Geometric Folding Algorithms

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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

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 2 3 4 5 6 Carbno, Collin (May 2009), "Review of Geometric Folding Algorithms", MAA Reviews, Mathematical Association of America, archived from the original on 2020-02-03, retrieved 2020-02-03
  2. 1 2 3 4 5 6 7 8 9 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. 1 2 mbec (2011), "Review of Geometric Folding Algorithms", EMS Reviews, European Mathematical Society, archived from the original on 2020-02-03, retrieved 2020-02-03
  4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fasy, Brittany Terese; Millman, David L. (March 2011), "Review of Geometric Folding Algorithms", SIGACT News, 42 (1), Association for Computing Machinery: 43–46, doi:10.1145/1959045.1959056, S2CID   6514501
  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 , retrieved 2020-02-02
  6. Hull, Tom (2012), "Other sources", Project Origami: Activities for Exploring Mathematics (2nd ed.), CRC Press, p. xviii