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. The circle method of origami design consists of representing the subject as a collection of flaps and allocating a unique circular region of paper for each flap. For the purposes of origami design, there are three different types of flaps: ’corner’ flaps, ’edge’ flaps, and/or ’interior’ flaps, or ’middle’ flaps, as some call them. The different types of flaps are named for the point where the tip of the flap falls on the square. If you take a model that has a lot of flaps, color the tip of each flap, and then unfold the model, you’ll get a square with a pattern of dots on it. Some dots will fall on the corners of the square; others on the edges; still others will be in the interior of the square. Since each dot corresponds to a flap of the model, we can classify the flap by the location of the dot, which is the location of the tip of the flap. A corner flap has its tip come from a corner of the square, an edge flap has its tip lie somewhere along an edge, and a middle flap, as you would expect, comes from the middle of the paper. For example, the four large flaps on a Frog Base are corner flaps; the four stubby flaps are edge flaps; and the thick flap at the top is a middle flap. The reason for the distinction between the three different types of flap is that for a given length, each of the three types of flaps consumes a different amount of paper. One way to see this difference is to fold corner, edge, and interior flaps of exactly the same size from three different squares as shown in Figure 1 below, where I illustrated the folding of a corner flap. If you imagine (or fold) a boundary across the base of the flap, then that boundary divides the paper into two regions: the paper above the boundary is part of the flap, and the paper below the boundary is everything else. The paper that goes into the flap is for all intents and purposes consumed by the flap; any other flaps must come from the rest of the square. So, as Figure l(a) shows, if you fold a flap of length L from a square so that the tip of the flap comes from the corner of the square, when you unfold the paper to the original square, you see that the region of the square that went into the flap is roughly a quarter of a circle. Well, technically, it’s a quarter of an octagon. Suppose we made the flap half the width, as shown in Figure l(b) before we unfolded it; MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 117 then the flap becomes a quarter of a 16-gon. If we kept making the flap thinner and thinner (using infinitely thin paper!), the boundary of the flap would approach a quarter-circle. Since we may not know ahead of time how thin a given flap will be, we’ll take the circle as a reasonable approximation of the boundary of the region of the paper consumed by the flap. A comer flap of length L, therefore, requires a quarter-circle of paper, and the radius of the circle is L, the length of the flap.
L
Figure l(a): Folding a corner flap of length L from a square.
Figure l(b): (Left) Making a narrower flap makes the boundary a quarter of a 16-gon. (Right) The limit of the boundary as the flap becomes infinitely thin approaches a semicircle. Therefore, all of the paper that lies within the quarter-circle is consumed by the flap, and the paper remaining is ours to use to fold the rest of the model. Now, suppose we are making a flap from an edge. How do we do that? Well, if we fold the square in half, then the point where the crease hits the edge become comers, and we can fold corner flaps out of one of these new corners, as shown in Figure 2. If we fold and unfold across the flap to define a flap of length L and then unfold to the square, you see that an edge flap of length L consumes a half-circle of paper, and again, the radius of the circle is L, the length of the flap. 1~ I. I_.ANG 118
Figure 2: Folding an edge flap of length L from a square. Similarly, we can make a flap from some region in the interior of the paper (it doesn’t have to be the very middle, of course). Figure 3 shows how such a flap is made. When you unfold the paper, you seen that an interior flap requires a full circle of paper, and once again, the radius of the circle is the length of the flap.
I
I I
Figur~ 3: Folding a middle flap of length L from a square.
Figure 4: Unfolding a crane (which has four major flap~) reveals that the quarter-circles corresponding to the four [lap~ consume almost all of the paper in the model. So, any given flap in a model consumes a quarter, half, orJfull circle of paper, depending upon whether it is a corner flap, edge flap, or interior flap. It is an interesting and illuminating exercise to unfold an existing model and draw in the MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 119 circles corresponding to the various flaps of the model. For example, the traditional crane, made from the Bird Base, has four major flaps. As shown in Figure 4, they are all corner flaps, and each flap consumes a quarter-circle of the square. (If we count the pyramid in the middle of the back as a flap, we would have another, smaller circle in the center of the square. However, since real cranes don’t have pyramids in their back, I consider that an ’accidental’ flap and we won’t count it in our tally.) Since well over 3/4 of the area of the square goes into the four major flaps, we would say that the crane shows an efficient use of paper. What we are doing here is to build up a set of mathematical tools that can be used to design origami models, making origami design a scientific process. One of the goals of all scientific endeavors is the concept of unification: describing several disparate phenomena as different aspects of a single concept. Rather than thinking in terms of quarter-circles, half-circles, and full-circles for different kinds of flaps, we can unify our description of these different types of flaps by realizing that the quarter-circles, half-circles, and full circles are all formed by the overlap of a full circle with the square, as shown in Figure 5. The concept common to all three types of flaps is that the paper for each can be represented by a circle with the center of the circle lying somewhere within the square. With middle flaps, the circle lies wholly within the square. However, with corner and edge flaps, part of the circle laps over the edge of the square. (The center of the circle still has to lie within the square, though.) Thus, any type of flap can be represented by a circle whose center, which corresponds to the tip of the flap, lies somewhere within the square.
Figure 5: All three types of points can be represented by a circle if we allow the circle to overlap the edges o~ the square. The examples above showed how to fold a single flap from a square. Suppose we want to make more than one flap at a time from the square (which is usually the case, unless you are designing a worm), and you draw the circles corresponding to each flap. Is there anything we can say about the circles even before we start? Well, since no part of the paper can be used in two different flaps simultaneously, and each circle delineates the paper used in each flap, no two parts of the paper can lie inside two different circles. Therefore, no two circles corresponding to different flaps can overlap on the unfolded square. Although this property seems pretty general, it is in fact quite restrictive. If you want to fold a model with ten flaps, you know that if you unfold the model and draw the circles corresponding to each flap, no two of the circles will overlap. So if you eliminate all arrangements of points for which the circles overlap, you must be closer to a design solution. In fact, the ramifications of the non-overlapping property are a great deal stronger; if you draw ten non-overlapping circles on a square, it is guaranteed (in a mathematical sense) that the square can be folded into a base with ten flaps whose tips come from the centers of the circles. So merely by shuffling circles around on a square, you can construct an arrangement of points that can be folded into a base with the same number of points, no matter how complex! Therefore, here is an algorithm for origami design: (1) Count up the number of appendages in the subject and note their lengths. (2) Represent each flap of the desired base by a circle whose radius is the length of the flap. (3) Position the circles on a square such that no two overlap and the center of each circle lies within the square. (4) Connect adjacent centers to one another with crease lines. The resulting pattern can be foldable into a base with the number and dimension of flaps that you started with.
This is a powerful property for origami designers. If you lay out circles corresponding to the flaps of your subject on a square so they don’t overlap, you are guaranteed of the existence of a folding sequence that can transform the pattern into the desired base. Finding the folding method may still be a bit of a trick, of course, but by beginning from a valid circle pattern, you certainly eliminate a lot of blind alleys. One thing that is immediately apparent from the circle method of design is that corner flaps consume less paper than edge flaps, which consume less paper than interior flaps. Turn this property around, and you find that for a given size square, you can fold a larger model (with fewer layers of paper) if you use corner flaps rather than edge flaps, and edges flaps rather than interior flaps. Seen in the light of the circle method, the traditional crane - and the Bird Base from which it comes - is an extremely efficient design, since all four flaps are corner flaps, and almost all of the paper goes into one of the four flaps. However, add one or two more MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 121 flaps, and you are forced to use edge flaps. Once you start mixing edge flaps and interior flaps, you begin to run into tradeoffs in efficiency. Sometimes, it is even better not to use the corners for flaps if there are additional flaps to be placed on the square! So, for example, suppose we want to fold a base withfive, rather than four, equal- length flaps. A little doodling with a pencil and paper (or alternatively, you can cut out some cardboard circles and shuffle them around) will reveal two particularly efficient arrangements of circles, as shown in Figure 6 below.
