# Geometric Folding Algorithms

Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.

Did you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even 'sign your name'? Or that a 'Latin cross' unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past decade, there has been a surge of interest in such problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this treatment gives hundreds of results and over 60 unsolved 'open problems' to inspire further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.

# How to Fold It

The book's website, http://www.howtofoldit.org, has dynamic animations of many of the foldings and downloadable templates for readers to fold or cut out.

What do proteins and pop-up cards have in common? How is opening a grocery bag different from opening a gift box? How can you cut out the letters for a whole word all at once with one straight scissors cut? How many ways are there to flatten a cube? With the help of 200 colour figures, author Joseph O'Rourke explains these fascinating folding problems starting from high school algebra and geometry and introducing more advanced concepts in tangible contexts as they arise. He shows how variations on these basic problems lead directly to the frontiers of current mathematical research and offers ten accessible unsolved problems for the enterprising reader. Before tackling these, you can test your skills on fifty exercises with complete solutions. The book's website, http://www.howtofoldit.org, has dynamic animations of many of the foldings and downloadable templates for readers to fold or cut out.

# Algorithms and Computation

This research was continued during the open-problem sessions organized around MIT class 6.849: Geometric Folding Algorithms in Fall 2010. We thank the other participants of these sessions—Scott Kominers, Jason Ku, Thomas Morgan, Jie Qi, ...

This book constitutes the refereed proceedings of the 22nd International Symposium on Algorithms and Computation, ISAAC 2011, held in Yokohama, Japan in December 2011. The 76 revised full papers presented together with two invited talks were carefully reviewed and selected from 187 submissions for inclusion in the book. This volume contains topics such as approximation algorithms; computational geometry; computational biology; computational complexity; data structures; distributed systems; graph algorithms; graph drawing and information visualization; optimization; online and streaming algorithms; parallel and external memory algorithms; parameterized algorithms; game theory and internet algorithms; randomized algorithms; and string algorithms.

# WALCOM Algorithms and Computation

The folding itself may be recovered by storing, for each level assignment a+ considered by the algorithm, one of the level assignments a− such that a− ∈ A ... O'Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

This book constitutes the thoroughly refereed conference proceedings of the 9th International Workshop on Algorithms and Computation, WALCOM 2015, held in Dhaka, Bangladesh, in February 2015. The 26 revised full papers presented together with 3 invited talks were carefully reviewed and selected from 85 submissions. The papers are organized in topical sections on approximation algorithms, data structures and algorithms, computational geometry, combinatorial algorithms, distributed and online algorithms, graph drawing and algorithms, combinatorial problems and complexity, and graph enumeration and algorithms.

# Computational Geometry Graphs and Applications

In: Proceedings of the 16th Canadian Conference on Computational Geometry, Montréal, Canada, pp. 64–67 (August 2004) 6. Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

This book constitutes the thoroughly refereed post-conference proceedings of the China-Japan Joint Conference on Computational Geometry, Graphs and Applications, CGGA 2010, held in Dalian, China, in November 2010. The 23 revised full papers presented were carefully selected during two rounds of reviewing and improvement from numerous submissions. All aspects of computational and discrete geometry, graph theory, graph algorithms, and their applications are covered.

# Algorithms and Computation

Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms. Cambridge University Press, Cambridge (2007) 6. Demaine, E.D., O'Rourke, J.: Open problems from CCCG 2008. In: Proc. 21st Canadian Conference on Computational Geometry, ...

