Origami 3 Third International Meeting of Origami Science, Mathematics, and Education

342 Pages·2002·5.394 MB·English

Origami3

Origami3
Third International Meeting
of Origami Science, Mathematics, and Education
Sponsored by OrigamiUSA
Thomas Hull, Editor
A K Peters
Natick, Massachusetts
Editorial, Sales, and Customer Service Office
A K Peters, Ltd.
63 South Avenue
Natick, MA 01760
www.akpeters.com
Copyright © 2002 by A K Peters, Ltd.
All rights reserved. No part of the material protected by this copyright notice may be reproduced
or utilized in any form, electronic or mechanical, including photocopying, recording, or by any
information storage and retrieval system, without written permission from the copyright owner.
Library of Congress Cataloging-in-Publication Data
International Meeting of Origami Science, Math, and Education (3rd : 2001 : Asilomar, Calif.)
Origami3 : Third International Meeting of Origami Science, Math, and Education /
Thomas Hull, editor.
p. cm.
Includes index.
ISBN 1-56881-181-0 (pbk.)
1. Origami--Mathematics--Congresses. 2. Origami in education--Congresses. I. Title:
Origami3. II. Title: Third: Third International Meeting of Origami Science, Math, and Education.
III. Hull, Thomas. IV. Title.
QA491 .I55 2001
736í.982--dc21
2002023871
Printed in Canada
06 05 04 03 02 10 9 8 7 6 5 4 3 2 1
Table of Contents