(a) (b)
L = 0.707 L = 0.647 Figure 6: Two c~rcle patterns corresponding to bases with five equal-length points. Now we have two possible circle patterns. Which one is better? Is there any way to quantify the ’quality’ of a crease pattern? One way of comparing different ways of folding the same base is to compare their efficiency; that is, from a given size square, how large is the base? A useful measure of efficiency is to compare the size of some standard feature of the base - such as the length of a flap - to the size of the original square. To facilitate this comparison, let’s assume our square is one ’unit’ on a side. If you’re using standard origami paper, a unit is 10 inches. For the crease patt.erns shown in Figure 6(a), if all of the circles are the same size, it is fairly easy to work out that the radius of each circle, and thus the length of each of the five flaps, is 1/v~, or 0.707. For the pattern in Figure 6(b), it is somewhat harder to calculate but the radius of each circle is 0.647, or about 10% smaller. Thus, a five-flap base made from pattern 6(a) will be slightly larger, and slightly more efficient than the pattern made from Figure 6(b). These two circle patterns are relatively simple. By connecting the centers of the circles with creases and adding a few more creases, you can collapse the model into 122 R ].. LANG a base that has the desired number of flaps. As it turns out, there already exists in the origami literature two bases that correspond to these circle patterns, shown in Figure 7. Figure 7(a) is the circle pattern for the Frog Base, while Figure 7(b) is the circle pattern for John Montroll’s ’Five-Sided Square’ (Montroll, 1985).
(a) (b)
Figure 7: Full crease patterns corresponding to the two circle patterns. You can also see a difference between the two bases. In the Frog Base, the fifth flap is a thick middle flap and points in the opposite direction from the four corner flaps; whereas in the Five-Sided Square, the four edge flaps and the corner flap go in the same direction and can easily be made to appear identical (which, of course, was the original rationale for John’s design). It’s worth a slight reduction in size to obtain the similarity in appearance for all five flaps. While the two solutions for five equal-sized flaps correspond to published bases, I find it remarkable that the most efficient base for six equal flaps is not yet published. You might wish to try your hand at the following two problems: (1) Find a circle pattern for the largest possible base that has six equal-length flaps and fold it into the base. (2) Find a circle pattern for the largest possible base that has six equal-length flaps, 3 on each side of a line of bilateral symmetry, and fold it into the base. (The surprising solution has two middle flap!!) As I said before, although the circle method guarantees that a folding sequence exists to convert the skeletal crease pattern into a base, it doesn’t necessarily provide any guidance as to what that folding sequence actually is! (Meguro’s ’molecular’ approach of fitting together pre-existing crease patterns, however, helps fill in this gap.) So even if you work out a circle pattern, with or without a computer, you (and the computer) still have some work ahead of you to figure out how to fold the crease pattern into a base. However, it is a big help to start with a crease pattern that is guaranteed to work - you can avoid using a basic symmetry that is doomed to failure from the start! MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN 123
Interestingly, there is a strong connection here between origami design and a well- known branch of geometry, that of packing circles. (For an excellent introduction to the latter, see (Gardner, 1992), Chapter 10, Tangent Circles.) For every pattern of circle packings in a square there is a corresponding origami base and vice-versa; conversely, many origami design problems my be solved by published solutions to different circle-packing problems.) The circle method as described above works very well for models with many flaps that all come from the same part of the subject’s body. The legs of insects and spiders, for example, all emanate from a single body segment. However, the circle method has one enormous liability. Since we consider only the total number of flaps and their lengths in the circle method of design, we have no way of incorporating information about how those flaps are connected to one another into the design. In fact, the circle method implicitly assumes that all the flaps are connected to each other at a single point! In the subject, the head bone may be connected to the neck bone, and the neck bone’s connected to the chest bone, but with the circle method, every bone’s connected to every other bone at one spot. That is a severe limitation for origami design. For example, a typical mammal has three flaps (head and forelegs) at one end, three (tail and hind legs) at the other, and a body in between. The circle method can produce the head, legs, and tail, but there is no mechanism to include extra paper between the forelegs and hind legs to form a body. The circle method is not the whole story of origami design, however. There is a more sophisticated algorithm that includes the body, and in fact does indeed work for arbitrary arrangements of flaps and their connections. Now that we have established some basic concepts of origami design, we are ready to move on to the next level of origami design and introduce the ’tree method’. This new algorithm will be described in the next section.
2 TREE METHOD OF DESIGN The circle method as described above works best for.models that have all of their flaps emanating from nearly the same place, models whose basic shape is star-like. Quite a few subjects fall into this category - particularly insects, which have legs and wings all emanating from a single body segment, the thorax. (Antenna cause problems, since they come from the head.) However, the circle method gives less- than-satisfactory results for subjects that don’t have a simple star shape - like most terrestrial vertebrates. A typical mammal, for example, has a cluster of three flaps (head, forelegs) separated from another cluster of three flaps (tail, hind legs) by an additional segment (the body). The circle method only deals in flaps and clusters of flaps; we have no way of including segments that connect different clusters together. 124 R. I. I../~VG
An extension of the circle method works for a much larger class of models that can have more complex structure. I call the extended method the ’tree method’. Although the tree method is built upon the ideas of the circle method, it takes a somewhat different form. The fundamental concept of the tree method of origami design is that you represent the model by a stick figure (the tree) that has a branch for each arm, leg, wing, or other appendage. Each branch has a certain length, which you have chosen to be the length of the appendage in the final model. By mapping the tree onto a square according to a small number of rules, you can construct the skeleton of a crease pattern that, like the one you get from the circle method, is guaranteed to be capable of being folding into the stick figure and, by extension, into the desired model. Unlike the circle method, which only applied to stick figures that were fundamentally star-like, the tree method works for arbitrarily connected graphs. To understand the rules for the mapping, I’ll show how it applies to a realistic design problem, a lizard. A lizard is simple enough that we won’t get bogged down in a lot of details, but it’s complicated enough to illustrate the basic approach, and because of its long body, it is precisely the type of model that the circle method has trouble with. So, Figure 8(a) shows a drawing of a lizard. We would like to fold an origami model of a lizard, that might look something like Figure 8(b). (Actually, I’d hope it looks a lot better than Figure 8(b); I can fold a lot better than I can draw.) I’ll start by drawing the ’tree’ that represents the lizard, as shown in Figure 8(c). A tree is a stick figure. On this stick figure, I have labeled each branch of the tree - which corresponds to an appendage or body segment of the lizard - with its desired length. The tail and body are 2 units long, the legs are 1 unit long each, and the head is also 1 unit long. Since I don’t know what size square I’m going to be folding from and I don’t know (yet) how large the lizard is with respect to the square, I’ll defer, for the moment, the question of just how long a ’unit’ is.