Thepapersinthisvolumewerepresentedatthe20thAnnualInternationalS- posiumonAlgorithmsandComputation,heldDecember 16–18,2009,in Hawaii, USA. Inresponseto the Call-for-Papers,279papersweresubmitted. Eachpaper receivedatleastthreereviewsbyeitherProgramCommitteemembersorexperts selectedby ProgramCommittee members. In all, 120paperswereselectedbased on the review reports and are included in this volume. We wish to thank all who submitted papers for consideration and all Program Committee members and reviewers for their excellent and hard work. To our invited speakers, we wish to thank you for sharing your expertise. Finally, we wish to thank our colleagues who contributed to the success of the symposium and the sponsors for their assistance and support. December 2009 Yingfei Dong Ding-Zhu Du Oscar Ibarra Organization Program Committee Co-chairs Yingfei Dong University of Hawaii, USA Ding-Zhu Du University of Texas at Dallas, USA Oscar Ibarra University of California, Santa Barbara, USA Advisory Committee Chair Takeshi Tokuyama Tohoku University, Japan Advisory Committee Tetsuo Asano Japan Advanced Institute of Science and Technology, Japan Kyung-Yong Chwa Korea Advanced Institute of Science and Technology, Republic of Korea Francis Y. L.

# Combinatorial and Computational Geometry

( Demaine and Mitchell 2001 ) E. D. Demaine and J. S. B. Mitchell , “ Reaching folded states of a rectangular piece of ... ( Demaine and O'Rourke > 2005 ) E. D. Demaine and J. O'Rourke , Geometric folding algorithms : linkages , origami ...

This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry.

# Surveys on Discrete and Computational Geometry

is a (1x1) algorithm, instead conflicts must be resolved by careful ordering of the recursive calls, and other techniques. ... 167–211. , Geometric folding algorithms: Linkages, origami, polyhedra, Cambridge University Press, ...

This volume contains nineteen survey papers describing the state of current research in discrete and computational geometry as well as a set of open problems presented at the 2006 AMS-IMS-SIAM Summer Research Conference Discrete and Computational Geometry--Twenty Years Later, held in Snowbird, Utah, in June 2006. Topics surveyed include metric graph theory, lattice polytopes, the combinatorial complexity of unions of geometric objects, line and pseudoline arrangements, algorithmic semialgebraic geometry, persistent homology, unfolding polyhedra, pseudo-triangulations, nonlinear computational geometry, \$k\$-sets, and the computational complexity of convex bodies.

# Thirty Essays on Geometric Graph Theory

S. Basu, R. Pollack, M.-F. Roy, Algorithms in real algebraic geometry, in Algorithms and Computation in Mathematics, ... E.D. Demaine, J. O'Rourke, Geometric Folding Algorithms (Cambridge University Press, Cambridge, 2007); Linkages, ...

In many applications of graph theory, graphs are regarded as geometric objects drawn in the plane or in some other surface. The traditional methods of "abstract" graph theory are often incapable of providing satisfactory answers to questions arising in such applications. In the past couple of decades, many powerful new combinatorial and topological techniques have been developed to tackle these problems. Today geometric graph theory is a burgeoning field with many striking results and appealing open questions. This contributed volume contains thirty original survey and research papers on important recent developments in geometric graph theory. The contributions were thoroughly reviewed and written by excellent researchers in this field.

# Algorithms and Data Structures

This work was initiated during a series of open-problem sessions for an MIT class on Geometric Folding Algorithms (6.885 in Fall 2007), and continued at the 23rd Bellairs Winter Workshop on Computational Geometry organized by Godfried ...

This book constitutes the refereed proceedings of the 11th Algorithms and Data Structures Symposium, WADS 2009, held in Banff, Canada, in August 2009. The Algorithms and Data Structures Symposium - WADS (formerly "Workshop on Algorithms and Data Structures") is intended as a forum for researchers in the area of design and analysis of algorithms and data structures. The 49 revised full papers presented in this volume were carefully reviewed and selected from 126 submissions. The papers present original research on algorithms and data structures in all areas, including bioinformatics, combinatorics, computational geometry, databases, graphics, and parallel and distributed computing.

