Table Of ContentORIGAMICS
MathematicalExplorationsthroughpaperFolding
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ORIGAMICS
Mathematical Explorations through paper Folding
Kazuo Haga
University of Tsukuba, Japan
edited and translated by
Josefina C Fonacier
University of Philippines, Philippines
Masami Isoda
University of Tsukuba, Japan
World Scientific
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ORIGAMICS
Mathematical Explorations Through Paper Folding
Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN-13 978-981-283-489-8
ISBN-10 981-283-489-3
ISBN-13 978-981-283-490-4 (pbk)
ISBN-10 981-283-490-7 (pbk)
Printed in Singapore.
ZhangJi - Origamics.pmd 1 8/13/2008, 7:52 PM
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Introduction
Theartoforigami,orpaperfolding,isagreattraditioninJapan. Inits
simplest form, the folding is carried out on a square piece of paper to
obtain attractive (cid:2)gures of animals, (cid:3)owers or other familiar (cid:2)gures.
Theartenjoysgreatpopularityandappealamongbothyoungandold,
andithasspreadtoothercountriesbeyondJapan.
Itis easytoseethat origamihas linkswithgeometry. Creasesand
edgesrepresentlines,intersectingcreasesandedgesmakeangles,the
intersectionsthemselvesrepresentpoints. Becauseofitsmanipulative
andexperientialnature, origami couldbecomean effectivecontextfor
thelearningandteachingofgeometry.
Inthisbook,origamiisusedtoreinforcethestudyofgeometry,with
the hope that the popularity and appeal for the former will stimulate
the latter. The activities in this book differ from ordinary origami in
that no (cid:2)guresof objects result. Rather, they leadthe readerto study
theeffectsofthefoldingandseekpatterns.
Theauthor,Dr. KazuoHaga,isaretiredprofessorofbiologyatthe
University of Tsukuba, Japan. His interest in science has been chan-
neled to the broader (cid:2)eld of science education. He mentioned in his
bookthatduringhiscareerasabiologyprofessor,whilewaitingforhis
experiments to progress, he used to while away the time doing paper
folding(ormorespeci(cid:2)cally,mathematicsthroughpaperfolding).
The experimental approach that characterizes much of science ac-
tivity(andpossiblymuchofProfessorHaga’sworkasabiologist)canbe
recognized throughout the book. The manipulative nature of origami
allows much experimenting, comparing, visualizing, discovering and
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Origamics:MathematicalExplorationsThroughPaperFolding vi
conjecturing. Ineverytopic,theexuberancethattheauthorfeltwhen-
everhe arrivedat mathematical ideasis re(cid:3)ectedin his writingstyle.
Toparaphrasetheauthor,(cid:147)morewondersemerge!(cid:148)
Admittedly proof is a necessary part of mathematical discourse.
However,proofsarenotemphasizedinthisbook. Theauthorisaware
that many students do not appreciate formal proofs. So while some
proofs are given after the paper folding, not all mathematical discov-
eries are proven. The reader is encouragedto (cid:2)ll in all the proofs, for
his/her own satisfaction and for the sake of mathematical complete-
ness.
This then is a resourcebook for mathematics teachers and mathe-
maticsteachereducators. Itishopedthatgoingthroughthisbookwill
givethemalternativeapproachesforreinforcingandapplyingthethe-
oremsofhighschoolgeometryandforprovokingmoreenthusiasmfor
mathematicsstudy.
Jose(cid:2)na.C.Fonacier
FormerDirector,
NationalInstituteforScienceandMathematicsEducation
Development,
UniversityofPhilippines
August11,2008 11:23 WorldScienti(cid:12)cBook-9inx6in Origamics_noline
Until the Publication of the English
Edition
When I was an undergraduate student almost 30 years ago, our stu-
dent’s mathematics research club, which aimed for understanding
mathematics through different ways, held a mathematics exhibition.
One of the exhibits was on Origami, paper folding: the mathemat-
ics in Orizuru (crane construction), based on the work of Professor
Koji Fushimi, a physicist, then the President of the Science Council
ofJapan. WewellrememberthatProfessorKazuoHagavisitedourex-
hibition,andheexplainedtoustheHagatheorems. Hewasabiologist,
and we were surprised that these works on mathematics and origami
hadbeendonebyscientists(Fushimiaphysicist,andHagaabiologist)
and not by mathematicians. Nowwe are working as mathematics ed-
ucators in universities, middle schools and high schools, and it was a
partofimportantexperienceforusinbecomingteachers.