rear front (a) (b) (c) foot foot
: head
rear front foot foot
Figure I1: (Left) A real lizard. (Middle) A hypothetical origami lizard. (Right) The tree, or stick figure, corresponding to our hypothetical lizard. My end goal is to lind a crease pattern on a square that can be folded into the lizard. My intermediate goal will be to find a crease pattern that can be folded into MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 125 the tree. At first, it seems like I’ve made my life harder by setting the tree as my target. Since the branches of the tree are infinitely thin, it would take an infinite amount of folding (and infinitely thin paper) to fold it exactly. However, think of the tree as the lizard stripped of confusing detail. The tree, in its stark simplicity, represents an easier target for origami design than the original subject. And anyhow, I don’t have to fold the tree exactly; if I fold a shape that closely resembles the tree, I’ll have a shape - a base - that is suitable for folding a lizard. In the tree method of design, each branch on the tree corresponds to some flap on the square. Tree branches that end at a point - which I call ’leaves’ (or terminal nodes, if you prefer) - correspond to appendages of the subject, like the head, tail and legs. Tree branches that are connected to other branches at both ends correspond to body segments that join groups of appendages. Since I’m trying to establish a link between the square and the hypothetical lizard, let’s start at the lizard and work backwards. If I already possessed a folded version of the lizard and I made dots at important points - the tips of the legs, head, and tail, and where legs and body come together - when I unfolded the paper to a square, I could keep track of where those significant points fall on the square. The points where branches terminate or where several branches come together are important; I’ll call each of those points a ’node’ and label it with a name, corresponding to its position in the subject. Obviously, all of the nodes must lie somewhere on the square, and a great deal of the structure of the base is tied up in where the different nodes fall on the square. We ought to be able to draw the entire tree on the square so that the nodes match up with their corresponding points. There are lots of different possibilities for the position of the nodes with respect to the square; a few of them are shown in Figure 9.
/..." Figure 9: Three different possible arrangements of the lizard tree on the square. It sort of makes sense that we would want the head and tail at opposite corners of the square, so let’s suppose for the sake of argument, that we already had a successfully folded lizard-like shape, that we marked the locations of the nodes and branches on the shape, and then unfolded it to a square, giving the arrangement shown in Figure 10.
front foot
~’ront foot
Figure 10: One possible arrangement of nodes for the lizard tree. An important issue (actually, one of the most important) is how large the tree is compared to the size of the square. One of the hallmarks of good origami design is efficiency; the best designs generally are those that waste very little paper. For example, if you are folding a 3-pointed shape, you could start from a 4-pointed base and crumple up one of the flaps to hide it; but that wouldn’t be nearly as esthetically pleasing as to work from a 3-pointed base from the very beginning. It’s poor form to have to hide an unwanted flap, or even to have to make a flap drastically shorter by folding it in half. In our design, we don’t want to have points or flaps that serve no purpose. So, in our design, we should use as much of the paper as possible for the designed parts of the figure and have no extra paper left over. Also, the most efficient designs generally have the fewest layers, at least, when compared to designs of comparable complexity. Generally, the smaller a shape is, the more layers it has. That means that for a given size square, the base that we design should be as large as possible; consequently, the size of a unit of length of the tree should be maximized. MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN 127
Let’s figure out how big the tree is for Figure 10. The distance from the head to the tail is 5 units (2 for the tail, 2 for the body, and 1 for the head). The diagonal of a square is about 1.4 times the side of the square, and as I’ve shown, the diagonal is equal to 5 tree units; thus one tree unit is 1.4/5 = 0.283 times the side of the square. This quantity - the ratio between a tree unit and the side of the square - is an important measure of the efficiency of a design, and we’ll call this the ’scale’ of the crease pattern. The larger the scale is, for a given size square, we get a larger base with fewer layers, a base that is more efficient and (one hopes) more esthetically pleasing. The main problem with the arrangement of nodes depicted in Figure 10 is that it doesn’t work. If you try to fold a lizard using this arrangement, you will find that, although the body and tail are easy to make, the legs come out rather shorter than we intended. In fact, the back legs will be almost nonexistent. Furthermore, quite a lot of paper at the top and bottom corners goes essentially unused. It doesn’t seem right that the base comes out with the wrong proportions and there is unused paper as well! So we can’t just draw the tree to scale on the square and expect things to work out. Obviously, we’re overlooking some crucial concept. There must be some additional rule to be applied that limits the size of the tree when it is mapped onto the square. Well, of course with all the folding that goes on between the square and the base, the tree pattern on the square could get rather distorted. We can establish some limits on the amount of distortion, though. Consider the following thought experiment. Suppose an ant wishes to walk from the tail of the lizard to one of the rear feet. On the lizard, she starts at the tip of the tail, walks up the tail to the flap where the tail and legs meet, turns, and walks down the leg. The distance the ant has waJked is the length of the tail plus the length of the leg, or, as I’ve drawn in Figure 11, a total of 3 units.
front foot
tail ...... xhead 2
rear front foot foot
Figure 11: Picture an ant walking from the tail to the rear leg. She can’t go by the shortest route (as the crow flies); instead, she must walk up the tail and back down the leg, for a total of three units. R..I. L4NG 128
Suppose that just before the ant set out, we dipped her in ink, so that when she walked, she left a trail of ink soaking through the paper. Now we unfold the base and look at the various trails left by the ant. Since in most origami bases, each flap consists of several layers of paper, the ant will probably have left several trails between the two nodes. Depending on the folding pattern for the base, some of the trails might weave around a bit, while others go more directly from the tail to the foot. Several possible trails are shown in Figure 12.
front foot
tail
front foot
Figurt 12: Three paths from tail to rear foot. Although we may not know what ground was covered by the ant, we do know that the ant walked exactly 3 units on the tree. If in the folded base some paper was doubled back on itself, then some of the paths on the square might be longer than 3 units, but no path can be shorter than 3 units. In particular, the shortest possible ink trail is the one that runs directly between the tail node and the leg node (trail (b) in Figure 5), and it, too, must be at least 3 units long. Thus, we know that a successful crease pattern must have the tail node and each leg node separated by a minimum of 3 units. A similar condition exists for every possible pair of nodes. If the ant goes from the tip of the tail to the tip of the head, he travels 2+2+1=5 units; thus the tip of the tail must be separated from the head on the square by 5 units. From one front leg to the other is 1+1=2 units, so the two front leg nodes must be separated by 2 units; and so on and so forth. This condition must hold for any pair of nodes: the distance between two nodes on the square must be at least the distance between the two nodes measured along the MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 129 branches of the tree. Nodes connected by a single branch must be separated by at least the length of the branch; nodes connected by two branches must be separated by the sum of the lengths of the two branches; and so on, for every possible pair of nodes. Now you see the problem with the tree structure of Figure 10. The way I’ve drawn it, although all nodes connected by a single branch are separated by the proper amount, the shortest path between the nodes corresponding to the tail and rear legs is only v~, or about 2.2 units long, when it should in fact be at least 3 units. The same type of shortfall affects the paths between the front and rear legs. As I’ve drawn them in Figure 10, the front and rear feet on one side of the body are separated from each other by the body length, or 2 units; but in fact, they need to be separated from each other by 4 units, since our proverbial ant must walk down one leg, along the body, and back out the other leg to go from toe to toe. Similar shortfalls afflict the paths between the head and front legs, between the head and rear legs, between the tail and front legs, and between the front and back legs. Figure 13 shows all possible paths drawn in on the tree and all of their minimum lengths.