# Handbook of Discrete and Computational Geometry

The protein folding problem . Physics Today , 46 : 24–32 ... Linking , and Folding of Geometric Objects in 3 - space , pages 287–311 , AMS , Providence , 2002 . ... 6.849 : Geometric folding algorithms : Linkages , origami , polyhedra .

The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

# Fun with Algorithms

Discrete & Computational Geometry 47(1), 150–186 (2012) Abellanas, M.: Conectando puntos: poligonizaciones y otros problemas ... March 24-28 (2010) Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

This book constitutes the refereed proceedings of the 7th International Conference, FUN 2014, held in July 2014 in Lipari Island, Sicily, Italy. The 29 revised full papers were carefully reviewed and selected from 49 submissions. They feature a large variety of topics in the field of the use, design and analysis of algorithms and data structures, focusing on results that provide amusing, witty but nonetheless original and scientifically profound contributions to the area. In particular, algorithmic questions rooted in biology, cryptography, game theory, graphs, the internet, robotics and mobility, combinatorics, geometry, stringology, as well as space-conscious, randomized, parallel, distributed algorithms and their visualization are addressed.

# Discrete and Computational Geometry and Graphs

Apply a valley fold to the line segment v3h to satisfy dist(vt8, vt9 )=2rt sin(π/5) defined in Step 3 for each 0 ≤ t ≤ 1. ... Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms, Linkages; Origami, Polyhedra.

This book constitutes the thoroughly refereed post-conference proceedings of the 18th Japanese Conference on Discrete and Computational Geometry and Graphs, JDCDGG 2015, held in Kyoto, Japan, in September 2015. The total of 25 papers included in this volume was carefully reviewed and selected from 64 submissions. The papers feature advances made in the field of computational geometry and focus on emerging technologies, new methodology and applications, graph theory and dynamics. This proceedings are dedicated to Naoki Katoh on the occasion of his retirement from Kyoto University.

# Project Origami

Instructors interested in using the Folding a Parabola, Can Ori— gami Trisect an Angle?, and Solving Cubic Equations activities in an advanced algebra class should consult this book. Geometric Folding Algorithms: Linkages, Origami, ...

Project Origami: Activities for Exploring Mathematics, Second Edition presents a flexible, discovery-based approach to learning origami-math topics. It helps readers see how origami intersects a variety of mathematical topics, from the more obvious realm of geometry to the fields of algebra, number theory, and combinatorics. With over 100 new pages, this updated and expanded edition now includes 30 activities and offers better solutions and teaching tips for all activities. The book contains detailed plans for 30 hands-on, scalable origami activities. Each activity lists courses in which the activity might fit, includes handouts for classroom use, and provides notes for instructors on solutions, how the handouts can be used, and other pedagogical suggestions. The handouts are also available on the book’s CRC Press web page. Reflecting feedback from teachers and students who have used the book, this classroom-tested text provides an easy and entertaining way for teachers to incorporate origami into a range of college and advanced high school math courses. Visit the author’s website for more information.

# 3D Origami Art

Mathematical knowledge, such as conditions for mountains and valleys to be folded flatwise (Maekawa's theorem and ... The book by Demaine and O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra* covers these topics.

Easily Create Origami with Curved Folds and Surfaces Origami—making shapes only through folding—reveals a fascinating area of geometry woven with a variety of representations. The world of origami has progressed dramatically since the advent of computer programs to perform the necessary computations for origami design. 3D Origami Art presents the design methods underlying 3D creations derived from computation. It includes numerous photos and design drawings called crease patterns, which are available for download on the author’s website. Through the book’s clear figures and descriptions, readers can easily create geometric 3D structures out of a set of lines and curves drawn on a 2D plane. The author uses various shapes of sheets such as rectangles and regular polygons, instead of square paper, to create the origami. Many of the origami creations have a 3D structure composed of curved surfaces, and some of them have complicated forms. However, the background theory underlying all the creations is very simple. The author shows how different origami forms are designed from a common theory.