WhenI cameback to theUniversityofTsukuba 15 years ago, Pro-
fessorHagabegantoteachschoolteachershismathematicaltheoryof
Origami under the name of (cid:147)ORIGAMICS(cid:148). I recommended the pub-
lisher of the Teachers’ Journal on Mathematics Education at Meiji
Tosho-ShuppantohavetheserialofProfessorHaga’s(cid:147)Origamics(cid:148), be-
cause we knew the importance of his activity for mathematics educa-
tionandteachereducation. Based ontheseries,he publishedhis (cid:2)rst
book, which would become the major resource of this English trans-
lation. Since then he has published two more books. This English
translationincludesonlyonethirdofhisworksonOrigamics.
There are several unique points in his Origamics. The (cid:2)rst one
comes from the object itself. Everyonehas experiencein folding a pa-
per,butheexploreditbasedonhis uniquegeometricalideas. Another
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Origamics:MathematicalExplorationsThroughPaperFolding viii
point is his approaches in mathematics. He used school mathemat-
ics that could be understood by anyone who has studied mathematics
at school. Through his mathematical viewpoint, we can learn how to
exploreandenjoydailysituationsgeometrically,anddevelopourmath-
ematicalviewsandmindsintheworld.
Today, through international conferences, origamics has become a
well-known research (cid:2)eld throughout the world. Some of Professor
Haga’sworksarelecturedbyhimselfattheseconferences;atthesame
time many of his works have been spread through teachers. In the
case of the Philippines, Mr. Mikio Masuda, who had been a teacher
at the Junior high school (middle school) attached to the University
of Tsukuba, was dispatched to the University of the Philippines Na-
tional Institute for Science and Mathematics Education Development
(UP-NISMED) as a specialist of the Japan International Cooperation
Agency(JICA)ontheappointmentofProfessorShizumiShimizu,Uni-
versity of Tsukuba. Among the people he worked with was Profes-
sor Jose(cid:2)na C. Fonacier; she was especially impressed with Professor
Haga’s work. The major part of this English edition of his book origi-
natesfromresultsoftheircollaborations.
Based on the experience of international cooperation with UP-
NISMED through JICA, as well as other international cooperation
projects/experiences, the University of Tsukuba established the Cen-
terforResearchonInternationalCooperationinEducationalDevelop-
ment(CRICED)onbehalfoftheMinistryofEducation,Japan. Forde-
veloping materials for international cooperation, CRICED staff mem-
bershavebegunfullysupportforpublication. Itismypleasuretoedit
the English edition of Professor Haga’s book with Professor Fonacier,
on behalfofmylongexchangewith ProfessorHaga andthecollabora-
tionexperiencewithUP-NISMED.
MasamiIsoda
CenterforResearchonInternationalCooperationinEducational
Development(CRICED)
UniversityofTsukuba,JAPAN
August11,2008 11:23 WorldScienti(cid:12)cBook-9inx6in Origamics_noline
Acknowledgments
We would like to acknowledge the following contributors and Institu-
tions:
Dr. SoledadUlep,DeputyDirector ofUP-NISMED(theUniversity
ofthePhilippinesNationalInstituteforScienceandMathematicsEd-
ucationDevelopment)forhersupportofoureditorialworks;
Dr. Yasuo Yuzawa, researcher of CRICED (Center for Research on
International Cooperation in Educational Development, University of
Tsukuba) for his major contribution for developing editorial version
from his mathematical expertise including pictures using Software,
CabriGeometryII+andLATEX;
Dr. Rene Felix, professor at the mathematics department of the
UniversityofthePhilippines, forhis mathematical expertiseandcare
inreadingandreviewingthewholemanuscript;
Mr. Mikio Masuda, retired teacher of the Junior High School at-
tachedtotheUniversityofTsukuba,forhisearlierassistanceoftrans-
lationbetweenProfessorsHagaandFonacier;
ProfessorShizumiShimizu, thegraduateschoolofhumancompre-
hensivescience, the University of Tsukuba, for his support to develop
international relationship between University of the Philippines and
theUniversityofTsukuba;
Ms. FooChuanEng,EducationOf(cid:2)cer,BruneiDarussalam forher
supportofdevelopingcaptions;
Dr. Hiroshi Yokota, researcher of CRICED for his advice of LATEX
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