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0521857570pre CUNY758/Demaine 0 521 81095 7 February 25, 2007 7:5
GEOMETRIC FOLDING ALGORITHMS
Folding and unfolding problems have been implicit since AlbrechtDürer
in the early 1500s but have only recently been studied in the mathemat-
ical literature. Over the past decade, there has been a surge of interest
in these problems, with applications ranging from robotics to protein
folding. With an emphasis on algorithmic or computational aspects,
this comprehensive treatment of the geometry of folding and unfolding
presents hundreds of results and more than 60 unsolved “open prob-
lems” to spur further research.
The authorscoverone-dimensional (1D) objects (linkages), 2D objects
(paper), and 3D objects (polyhedra). Among the results in Part I is that
there is a planar linkage that can trace out any algebraic curve, even “sign
your name.” Part II features the “fold-and-cut” algorithm, establishing
that any straight-line drawing on paper can be folded so that the com-
plete drawing can be cut out with one straight scissors cut. In Part III,
readers will see that the “Latin cross” unfolding of a cube can be refolded
to 23 different convex polyhedra.
Aimed primarily at advanced undergraduate and graduate students
in mathematics or computer science, this lavishly illustrated book will
fascinate a broad audience, from high school students to researchers.
Erik D. Demaine is the Esther and Harold E. Edgerton Professor of Elec-
trical Engineering and Computer Science at the Massachusetts Institute
of Technology, where he joined the faculty in 2001. He is the recipient of
several awards, including a MacArthur Fellowship, a Sloan Fellowship,
the Harold E. Edgerton Faculty Achievement Award, the Ruth and Joel
Spira Award for Distinguished Teaching, and the NSERC Doctoral Prize.
He has published more than 150 papers with more than 150 collabora-
tors and coedited the book Tribute to a Mathemagician in honor of the
influential recreational mathematician Martin Gardner.
Joseph O’Rourke is the Olin Professor of Computer Science at Smith
Collegeandthefounding Chair of the Computer Science Department. He
has received several grants and awards, including a Presidential Young
Investigator Award, a Guggenheim Fellowship, and the NSF Director’s
Award for Distinguished Teaching Scholars. His research is in the field
of computational geometry, where he has published a monograph and
a textbook, and coedited the Handbook of Discrete and Computational
Geometry.
i
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0521857570pre CUNY758/Demaine 0 521 81095 7 February 25, 2007 7:5
Geometric Folding
Algorithms
Linkages, Origami, Polyhedra
ERIK D. DEMAINE
Massachusetts Institute of Technology
JOSEPH O’ROURKE
Smith College
iii
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cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
32 Avenue of the Americas, New York, NY 10013-2473, USA
www.cambridge.org
Information on this title: www.cambridge.org/9780521857574
C
Erik D. Demaine, Joseph O’Rourke 2007
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2007
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Demaine, Erik D., 1981–
Geometric folding algorithms : linkages, origami, polyhedra / Erik D.
Demaine, Joseph O’Rourke.
p. cm.
Includes index.
ISBN-13: 978-0-521-85757-4 (hardback)
ISBN-10: 0-521-85757-0 (hardback)
1. Polyhedra – Models. 2. Polyhedra – Data processing.
I. O’Rourke, Joseph. II. Title.
QA491.D46 2007
516
.156 – dc22 2006038156
ISBN 978-0-521-85757-4 hardback
Cambridge University Press has no responsibility for
the persistence or accuracy of URLs for external or
third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such
Web sites is, or will remain, accurate or appropriate.
iv
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To my father, Martin Demaine To my mother, Eleanor O’Rourke
–Erik –Joe
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Contents
Preface page xi
0 Introduction .........................
1
0.1 Design Problems 1
0.2 Foldability Questions 3
Part I. Linkages
1 Problem Classification and Examples ............
9
1.1 Classification 10
1.2 Applications 11
2 Upper and Lower Bounds .................17
2.1 General Algorithms and Upper Bounds 17
2.2 Lower Bounds 22
3 Planar Linkage Mechanisms ................
29
3.1 Straight-line Linkages 29
3.2 Kempe’s Universality Theorem 31
3.3 Hart’s Inversor 40
4 Rigid Frameworks .....................43
4.1 Brief History 43
4.2 Rigidity 43
4.3 Generic Rigidity 44
4.4 Infinitesimal Rigidity 49
4.5 Tensegrities 53
4.6 Polyhedral Liftings 57
5 Reconfiguration of Chains .................
59
5.1 Reconfiguration Permitting Intersection 59
5.2 Reconfiguration in Confined Regions 67
5.3 Reconfiguration without Self-Crossing 70
6 Locked Chains .......................
86
6.1 Introduction 86
6.2 History 87
vii
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