# Twentieth Anniversary Volume Discrete Computational Geometry

Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007) 12. Driscoll, J.R., Sleator, D.D., Tarjan, R.E.: Fully persistent lists with catenation.

This commemorative book contains the 28 major articles that appeared in the 2008 Twentieth Anniversary Issue of the journal Discrete & Computational Geometry, and presents a comprehensive picture of the current state of the field. The articles in this volume, a number of which solve long-outstanding problems in the field, were chosen by the editors of DCG for the importance of their results, for the breadth of their scope, and to show the intimate connections that have arisen between discrete and computational geometry and other areas of both computer science and mathematics. Apart from the articles, the editors present an expanded preface, along with a set of photographs of groups and individuals who have played a major role in the history of the field during the past twenty years.

# A History of Folding in Mathematics

A K Peters, Natick, pp 3–16 Demaine ED, O'Rourke J (2007) Geometric folding algorithms: linkages, origami, polyhedra. Cambridge University Press, Cambridge Demaine ED, Demaine ML, Lubiw A (1999) Folding and one straight cut suffice.

While it is well known that the Delian problems are impossible to solve with a straightedge and compass – for example, it is impossible to construct a segment whose length is cube root of 2 with these instruments – the discovery of the Italian mathematician Margherita Beloch Piazzolla in 1934 that one can in fact construct a segment of length cube root of 2 with a single paper fold was completely ignored (till the end of the 1980s). This comes as no surprise, since with few exceptions paper folding was seldom considered as a mathematical practice, let alone as a mathematical procedure of inference or proof that could prompt novel mathematical discoveries. A few questions immediately arise: Why did paper folding become a non-instrument? What caused the marginalisation of this technique? And how was the mathematical knowledge, which was nevertheless transmitted and prompted by paper folding, later treated and conceptualised? Aiming to answer these questions, this volume provides, for the first time, an extensive historical study on the history of folding in mathematics, spanning from the 16th century to the 20th century, and offers a general study on the ways mathematical knowledge is marginalised, disappears, is ignored or becomes obsolete. In doing so, it makes a valuable contribution to the field of history and philosophy of science, particularly the history and philosophy of mathematics and is highly recommended for anyone interested in these topics.

# Automated Deduction in Geometry

Bowers, J.C., Streinu, I.: Lang's universal molecule algorithm. ... Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and one straight cut suffices. In: Proc. ... Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms: Linkages, ...

This book constitutes the thoroughly refereed post-workshop proceedings of the 9th International Workshop on Automated Deduction in Geometry, ADG 2012, held in Edinburgh, UK, in September 2012. The 10 revised full papers presented together with 2 invited papers were carefully selected during two rounds of reviewing and improvement from the lectures given at the workshop. The conference represents a forum to exchange ideas and views, to present research results and progress, and to demonstrate software tools at the intersection between geometry and automated deduction; the scope of the ADG 2012 moreover has been expanded to cover topics in dynamic geometry.

# Discrete and Computational Geometry

The linear-time algorithm for computing the medial axis of a polygon successively partitions the polygon into three ... of the straight skeleton to origami and flattening are described in Geometric Folding Algorithms: Linkages, Origami, ...

Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also features numerous exercises and unsolved problems. The essential introduction to discrete and computational geometry Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and unsolved problems Suitable for sophomores in mathematics, computer science, engineering, or physics Rigorous but accessible An online solutions manual is available (for teachers only). To obtain access, please e-mail: [email protected]

# Automated Deduction in Geometry

Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007) 6. Dillencourt, M.B.: Realizability of Delaunay triangulations. Inf. Process. Lett.

This book constitutes the thoroughly refereed post-workshop proceedings of the 10th International Workshop on Automated Deduction in Geometry, ADG 2014, held in Coimbra, Portugal, in July 2014. The 11 revised full papers presented in this volume were carefully selected from 20 submissions. The papers show the trend set of current research in automated reasoning in geometry.