Figure t3: Paths and lengths for all nodes on the lizard tree. In general, many pairs of nodes on the square must be farther apart from one another than they are ’as the crow flies’ on the tree. There are two ways we could overcome this problem. We could keep the arrangement of nodes as we have them in Figure 11 and multiply all distances by some fixed value. A quick check of all possible paths shows that the paths in Figure 10 in the worse shape are the ones between front and rear legs on one side, which must be separated by 4 units. If we simply double all distances so that the tail node is separated from the hip node by 4 units, the leg nodes are separated from the hip nodes by 2 units and so forth, then the leg nodes are separated from one another by the required 4 units, and in fact all pairs of nodes meet their minimum separation requirement. 1.30 R..1. L4NG
But doing this means making our tree unit smaller. After scaling down the tree by a factor of 2, the diagonal of the square is 10 tree units long, so that the scale of the crease pattern has fallen from 0.28 to 0.14. But this is awfully wasteful! Although now the path between front and rear legs are equal to their minimum length, all the other paths are longer than they have to be, which means that we’ll be wadding up excess paper to get each point down to its proper length. What we really ought to do is to increage the size of a unit and rearrange the nodes to allow a larger unit size. Since the lizard is a pretty simple shape, it’s easy to see that we can improve the design by moving the nodes corresponding to the feet out to the edges of the square and, since the tail is longer than the head, moving the rear feet farther from the tail corner than the front feet are from the head. Figure 14 shows an optimum distribution of nodes for the lizard. Even for the optimum, most of the paths turn out to be longer than their allowed minima. (In fact, you have considerably freedom in your placement of the nodes corresponding to the hips and shoulders.) The paths that turn out to be the limiting paths - which I call the ’critical paths’ - are the ones from head to forelegs, from forelegs to rear legs, and from rear legs to tail. I’ve made the critical paths heavy in Figure 7. If you work out the geometry (or just draw it to scale and measure), you find that one tree unit is (vr3-1-5)/3 times the side of the square, which works out to a scale of 0.189 - smaller than what you get from Figure 10, but now it’s foldable. Since we know the scale, we can also figure out how big the final model will be: the base we fold from this pattern will make the length of each leg about one-fifth the side of the square, and from nose to tail, the model will be almost as long as the side of the square.
foot front foot 2
foot foot
14: An optimum distribution of nodes on the square for the lizard tree. MA THEhfA TICA]_, ALGORITHMS FOR ORIGAMI DESIGN 131
Now, we have a crease pattern that satisfies all of the criteria we have set so far. Can it really be folded into a base of the proportions we have set? The answer is yes, it can. Figure 15 shows all the creases for one of many possible ways to fold this into a base. While the base itself is probably what you would fold a lizard from, by repeated box-pleated sinks, you can transform the base into a pretty fair approximation of the stick figure, the tree, itself. Of course, you don’t need to go that far. The useful result of this little exercise is the second figure in Figure 15; the base, which we construct as a byproduct bf folding the tree.
(b) front foot ,head front foot tatl rear foot front ~z-_.l foot (c) rear 1 2 ~ "~head
z tail - rear~" " foot foot
Figure 15: (a) Crease pattern for the lizard base. (b) The lizard base. (c) By repeatedly sinking the lizard base, you can even make a close approximation of the tree. In fact, just as was the case with the circle method, it can be shown that any pattern of nodes you construct according to the rules of the tree method is foldable into the stick figure you started from. What we have here is a rudimentary algorithm for designing a large class of origami models. Any subject that can be approximated by a tree diagram - a stick figure - can be designed by (1) identifying the nodes, paths, and their lengths on the tree; (2) Laying out the nodes on a square such that the distances between any pair of nodes is larger than the corresponding path on the tree; (3) Folding the resulting pattern of nodes into a base. I don’t wish to gloss over the great difficulty in step 3, of course. This algorithm gives a skeletal crease pattern, not a folding sequence. That you still have to lind yourself. However, the search is always easier if you know for certain that the answer exists. R. J. LANG
3 SYMMETRY AND THE TREE METHOD
If getting the right number and length of flaps were all that was needed to make a successful origami model, then the tree method algorithm as described above would be sufficient to do all origami design. However, nothing is perfect, and there are some limitations to the tree method - some fixable, some not. Although the pattern of nodes you get from the tree method is guaranteed to be foldable into a base with the right number and size of flaps, there is no guarantee that the folding method is (a) symmetric, (b) elegant, or (c) obvious. In fact, in many cases, the tree method, applied blindly, leads to wild, asymmetric patterns that don’t give very nice-looking models. For example, if we take a simple 5-node tree corresponding to a 4-legged creature, the most efficient skeleton puts the four nodes at the four corners of the square. This crease pattern can be folded into a base in several ways, but one of the most elegant is the traditional Bird Base as illustrated in Figure 16. The Bird Base is nicely symmetric, both from front-to-back and from side-to-side, and thus it can be used to fold animals that are bilaterally symmetric - animals whose left and right halves are mirror images of one another. 2 b a b
2 e de Figure 16: Tree, node pattern, and base for a base with four equal-length points emanating from a common point. Now, suppose we want the same four flaps in our subject, but we want them separated by a segment (a body?) one unit long, as illustrated in Figure 17. If you place the nodes on a square and start figuring out path lengths, you are likely to find the pattern shown in Figure 17 (or one very similar) as an efficient placement of nodes. The pattern in Figure 17 has a scale [scale=(1 unit of tree)/(side of square)] of 1/3=0.333. However, this isn’t the most efficient node pattern possible. The most efficient distribution of nodes is shown in Figure 18; it has a scale of (vri~-2)/5=0.348, about 5% larger. This means that the base you fold from MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 133
Figure 18 would be 5% larger than the base folded from Figure 17 for the same size square, and thus each flap would contain proportionately less paper.
a e
/ ~c 1 d~ d
bq
Figure 17: Tree and node pattern for a base with two pairs of equal-length points separated by a one-unit segment.
d
f
b Figure 18: The most efficient node pattern for the tree of Figure 17. Efficiency is not the same as elegance, however. Although Figure 18 is more efficient, Figure 17 makes a better origami animal (that is, if there is an animal out there with four appendages but no head or tail). Most animals have bilateral symmetry, but the crease pattern in Figure 18 does not. Figure 18 would be very difficult to fold into a symmetric-appearing base. Now, for most origami designers, starting from a bilaterally-symmetric crease pattern to fold a bilaterally-symmetric base would come instinctively. But if we are constructing an algorithm for design, then we must build symmetry into the algorithm explicitly. Although the most efficient crease pattern for the lizard came out symmetric, we will not always be so lucky. I have found tb.at the node patterns for more complicated subjects - spiders, sea urchins, mating mosquitoes - usually are asymmetric and inelegant. Forcing symmetry of the crease pattern is an absolute requirement for an elegant design. Of course, there are exceptions to every rule. Sometimes you are actually better off using an asymmetric base, either because your subject is inherently asymmetric, such as a fiddler crab, or because the asymmetry can be disguised, resulting in a larger overall base. The exception probes the rule, but by definition is less common; so we need to enforce symmetry in most cases. The symmetry shortfall is remedied by recognizing two special types of nodes - those that come in mirror-image pairs (like arms, legs, and feet) and those that lie along a symmetry plane (like the head, shoulders, hips, and tail). We will require that for bilaterally symmetric subjects, the nodes corresponding to mirror pairs of appendages must lie symmetrically about a symmetry line of the square. So, for the lizard, we would require that the node corresponding to the left rear foot be the mirror image of the node corresponding to the right rear foot and the node corresponding to the left front foot would be the mirror image of the node corresponding to the right front foot. There are also nodes that correspond to body parts that lie along a line of symmetry of the subject, and we will simply require that these nodes lie on top of a line of symmetry of the square. Note that a square has two different types of lines of mirror symmetry - one from corner to corner, and one from edge to edge. For the same tree structure and assignment of symmetry relationships, the two different choices of mirror symmetry will give different node patterns and consequently, different ways of folding the same subject. Figure 15 shows the crease pattern form a lizard when we use a diagonal line of symmetry; Figure 19 shows the crease pattern for a similar lizard, but based on a side-to-side line of symmetry. You might find it interesting to try to work out a folding sequence to transform the crease pattern in Figure 19 into a base.
