Geometric Folding Algorithms

Short description: 2007 mathematics book by Demaine and O'Rourke

Geometric Folding Algorithms  *+[[Biography:Erik+Demaine|Erik+Demaine]] *+Joseph+O'Rourke

&rft.date=2007&rft.pub=Cambridge+University+Press">

Author|

* Erik Demaine
* Joseph O'Rourke

Language| English
Genre| Mathematics
Publisher| Cambridge University Press

Publication date

| 2007
ISBN| 978-0-521-85757-4

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]

## Contents

* 1 Audience
* 2 Topics and organization
* 3 References
* 4 External links

## Audience

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]

## Topics and organization

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]

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Carbno, Collin (May 2009), "Review of Geometric Folding Algorithms", MAA Reviews (Mathematical Association of America), https://www.maa.org/press/maa-reviews/geometric-folding-algorithms-linkages-origami-polyhedra, retrieved 2020-02-03
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Paquete, Luís (November 2009), "Review of Geometric Folding Algorithms", European Journal of Operational Research 199 (1): 311–313, doi:10.1016/j.ejor.2008.06.009
3. ↑ 3.0 3.1 mbec (2011), "Review of Geometric Folding Algorithms", EMS Reviews (European Mathematical Society), https://euro-math-soc.eu/review/geometric-folding-algorithms-linkages-origami-polyhedra, retrieved 2020-02-03
4. ↑ 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13 Fasy, Brittany Terese; Millman, David L. (March 2011), "Review of Geometric Folding Algorithms", SIGACT News (Association for Computing Machinery) 42 (1): 43–46, doi:10.1145/1959045.1959056
5. ↑ Uehara, Ryuhei, 幾何的な折りアルゴリズム リンケージ・折り紙・多面体, http://www.jaist.ac.jp/~uehara/gfalop/, retrieved 2020-02-02
6. ↑ Hull, Tom (2012), "Other sources", Project Origami: Activities for Exploring Mathematics (2nd ed.), CRC Press, p. xviii, https://books.google.com/books?id=blvSBQAAQBAJ&pg=PR18

## External links

* Authors' web site for Geometric Folding Algorithms including contents, errata, and advances on open problems

* v
* t
* e

Mathematics of paper folding

Flat folding|

* Big-little-big lemma
* Crease pattern
* Huzita–Hatori axioms
* Kawasaki's theorem
* Maekawa's theorem
* Map folding
* Napkin folding problem
* Pureland origami
* Yoshizawa–Randlett system

Strip folding|

* Dragon curve
* Flexagon
* Möbius strip
* Regular paperfolding sequence

3d structures|

* Miura fold
* Modular origami
* Paper bag problem
* Rigid origami
* Sonobe

Polyhedra|

* Alexandrov's uniqueness theorem
* Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
* Net
* Star unfolding

Miscellaneous|

* Fold-and-cut theorem
* Lill's method

Publications|

* Geometric Exercises in Paper Folding
* Geometric Folding Algorithms
* A History of Folding in Mathematics
* Origami Polyhedra Design

People|

* Margherita Piazzola Beloch
* Erik Demaine
* Martin Demaine
* Britney Gallivan
* Rona Gurkewitz
* David A. Huffman
* Tom Hull
* Kôdi Husimi
* Humiaki Huzita
* Toshikazu Kawasaki
* Robert J. Lang
* Anna Lubiw
* Jun Maekawa
* Joseph O'Rourke

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Geometric Folding Algorithms. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Geometric_Folding_Algorithms&oldid=4347601"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template