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The Miura-Ori Opened out Like a Fan INTERNATIONAL SOCIETY for the INTERDISCIPLINARY STUDY of SYMMETRY (ISIS-SYMMETRY)

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The Miura-Ori Opened out Like a Fan INTERNATIONAL SOCIETY for the INTERDISCIPLINARY STUDY of SYMMETRY (ISIS-SYMMETRY) The Quarterly of the Editors: International Society for the GyiSrgy Darvas and D~nes Nag¥ interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 5, Number 2, 1994 The Miura-ori opened out like a fan INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) President ASIA D~nes Nagy, lnslltute of Apphed Physics, University of China. t~R. Da-Fu Ding, Shangha~ Institute of Biochemistry. Tsukuba, Tsukuba Soence C~ty 305, Japan Academia Stoma, 320 Yue-Yang Road, (on leave from Eotvos Lot’find Umve~ty, Budapest, Hungary) Shanghai 200031, PR China IGeometry and Crystallography, H~story of Science and [Theoreucal B~ology] Tecbnology, Lmgmsucs] Le~Xiao Yu, Department of Fine Arts. Nanjmg Normal Umvers~ty, Nanjmg 210024, P.R China Honorary Presidents }Free Art, Folk Art, Calhgraphy] Konstantin V. Frolov (Moscow) and lndta. Kirti Trivedi, Industrial Design Cenlre, lndmn Maval Ne’eman (TeI-Avw) Institute of Technology, Powa~, Bombay 400076, India lDes~gn, lndmn Art] Vice-President Israel. Hanan Bruen, School of Education, Arthur L. Loeb, Carpenter Center for the V~sual Arts, Umvers~ty of Hallo, Mount Carmel, Haffa 31999, Israel Harvard Umverslty. Cambridge, MA 02138, [Educanon] U S A. [Crystallography, Chemical Physics, Visual Art~, Jim Rosen, School of Physics and Astronomy, Choreography, Music} TeI-Av~v Umvers~ty, Ramat-Avtv, Tel-Av~v 69978. Israel and [Theoretical Physms] Sergei V Petukhov, Instnut mashmovedemya RAN (Mechamcal Engineering Research Institute, Russian, Japan. Yasushi Kajfl~awa, Synergel~cs Institute. Academy of Scmnces 101830 Moskva, ul Griboedova 4, Russia (also Head of the Russian Branch Office of the Society) 206 Nakammurahara, Odawara 256, Japan }Design, Geometry] }B~omechanlcs, B~ontcs, Informauon Mechamcs] Koichtro Mat~uno, Department of BioEngineering. Nagaoka Umvers~ty of Technology, Nagaoka 940-21, Japan Executive Secretary [Theoretical Physms, Blophys*cs] Gybrgy Darvas, Symmetrion - The lnsmute for Advanced Symmetry Studies AUSTRALIA AND OCEANIA Budapest, PO Box 4, H-1361 Hungary Austraha Leslie A, Bursill, School of Physics, }Theoretical Physms, Philosophy of Science} Umvers~ty of Melbourne, Parkwlle, Vtctorm 3052, Austraha [Physics, Crystallography] Associale Edttar. Jobn Hosack, Department of Mathematics and Computing Science, Unlverstty of the South Pacific, PO Box 1168, Suva, FIji F01: Jan Tent, Department of L~leratum and Language, }Mathematical Analysts, Phflosophyl University of the South Pacific, PO Box 1168, Suva, F0t [Lmgmsttcs] Regional Chat,persons / Representatives. New Zealand. Michael C. Corballis, Department of Psychology, Umversily of Auckland, Private Bag, Auckland I, New Zealand [Psychology] AFRICA Mozambique Paulus Gerdes, Inst~tuto Tonga. ’Ilaisa Futa-i-Ha’angana Helu, Director, Superior Pedag6gico, Ca~xa Postal 3276, Maputo, ’Atems~ (Athens) Institute and Umverslly, Mozambique PO. Box 90, Nuku’alofa, Kingdom of Tonga |Geometry, Ethomatb.emaucs, History of Science} [Phdosophy, Polynesian Culture] AMERICAS EUROPE Brazd: Ubiratan D’Ambrosio, Rua Pe~xoto Gomide 1772, up. 83, Benelux" Pieter Huybers, Facultett der Civlele Techniek, BR-01409 S~o Paulo, Brazd Techmsche Untverstte~t Delft [Ethnomathemat~cs] (C~wl Engineering Faculty, Delft Utavers~ty of Technology), Stevmweg I, NL-2628 CN Delft, The Netherlands [Geometry of Structures, Budding Technology} Canada: Roger V. Jean, DEpartement de mathfimatlques et mformauque, Um~ers~t~ du QuEbec ~ gamouskt, Bulgaria: Ruslan I. Kostov, Geologtcheski lnstltut BAN 300 allEe des Ursuhnes, R~mouskL QuEbec, Canada G5L 3AI (Geological Institute, Bulgarian Academy of Sciences), [ B~omathemattcs] ul Akad G. Bonchev 24, BG-III3 Sofia, Bulgaria [Geology, M meralog3’] U.S.A " William S. Huff, Departlnent of Architecture, State Umversfly of New York at Buffalo, Buffalo, Czech Republic: X’bjt~h KopskJ;, Fyz~k~lnt t~stav (~AV NY 14214, USA. (Institute of Physics, Czech Academy of Sciences), CS-180 40 }Architecture. Des~gnl Praha 8 (Prague), Na Slovance 2 (POB 24), Nicholas Toth, Department of Anthropology, Czech Republic [Sohd S~te Physics} Indiana Utaverslty, Rawles Hall 108, Bloomington, IN ~,7405, U.S A. France: Pierre Sz~kely, 3bts, impasse Vflliers de I’lsle Adam, [Preh~storta Archaeology, Anthropology] F-75020 Paris, France [Sculpture] continued inside back cover .!1.. i|CULTURE & SCIENCE I| The Quarterly of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry) Editors: GyOrgy Darvas and D~nes Nagy Volume 5, Number ~ 113-224, 1994 SPECIAL ISSUE: ORIGAMI, 2 Edited by D~nes Nagy and Gy0rgy Darvas CONTENTS SYMMETRY: CULTURE & SCIENCE ¯ Mathematical algorithms for origami design, RobertJ. Lang 115 ¯ Mathematical remarks about origami bases, Jacques Justin 153 ¯ Evolution of origami organisms, Jun Maekawa 167 ¯ Paper sculpture, Didier Boursin 179 SYMMETRIC GALLERY - ORIGAMI 189 ¯ Paper sculpture, Didier Boursin 190 ¯ Stag beetle 2, RobertJ. Lang 197 RESEARCH