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

monograph A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...

on the mathematics and computational geometry of mechanical linkages,

paper folding ) is the Japanese paper art, art of Paper folding (disambiguation), paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to trans ...

, and polyhedral nets, by

Erik Demaine Erik D. Demaine (born February 28, 1981) is a Canadian-American professor of computer science at the Massachusetts Institute of Technology and a former child prodigy. Early life and education Demaine was born in Halifax, Nova Scotia, to mathe ...

and Joseph O'Rourke. It was published in 2007 by

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

(). 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 A university () is an educational institution, institution of tertiary edu ...

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 The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion ...

for converting rotary motion into linear motion,

Kempe's universality theorem In algebraic geometry, Kempe's universality theorem states that any bounded subset of an algebraic curve may be traced out by the motion of one of the joints in a suitably chosen linkage. It is named for British mathematician Alfred B. Kempe, w ...

that any

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

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 and ...

, and the carpenter's rule problem on straightening two-dimensional

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

s. This part of the book also includes applications to

motion planning Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used ...

for

robotic arm A robotic arm is a type of mechanical arm, usually programmable, with similar functions to a human arm; the arm may be the sum total of the mechanism or may be part of a more complex robot. The links of such a manipulator are connected by join ...

s, and to

protein folding Protein folding is the physical process by which a protein, after Protein biosynthesis, synthesis by a ribosome as a linear chain of Amino acid, amino acids, changes from an unstable random coil into a more ordered protein tertiary structure, t ...

. The second section of the book concerns the mathematics of paper folding, and mathematical

origami ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a ...

. It includes the

NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...

ness 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 In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...

to automate the design of origami folding patterns, 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, origami-based angle trisection, rigid origami, and the work of David A. Huffman on curved folds. In the third section, on

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

, 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 polyhedron, convex polyhedra: they are exactly the vertex connect ...

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 Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each othe ...

stating that the three-dimensional shape of a convex polyhedron is uniquely determined by the

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...

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