorithms''
including contents, errata, and advances on open problems Linkages (mechanical) Paper folding Polyhedra Computational geometry Mathematics books 2007 non-fiction books 2009 non-fiction books {{Mathematics of paper folding