PROBLEMS ON SYMMETRY ¯ Research problem 1, D~nes Nagy 211 SYMMETR O-GRAPHY 213 SFS: SYMMETRIC FORUM OF THE SOCIETY 219 SYMMETRY: CULTUlCEAND SCIENCE is edited by the Board of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry) and published quar- terly by the International Symmetry Foundation. The views expressed are those of individual authors, and not necessarily shared by the Society or the Editors. Any correspondence should be addressed to the Editors: Gy6rgy Darvas Symmetrion - The Institute for Advanced Symmetry Studies P.O. Box 4, Budapest, H-1361 Hungary Phone: 36-1-131-8326 Fax: 36-1-131-3161 E-mail: [email protected] D6nes Nagy Institute of Applied Physics University of Tsukuba Tsukuba Science City 305, Japan Phone: 81-298-53-6786 Fax: 81-298-53-5205 E-mail: [email protected] The section SFS: Symmetric Forum of the Society has an E-Journal Supplement. Annual membership fee of the Society: Benefactors, US$780.00; Ordinary Members, US$78.00 (including the subscription to the quarterly); Student Members, US$63.00; Instituaonal Members, please contact the Executive Secretary. Annual subscription rate for non-members: US$96.00 + mailing cost. Make checks payable to ISIS-Symmetry and mail to Gy0rgy Darvas, Executive Sec- retary, or transfer to the following account number: ISIS-Symmetry, International Symmetry Foundation, 401-0004.827-99 (US$) or 407-0004-827-99 (DM), Hungarian Foreign Trade Bank, Budapest, Szt. Istv~in t6r 11, H-1821 Hungary (Telex: Hungary 22-6941 extr-h; Swift MKKB HU HB). ISIS-Symmetry. No part of this publication may be reproduced without written permission from the Society. ISSN 0865-4824 Cover layout: Gunter Schmitz Image on the front cover. Biruta Kresling The Miura-ori opened out like a.fan, simulates the mechanism responsible J’or the outstretching of the beetle’s membraneous hindwin g Ambigram on the back cove~. John Langdon (Wordplay, 1992) Logo on the title page: Kirti Trivedi and Manisha l.ele Fot6k~sz anyagr61 a nyomdai kivitelez6st vdgezte: 9421768 AKAPRINT Kft. F. v.: Dr. H~czey Lfiszl6n~ 3)rmmetry: Culture and Science Vot 5, No. 2, 115-152, 1994 MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN Robert J. Lang 7580 Olive Drive Pleasanton, CA 94588, USA E-mail: [email protected] Although hundreds of years old, the Japanese art of origami has only recently become the subject of mathematical scrutiny. In recent years, a number of mathematical aspects of origami have been published in books and journals. A sampling of the work of mathematical folders is to be found in recent mainstream publications, e.g., (Kasahara, 1988) and (Engel, 1989); however a large number of folders have attacked the problem of systematic/mathematical origami design. They include Peter Engel and myself in America, and many folders in Japan, including Husimi, Meguro, Maekawa, and Kawahata. As befits a young and expanding field, much of the scientific analysis is circulated informally (notably over the origami-I mailing list on the Internet: to join, send the message "subscribe origami-I yourname" to [email protected]). The goal of many origami aficionados is to design new origami figures and for many, the pursuit of origami mathematics is a search for tools leading to ever more complex or sophisticated designs. In this article, I will describe two powerful algorithms for origami design that I have successfully applied to the design of fish, crustacea, insects, and numerous other origami models. Although I will describe the algorithms in the form I am familiar with, similar techniques have been described by Dr. Toshiyuki Meguro in the (Japanese-language) publication Oru and in the newsletter of the Origami Tanteidan, a Japanese association of origami designers. 1. THE CIRCLE METHOD OF DESIGN The first design approach is represented by what I and others call the ’circle method’. In the circle method, each flap on the origami model is represented by a circle whose radius is equal to the length of the flap. The goal of the origami design process is to place circles representing each flap on the square in such a way that the centers of all circles lie within the square (although some part of the circle can extend over the edges of the square) and no two circles overlap one another. This R. J. LANG 116 approach is one that both I and Fumiaki Kawahata have used extensively, although - as happens so often in the sciences - we each developed our methods initially unaware of the other’s activities. I am not aware of other Westerners using the circle method, although the young American folder, Jimmy Schaefer, has developed a successful and related design method, which he has dubbed, ’the method of isolating squares’, based on concepts similar to the circle method. In Japan, these concepts of origami design are more widely known than in the West. The fundamental concepts of the circle method of design and i~s derivatives are_ two: first, since most origami models can be broken down into a number of flaps of various lengths, a successful design hinges upon constructing the right number and sizes of flaps. Second, paper must be conserved; any part of the square can be used in no more than 1 flap at a time.
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