Preface ix
Part One: Mathematics of Origami 1
Recent Results in Computational Origami
Erik D. Demaine and Martin L. Demaine.............................................. 3
A Disk-Packing Algorithm for an Origami Magic Trick
Marshall Bern, Erik Demaine, David Eppstein, and Barry Hayes . . 17
The Combinatorics of Flat Folds: A Survey
Thomas C. Hull....................................................................................... 29
A Mathematical Model for Non-Flat Origami
sarah-marie belcastro and Thomas C. Hull........................................... 39
The Definition of Iso-Area Folding
Jun Maekawa.......................................................................................... 53
The Geometry of Orizuru
Toshikazu Kawasaki................................................................................ 61
V
vi Table of Contents
The Validity of the Orb, an Origami Model
Jeannine Mosely .................................................................................. 75
Mathematical Origami: Another View of Alhazen's
Optical Problem
Roger C. Alperin..................................................................................... 83
Just Like Young Gauss Playing with a Square:
Folding the Regular 17-gon
Robert Geretschläger.............................................................................. 95
Paper-Folding Constructions in Euclidean Geometry:
An Exercise in Thrift
Benedetto Scimemi..................................................................................107
Part Two: Origami Science and Applications 119
Computer Tools and Algorithms for Origami Tessellation
Design
Alex Bateman..........................................................................................121
The Foldinator Modeler and Document Generator
John Szinger .........................................................................................129
The Application of Origami Science to Map and Atlas Design
Koryo Miura ........................................................................................137
Origami Pots
Tomoko Fuse, Akira Nagashima, Yasuhiro Ohara
and Hiroshi Okumura.............................................................................147
Polypolyhedra in Origami
Robert J. Lang.........................................................................................153
Origami with Trigonometric Functions
Miyuki Kawamura ...............................................................................169
Table of Contents vii
A Study of Twist Boxes
Noriko Nagata ......................................................................................179
To Fold or to Crumple?
B. A. DiDonna ......................................................................................187
Folded Tubes as Compared to Kikko ("Tortoise-Shell") Bamboo
Biruta Kresling.......................................................................................197
Square Cycles: An Introduction to the Analysis of Flexagons
Ethan Berkove and Jeffrey Dumont........................................................209
Self-Assembling Global Shape Using Concepts from Origami
Radhika Nagpal......................................................................................219
Part Three: Origami in Education 233
Applications of Origami to the Teaching of Sophisticated
Communication Techniques
Deborah Foreman-Takano......................................................................235
Origami and the Adult ESL Learner
Lillian Yee Ho.........................................................................................247
Exploring the Possibilities of a Module
Krystyna Burczyk and Wojciech Burczyk...............................................257
Using Graphs to Color Origami Polyhedra
Charlene Morrow...................................................................................269
Circular Origami: A Survey of Recent Results
Eva Knoll.................................................................................................283
In Praise of the Papercup: Mathematics and Origami at
the University
Emma Frigerio........................................................................................291
viii Table of Contents
Using Triangular Boxes from Rectangular Paper to
Enrich Trigonometry and Calculus
V’Ann Cornelius and Arnold Tubis .....................................................299
Fold Paper and Enjoy Math: Origamics
Kazuo Haga............................................................................................307
Origami as a Model for Development in Organisms
Norman Budnitz......................................................................................329
Instances of Origami within Mathematics Content Texts for
Preservice Elementary School Teachers
Jack A. Carter and Beverly J. Ferrucci.................................................337
Contributors 345
Index 349
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It is with much excitement and pleasure that I write the preface to Origami(cid:1),
ProceedingsoftheThirdInternationalMeetingofOrigamiScience,Math,and
Education(3OSME).Asprogramorganizerofthisconferenceandeditorofthis
proceedings, this book symbolizes the completetion of three years of work.
The first such conference, held in Ferrara, Italyin December of 1989, was
titled The First International Meeting of Origami Science and Technology.
Humiaki Huzita of the University of Padova was the father of this historic
event. Therewereonly17talks,butthefoundationwaslaidforpeoplearound
the globe to share their interests and research on the scientific, mathematical,
and educational aspects of origami. The proceedings from the first meeting
were ground-breaking, being the first publication to collect and disseminate
paper-folding research. To this day, these proceedings are much sought-after.
The Second International Meeting of Origami Science and Scientific
Origami was held in Otsu, Japan in December of 1994; this was the first
such meeting that I was able to attend. I was very impressed with the va-
riety and scope of the talks, which covered everything from Miura’s uses of
origami in designing solar panel arrays to Yoshizawa’s views on origami art;
fromKawasaki’s work on abstract, high-dimensional flat-foldingtoKresling’s
studiesoforigamipatternsinnature. Thejuxtapositionofideasinscience,art,
education, and math made the Otsu meeting unique in the history of origami.
This meeting also gave birth to a wonderful proceedings book; the organizers
of the meeting and editors of the proceedings (Miura, Fuse, Kawasaki, and
Maekawa) should be commended for their outstanding achievement.
(cid:2)(cid:14)
(cid:14) (cid:15)(cid:16)(cid:7)(cid:9)(cid:4)(cid:17)(cid:7)
Duringthe1994Otsumeeting,itwassuggestedthatthethirdsuchmeeting
should take place in the United States. The next year, I was in San Francisco
for a conference and was fortunate enough to have dinner with Peter Engel
(author of Folding the Universe) and Robert J. Lang. When I told them about
the Otsu conference I had been to and how the next meeting should be in the
USA, they replied that I should be the one to organize it. I was in graduate
school at the time, and couldn’t conceive of how Iwas goingtoorganize such
an event. Now that it has happened, I must confess that it feels like I have
fulfilled one of the purposes of mylife.
Of course, OrigamiUSA was the natural choice to sponsor and organize
the event. As an OrigamiUSA board member, I chose to take the lead in
this project, and they provided all the logistical help in putting the conference
together. One of OrigamiUSA’s missions is to disseminate the knowledge
and educational uses of origami to the public, and through this conference,
they certainly furthered that goal. For more information about OrigamiUSA,
or to learn about becoming a member, please write to them at OrigamiUSA,
15 West 77th Street, New York, NY 10024-5192 or visit their web site at:
http://www.origami-usa.org.
Wearealsoveryfortunatetohavefounda publisher fortheseproceedings,
A K Peters. Through their means as a publisher, these proceedings will find
theirwayintothelibrariesoftheworldandbecomeknowntomathematicians,
scientists, and educators. For the first time, scholarlywork on origami will be
easily available for those seeking it. I, for one, am very glad that I will now
have somewhere to direct all the people who ask me everyyear, “Where can I
learn more about origami and math/science/education?”
I also had personal reasons for wanting to make the third meeting as suc-
cessfulaspossible. SomeofthefirstresearchIencounteredonthemathematics
of origami came from the first proceedings book of the 1989 meeting. It is
not an exaggeration to say that reading these articles while I was in graduate
school studyingtobe a mathematician changed mylife. Theyopened myeyes
to the vast potential for serious work in the studyof paperfolding. Since then,
my career path and life’s dream has been to pursue study, both in math and
in education. It is my deep hope that through 3OSME, and through the pro-
ceedingsyounowholdinyourhands, wewillextendthetraditionofexploring
the joyof origami knowledgein all fields of studyand continuethewonderful
efforts of our predecessors.
A number of people were instrumental to the development of these pro-
ceedings. sarah-marie belcastro, V’Ann Cornelius, and Robert Lang provided
much-needed editorial help and advice on a number of the papers. Anne
LaVinwasanabsolutegodsendinhelpingtheeditordealwiththemultitudeof
graphicimagesthatweresenttotheauthorinavarietyofinternationalformats.