rear foot 3.5 front foot (a) rear front (b) foot foot tail J -’ head tail, head
rear front foot foot
front rear 3.5 foot foot
19: Tree and node pattern for the lizard with rectangular symmetry. MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN 135
Two features of this crease pattern are worthy of note: one is that two of the corners go unused, which just goes to show you that even simple figure don’t always use the corners. The other feature is that in Figure 19, the path from the front foot to the head is no longer a critical path as it was in Figure 15, which illustrates that as you change the orientation and/or lengths of paths, you need to carefully keep track of which paths are critical paths and which are not, because they can change. I mentioned that the crease pattern you derive from the tree method is guaranteed to be foldable into a base with the same number and length of flaps as the original stick figure tree. However, this property is only true for infinitely thin (zero-width) flaps made with infinitely many creases. When you cut down on the number of creases to some finite number - and the jump from infinity down to six or eight is considerable - and let the width of the flaps become nonzero, you’ll find that some flaps turn out shorter than they were ia the tree. That is, some of the paper that might have gone into ’length’ ends up going into ’width’ instead. This situation happens whenever two flaps lie side-by-side, rather than one atop the other. For example, if we were making a base for a four-legged animal with two legs on the left and two legs on the right and no body in between (like Figure 17, it’s a gedanken animal), we would start with the tree and crease pattern shown in Figure 16, which would lead to a base very like the Bird Base, in which the four flaps come from the four corners of the square. According to our tree, each flap would then be as long as half the side of the original square, and indeed, each of the four flaps of a Bird Base is half as long as the side of the square from which it is folded.
0.35
Figure :ZO: Making two symmetric pairs of flaps from a Bird Base. Now, if we want our legs in two side-by-side pairs, the Bird Base doesn’t quite work; it has two flaps side-by-side and the other two one atop the other. We can transform the four flaps into two pairs of mirror-symmetric flaps by spread-sinking two corners of the Bird Base, as shown in Figure 20. However, when we do this, we get an extra folded edge running across the two flaps that shortens the gap between them. Because the gap has been shortened, the ’free length’ of the flaps has been reduced. Instead of being 0.5 times the side of a square, the flaps turn out to be only 0.354 of the side - a loss of about 30% of their length. 1~ ~. L.ANG
We can lengthen the gap and thus lengthen the flaps with a little more folding. If I put twice as many pleats running to the tips of the flaps, using the folding sequence shown in Figure 21, I can increase their length to 0.42, at the expense of doubling the number of layers, halving the width of the flaps, and adding a whole lot more folding. You can recover more of the gap by using more and more petal folds and narrower and narrower flaps, but only in the limit of infinitely many pleats (and an infinite number of layers) do you recover the full length of the flaps.
0.42I
Figure 21: Making the gap deeper, at the expense of narrowing the flaps and introducing a lot more creases and layers. This situation will always occur when you want two flaps to lie side-by-side, as opposed to one atop the other (note that the two flaps in each pair on one side of the Bird Base are separated from each another by the proper amount). The way to avoid having to do all of this back-and forth folding is for any pair of flaps that will wind up side-by-side, you add a little extra paper between them. In other words, you increase the minimum length associated with the path between that pair of flaps. The amount you must add depends on how many back-and-forth pleats you are willing to tolerate; typically, extending the path by about 40% will do the trick for a single back-and-forth pleat. But, you’re not ready to place paths yet. After you have placed all the nodes and lengthened some paths, you may find that paths that correspond to important creases are oriented at angles close to natural symmetry lines - i.e., at multiples of 30° or 22.5* - but aren’t precisely at those symmetry lines. For esthetic reasons - MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 137 to avoid long, skinny crimps - it may be desirable to require those paths to lie exactly along the natural symmetry lines, even at the expense of a small decrease in efficiency. Thus, some of the nodes will have to be constrained to lie at certain angles with respect to one another. Also for reasons of esthetics, you might want to insure that a particular node lands in a particular place on the square. For example, the most efficient crease pattern might put a particular node in the interior of the paper, whereas you might want the node to land on an edge or corner so that it has fewer layers.
4 A NUMERICAL IMPLEMENTATION OF THE TREE ALGORITHM By the time you have drawn and labeled the tree, enumerated all of the paths and calculated their lengths, and noted which nodes need to lie symmetrically, and extended some of the paths, and forced some paths to run at certain angles and forced some nodes to lie on the edges of the square and so forth and so on, things can get pretty complicated. The biggest difficulty with applying the tree method of design is keeping track of all of the different paths and the constraints associated with each path. If the tree has N nodes, there are N(N- 1)/2 different paths between pairs of nodes, each of which has a specific minimum length. The lizard, which has 8 nodes, has 28 paths! However, you really only need to keep track of the critical paths - the paths that are at their minimum length - and there are only six of those in the lizard. The problem with complicated models is that as you try out different arrangements of the nodes, some paths stop being critical paths and other paths become critical paths. In addition, you must also keep track of which nodes lie symmetrically with respect to one another, and which paths you have extended the length of. This can get very difficult for a person to keep track of, and it is very easy to work out a design, only to discover that you overlooked a critical path somewhere in the criss- crossing network of paths that doesn’t meet its minimum length requirement and throws your entire design off. Ah, but what is difficult for a person to keep track of is a snap for a computer. The tree method of origami design that I have described above can be mathematically formalized and implemented as a computer algorithm. The algorithm is simple: You do this to define the shape: (1) Define a set of N nodes and the N-1 branches that connect them. Define the lengths of all of the branches. Give each node a set of coordinates on a square. (This is the setup input to the computer program.) (2) Define a quantity called the ’scale’, which is the length of a 1-unit branch on the square. Assign each node a pair of coordinates on the square. (This is an initial guess at the tree structure, which can be arbitrarily chosen.) You do this to find the crease pattern: (2) Construct all of the N(N- 1)/2 possible paths between pairs of nodes and their lengths. (3) Maximize the scale subject to the following constraints: (a) The length of each path on the square must be greater than or equal to its length measured along the tree.
(b) The coordinates of all of the nodes must lie within the square.
(c) Nodes that lie on a symmetry line of the tree must lie on a symmetry line of the square. (d) Nodes that are mirror-image pairs with each other on the tree must be mirror images of each other with respect to the symmetry line of the square.
(e) Paths close to natural symmetry lines of the square must lie exactly at the angles of those symmetry lines.
To computerize this algorithm, the verbal description of the algorithm needs to be converted to mathematical equations. I define xi and Yi as the Cartesian coordinates of the ith node, define lij as the length (in units) of the path between nodes i andj as measured along the tree, define m as my scale factor, and define the coordinates of my square as lying between -1 and +1 on each axis. The problem becomes: Maximize rn over the variables xi andyi subject to the constraints:
(a) (xi -x~’)2 + (Yi _y~)2 >_m2[2ij foralli, j (b) xi<_1, xi>_ -1, ~_<1, yi>_-i for alli (c) YiCOSa - ~sin. = 0, for each node i that lies on a symmetry line at angle with respect to the x axis. + (d) (xi - xj)co,sot - (Yi - Yj )sin. = 0, (x~ + ~j. )sinot + (Yi Y~ ) cos. = 0 for two nodes i and j that are symmetric about a line at angle, with respect to the x axis. (e) (Yi - y~)cosa - (x. - :~ )sin. = 0 for a path between two nodes i andj that lies at angle a with respect to thex axis. MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 139
This is a problem of constrained optimization. In fact, it would be a fairly simple problem in linear programming, if it weren’t for the presence of the quadratic terms in letter (a) above. There is no algebraic solution to this system of equations, but the numerical techniques for the solution of such systems have been known for years. In the early 1970s, several workers contributed their efforts and subsequently their names to the development of a powerful technique for solving general nonlinear constrained optimizations, known as the Powell-Hestenes-Rockafeller Augmented Lagrangian Multiplier algorithm (Rockafeller, 1973). Several months ago, I wrote a computer code that takes a description of a tree, constructs the equations in (a)-(e) above, solves for a local optimum, and displays the resulting skeletal crease pattern. The program is called TreeMaker, and it performs quite well. I’ve now used TreeMaker to design a number of non-trivial models, as well as to work out the examples presented in this series. Now admittedly, the lizard that I used throughout this series was a pretty simple model, and using a computer program to design it might seem like overkill. While simple models can be worked out entirely by hand, I have found that for complex subjects, the computer can find and evaluate efficient crease patterns much faster than I can by hand., including crease patterns that are not obvious to the unaided eye. In the next section, I’ll illustrate the design process for a challenging model - an insect with legs, jaws, and antennae - and end with the final folding sequence for the design.
5 A CASE STUDY IN ORIGAMI DESIGN I’ve always had a fascination for insects as origami subjects. For many years, most insects were considered too difficult to be realized in any but the most stylized form (or were realized with cuts and/or multiple sheets of paper). However, with the revolutions in modern origami design, insects have become commonplace. In origami, a six-legged, two-winged insect is no longer a challenge; it must have jaws, antenna, multiple color-changed wings, or something else to make it rise above the merely ordinary. Insects, with their skinny, jointed, multi-pointed bodies, are perfect candidates for the tree method - and thus, for computerized design. The insect I chose to make for this article was a stag beetle, a beetle with six legs, jaws, and antennae, for a total of ten flaps. I decided to make the jaws and antennae somewhat shorter than the legs and to have them emanate from the head. I put the legs emanating from the thorax along with the abdomen, and make the abdomen somewhat longer than the legs, to give myself some extra paper to make a round body and/or pleated wings. This choice of subject also gives me an opportunity to compare man and machine; I already invented a stag beetle several years ago (it is included in my upcoming book, Origami lnsects (Lang, 1994), and is shown in Figure 22. It will be interesting to see if TreeMaker can come up with a better design. 140 R. J. LANG
Figure 22: An origami stag beetle. The first step in a design is to take the subject we are working from and convert it into a stick figure - the tree. Figure 23 shows a drawing of a real stag beetle and the tree that represents it. I have some choice in how I construct the tree. I’ve already specified that I want jaws and antennae, and of course I need six legs coming from the thorax. If I were being really daring, I could put separate wings on as well, but since beetles almost always keep their wings and elytra (forewings) folded over their body, we’ll allocate a single flap to represent the wings and abdomen, and will plan on trying to suggest both features with pleats, crimps, and other detail folds.
Figure 23: A real stag beetle and its equivalent stick figure, or ’tree’. Now we need to assign lengths to the branches of the tree. On the real beetle, although each member of a pair of appendages are the same length, each pair - jaws, antennae, forelegs, midlegs, hind legs - is a slightly different length. When we MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 141 make an origami base, however, if all of the flaps are different lengths, then they’ll all be slightly different widths where they join the body and each other. These differences translate into lots of little crimps (if you’re careful) or little bodges (if you’re not) to make the model lie fiat; there will also undoubtedly be lots of misaligned edges. A base with lots of crimps, bodges, and misaligned edges looks messy, and a messy base makes a messy subject and a messy folding sequence. To keep the base and the folding sequence neat, it helps to make edges line up with each other whenever possible. Thus, we will stipulate that flaps that are approximately the same size on the subject should be exactly the same length on the base. For our stag beetle, we’ll make the six legs all the same length and make the jaws and antenna the same length as each other but a little shorter than the legs. We still need to quantify the lengths of the appendages. If we are trying to make an exact copy of the beetle in the photo, we could measure each leg, jaw, and antenna in the photo and assign that length to the corresponding branch of the tree, but that is a rather brute-force approach. Besides, it overlooks something: it is not enough just to find some folding sequence to make the base; we want to find an elegant folding sequence. ’Elegance’ is a hard-to-define concept; it’s easier to describe than to define. An elegant folding sequence is one in which all the folds flow naturally from one step to the next and the edges and creases line up with one another. An elegant model is one in which the lines are simple and clean. Elegant folding sequences arise when the creases and the base arise from natural symmetries of the paper. While TreeMaker will find the most efficient crease pattern for any given tree, it is up to the designer to pick the tree that best exploits the natural symmetries of the paper. And one of the ways in which natural symmetries appear is in certain combinations of distances and lengths. Most origami - and in my opinion, the most elegant folding sequences - arise from exploitation of the symmetries associated with an angle of 22.5°, which is 1/16 of a unit circle. This angle (and multiples thereof) show up repeatedly in elegant origami bases, as illustrated in Figure 24 in the four ’classic’ bases (the Kite, Fish, Bird, and Frog Bases). Along with this ubiquitous angle, there are ubiquitous distances, which are various algebraic combinations of integers and the number d;2. If you unfold a model based on the symmetries of 22.5° and calculate the lengths of the major creases, you’ll find that most of the distances work out to simple combinations of 1, 2, and d2, as shown in Figure 25. Figure 24: The four classic bases -- Kite, Fish, Bird, and Frog base. The angle between any two creases is a multiple of 22.5°.
Figure 2,5: Typical distances to be found in crease patterns based on the 22.5° symmetry. Since in symmetric, elegant crease patterns the same distances show up over and over, if we want our crease pattern to be symmetric and elegant, it makes sense to MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 143 use the distances that we know will already be present. Conversely, if we use strange values for distances, values that don’t correspond to 22.5° symmetries, it’s going to be really hard to find a folding method that results in an elegant base. We are more likely to find a symmetric base if we express all the dimensions in terms of ’symmetric’ distances. This is really not so hard to do, because there are a lot of symmetric distances to choose from. It turns out that for any ideal distance, there is an symmetric distance with nearly the same value. For example, Figure 26 displays the algebraic form of all the distances shown in Figure 25, along with their decimal value and for comparison, a line whose length is proportional to that value. No matter what length a given flap or appendage is, you can find a symmetric distance that differs from the target by only a few percent. Figure 26 is an incomplete list of symmetric distances associated with the 22.5° symmetries - there is a whole other family of distances associated with multiples of 15° - but you can extend this list to larger and smaller values simply by multiplying or dividing by factors of 2. Any numerical value you choose lies fairly close to one of these significant lengths.
Algebraic Decimal Linear ~ 1.414 (1+~/2)/2 1.207 1 1.000 (3-~/~)/2 0.793 (1/~ 0.707 2-’~ 0.586 1/2 0.500 "~-1 0.414 1-1/~- 0.293 (~--1)/2 0.207 Figure 26: Distanc~ found in the eightfold (22.5°) symmetry. The first column is the algebraic formula for a length; the second is its decimal value; and the third shows a line segment proportional to the length. So for our origami stag beetle, we will choose flap lengths that correspond to interesting lengths. We will define our legs to be 1 unit long each; the jaws and antenna are slightly smaller, so we’ll choose a length of 1/v~=0.707 for those. The abdomen should be somewhat longer than the legs, to give us some extra paper for crimps and pleats to form the wings and/or body segments; we’ll make it d’2= 1.414 units long. Finally, we need a short segment between the thorax and the head, so that the jaws and antennae don’t come from the same part of the model as the six legs (otherwise, we could just use the circle method). This short segment will be 144 R.J. LANG d-2-1=0.414 units long. We now have completely defined the target tree for our design efforts. Now that we have defined our tree, we could start enumerating paths and calculating their lengths, but since our tree has 13 nodes, there are 78 possible paths between nodes that must be considered, which is a lot to try to keep track of. So I will turn now to describe the use of my TreeMaker computer program. TreeMaker has a point-and-click user interface that allows you to enter a tree structure by clicking on a square to create nodes and branches, clicking and dragging to shift them around, and double-clicking on nodes and branches to change their properties. The tree as I entered it is shown in Figure 27. TreeMaker allows you to name nodes with the part of the subject they correspond to and displays the lengths of branches, so I’ve shown those numbers and names for clarity.
left jaw right jaw fight left antenna fight
fight left back back 1.4 t4 leg leg
, abdomen
Figure 27: The tree I~or a stag beetle, superimposed over a square. The numbers are the desired lengths of the branches; the names identify which part of the subject corresponds to each node. To give you an idea of the complexity of this model that has 13 nodes and 78 paths, I’ve shown all 78 paths in Figure 28, each represented by a straight line. Every line corresponds to a constraint on the distance between two nodes. Now, if you or I were working this design out by hand, we would figure out which paths are the important ones and which could be ignored, but the computer is not so clever, and needs to consider them all. MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 145
Figure 28: Every possible path between two nodes is represented by a line. There are 78 total paths between the 13 nodes. Once we have set up nodes and branches, the computer calculates all possible paths and the minimum length of each path. To find a possible arrangement of nodes, the computer needs to optimize (find the maximum value of) the scale of the tree while insuring that all paths meet their minimum distance constraints. At this point, we could just tell the computer to go through its optimization process. TreeMaker arrives at the following arrangement of nodes:
left aw fight jaw right front leg
right 5 middle leg
left front leg / / /
abdomen ~8 Figure 29: A node pattern that satisfies all of the path constraints for the tree. The circles identify the terminal nodes of the tree. The scale is 0.2012. 146
This first go-around was certainly less than successful. One back leg is a middle flap; the other is an edge flap. One jaw is a corner flap; the other is an edge flap. The head is twisted way around to the right. Folding this crease pattern into a base would be a nightmare. About the only positive thing I can say is that the scale - the length of one unit compared to the size of the square - is a very respectable 0.2012, meaning that the length of a leg is about 1/5 of the length of the side of the square. Nevertheless, there is no symmetry, no rhyme or reason, and most importantly, no easy or obvious way to fold this crease pattern into a base! So the first foray into computerized design is without doubt, a failure. Well, maybe not quite a complete failure. We haven’t yet imposed any symmetry requirements, and lack of symmetry is the main problem with the node pattern in Figure 29. We need the legs, jaws and antennae to be mirror images of each other with respect to a symmetry line of the square - and they aren’t in Figure 29. We also need the head, thorax, and abdomen to lie directly on top of the symmetry line, since they lie on the symmetry line of the subject. Since the shape in Figure 28 is oriented roughly along the diagonal, let’s choose the diagonal of the square to be our line of symmetry. TreeMaker allows you to select either a diagonal or a rectangular line of mirror symmetry for the square, to constrain individual nodes to lie on the symmetry line and to force pairs of nodes to be mirror images of each other about the symmetry line. So, when we establish these symmetry requirements and restart the optimization, we arrive at the node pattern shown in Figure 30. This pattern has a scale of 0.1853. The scaled has decreased by about 8%, which means that the base folded from this crease pattern will be about 8% smaller than the base folded from the crease pattern in Figure 28. But the pattern in Figure 30 is more likely to be foldable into a symmetric base and thus a symmetric model, and so the small decrease in scale is an acceptable price to pay. 13 5
4 2 Figure 30: Optimized node pattern for the stag beetle with mirror symmet~ invoked. The scale is 0.1853. MA THEMA TICAL ALGORITHMS FOR ORIGAMI DESIGN 147
There is one other chore before us; if we wish our finished base to be flattened from top to bottom (as is the beetle in Figure 22), then the legs come in side-by- side pairs. To avoid having to make infinitely many pleats between them, we need to allocate some extra paper by extending the paths between mirror pairs of legs and iaws. To keep the back-and-forth folding to a minimum, I’d like to use a single back-and-forth pleat between the legs, such as is accomplished in the Frog Base by petal-folding an edge. For this petal fold, the distance between the two flaps must be ::~ times as long as the minimum path, so between side-by-side pairs of legs, I’ll need to extend each path by a factor vr2. Just as you could edit individual nodes, TreeMaker lets you edit individual paths and to increase the minimum separation between node pairs. We do this for each pair of mirror flaps, increasing the separation between the jaws from 1.414 to 2.000, between the antennae from 1.414 to 2.000, and between pairs of legs from 2.000 to 2.828. After bumping up these paths, I re-optimize. The scale has decreased from 0.1853 to 0.1775, which is still not a large decrease, but the resulting pattern, shown in Figure 31 is now closer to something that can conceivably be folded into an elegant base.
Figure 31: Opttmized node pattern for the stage beetle with mirror symmetry and path extensions between side-by-side flaps. The scale is 0.1775. But wait - there’s more. The crease pattern shown in Figure 31 can in principle be folded into a beetle, but I doubt that the folding sequence is very clean or elegant" because the major crease lines are running at irregular angles. Not only do we want our distances to be symmetric; we want the angles of major creases also to lie at multiples of 22.5° whenever possible. Thus, we need to set some more constraints that force paths to lie at multiples of 22.5°. We can’t force all the paths to the same angles, though; just the ones that correspond to major creases of the base. Forcing paths to lie at particular angles is another capability of TreeMaker. For any given path, there are 8 possible multiples of 22.5° to set the angle to, but you might guess that you’ll perturb the crease pattern the least by looking for paths between adjacent nodes that are nearly at multiples of 22.5° and setting them to the nearest multiple. If I do this for the paths marked in Figure 31 and again re-optimize, I find the following crease pattern in Figure 32, which is more nicely symmetric, and which has a scale of 0.1726.
Figure 32: Optimized node pattern for the stag beetle with mirror symmetry, path extensions, and angular constraints. The scale is 0.1726. It’s interesting; each time we add some more constraints, the scale, and thus the size of the model, gets reduced. That is, we are trading off folding efficiency for folding elegance. This is an esthetic judgment, and requires the active participation of the designer. Computer programs like TreeMaker can simplify aspects of origami design, but they are no substitute for good design sense. In fact, it might even be considered a bit of a stretch to .say that TreeMaker is ’designing’ the base. It isn’t so much creating a useful arrangement of nodes as it is eliminating the enormous number of useless ones. For 13 nodes, there are 27 degrees of freedom in the placement of the nodes on the square (two coordinates for each node plus the scale~. A given configuration of nodes and scale can be described by a single point in a 27-dimensional space. The various constraints on path distances and angles can be written as equations that subdivide the 27- dimensional space into spheroidal regions of feasibility (for inequalities) or spheroidal surfaces (for equalities). The combination of all of the equations define an incredibly convoluted hyperdimensional blob of feasible space over which the scale - the merit function - varies in value, and all we are doing is looking for MATHEMATICAL ALGORITHMS FOR ORIGAMI DESIGN 149 those flaps with the maximum value of the scale. The convolution of the surface implies that there are numerous local minima on the surface and there is no guaranteed routine for finding a global minimum. Just because we found the best configuration of nodes from a given starting point, a different starting point might give us an even more efficient arrangement of nodes! So, one new starting point we might try is to move the abdomen into the center of the paper, rather than using a corner for the abdomen. This would have the effect of moving the legs closer to the corner opposite the head, but exactly what else is not clear until we re-optimize TreeMaker. This starting point gives the crease pattern shown in Figure 33, which has a slightly larger scale of 0.1745, and would give a barely larger base than the pattern of Figure 22. Significantly, the crease pattern to transform Figure 33 into a base would in all likelihood be entirely different from the pattern for Figure 22! Thus, simply by starting from a different initial configuration, you can discover entirely new ways of folding existing subjects simply by moving the initial distribution of nodes around.
Flgur~ 33: Node pattern with the abdomen in the center. The scale is 0.1745. Figure 33 has one other nice feature compared to Figure 32; all of the legs come out as edge flaps in Figure 33, whereas two of the legs are middle flaps in Figure 32. For an insect, we want the legs to be as thin as possible, and the extra thickness in a middle flap might be something to be avoided. (Then again, it might not; I know a lot of good insect designs that use middle flaps for legs.) If we start with Figure 32, force all of the legs to lie on the boundary of the square and set angles that are near 45° to exactly 45°, we get the crease pattern shown in Figure 34, which has a scale of 0.1715, still smaller than any of the preceding patterns, but acceptable. 1~o ~ ]. L4NG
Figure 34: Node pattern for the stag beetle with all legs constrained to lie on the edge of the square. The scale is 0.1715. We might also wish the jaws to be edge flaps and allow the legs to be middle flaps. This constraint, plus a few angular constraints gives the very symmetric crease pattern shown in Figure 35. Not only are all the major creases at 45°, but the center of the paper is an intersection of two important creases. The scale has decreased further, to 0.167, almost precisely 1/6 of the side of the square.
Figure 35: (Left) Node pattern for the stag beetle with all jaws constrained to lie on the edge and with the antenna and two legs on each side constrained to a 45* line. The scale is 0.1665. (Right) Resulting stag beetle. MATHEMATICAL ALGORITHMS FOR OR1GAMI DESIGN 151
In fact, to my taste, Figure 35 is too symmetric. For years, origami designs were made from bases such as the Bird and Frog base that are four-fold symmetric about the center of the square, which always resulted in a single thick flap somewhere in the model - a waste of paper, and a waste of symmetry. Only in rare circumstance does a four-fold symmetric base make a good match to a two-fold symmetric subject. The crease pattern shown in Figure 35 can be folded into the rather stubby stag beetle shown in Figure 35 - you might wish to try to work out a folding sequence - but the sequence I found is rather predictable, and is four-fold symmetric for much of the sequence. Of course, we could try something completely different by trying the alternate line of symmetry - that is, orienting the model along an axis parallel to the side of the square, rather than along the diagonal. Setting this line as our symmetry line, optimizing, and forcing a few major crease lines to verticals results in the following intriguing crease pattern, which clearly has a 30° symmetry built in. It also has a scale of 0.1808, the largest of any of the crease patterns we have examined so far.
Figure 36: Node pattern for the stag beetle with rectangular symmetiy and path angle constraints. The scale is 0.1808. Now, all of these crease patterns can be folded into a stag beetle, and the most interesting folding sequence for each is going to be very different from the sequence for any of the other crease patterns. So right here we have the beginnings of five or six new models. Of course, the question of how you develop a folding sequence that takes a crease pattern to a base is no small matter, and is in fact worthy of an entire set of articles in its own right. Setting aside the issue of how you fold up a crease pattern into the base, though, you see the boundless possibilities that you can avail yourself up even from a single tree structure! I’ve developed stag R. J. I.~4~G beetles from about half of the crease patterns I’ve shown here. I’ll close this article with my favorite, which is derived from Figure 34, and is illustrated in Figure 37.
Figure 37: Stag beetles, according to (left) human, (right) computer. You might wish to try to find your own folding sequence for one of these crease patterns. Better yet - change some dimensions of the tree and start over yourself! Give the beetle extra-long jaws this time, or get rid of the antennae, or add a pair of open wings. Whatever you do, you are bound to find new and undiscovered models lurking in the paper. Whether you use TreeMaker, write your own computer program, or work out your design by hand, mathematical origami design is a powerful technique that can lead you to new vistas of origami design.
REFERENCES:
Engel, P. (1989) Folding the Universe, New York: Random House. Gardner, M. (1992) FractalMusic, Hypercards, andMore, New York: Freeman. Kasahara, IC (1988) Origami Omnibus, New York: Japan~ Publications. Lang, R. J. 1994) Origami Insects, New York: Dover Publications, (to be published.) Montroll, J. (1985) Animal Origami for the Enthusiast, New York: Dover. Rockafeiler, R. T. (1973) A dual approach to solving nonlinear programming problems by unconstrained optiraization, MathematicalProgramming, 5, 354-373. Symmetry: Culture and Science Vot 5, No. 2, 153-165, 1994
MATHEMATICAL REMARKS ABOUT ORIGAMI BASES Jacques Justin
LITP INSTITUT BLAISE PASCAL 4 place Jussieu, 75252 Paris CEDEX 05, France*
FOREWORD This paper is based, with only minor modifications, adjunctions and suppressions, on an handwritten manuscript of 1982 which was sent at this time to some folders interested in mathematics. This led to correspondence with at least two of them: professors Kodi Husimi and James Sakoda. The first one has seemingly considered a particular case of what I call a ’perfect bird base’ in his book, in Japanese, Origami no Kikagaku (The Geometry of Origami, Tokyo: Nippon Hyoronsha, 1979). The second one has made use of perfect bird bases in his magazine Mac Origami and in his book Origami Flowers Arrangement (published by himself, 1992) and has devised a good approximate method to find the ’origin’ of the perfect bird base.
1 INTRODUCTION
When we fold the traditional bird base (Orizuru K/so, in Japanese) we find that it has agreeable properties, for instance the flaps are easy to move. But try with a rectangle: the result is less satisfactory. We shall examine the possibility of folding ’good’ bird bases with polygons of various shapes. Afterwards we shall try a similar approach for the frog and the windmill bases.
2 DEFINITIONS
The polygons to be studied need not be convex. Let Ar42...An (Figure 1) be a polygon, and O a point inside it such that the triangles AIOA2 and so on do not
* Mailing address: J. Justin, 19 rue de Bagneux, 92330 Sceaux, France. J. JUSTIN 154 overlap each other (the polygon is said to be star-shaped from O; if it is concave, some points O do not work). We call preliminary base the result of bringing OA1, OA2 .... , OAn together by mountain folds, and origin of the base the point O (Figure 2). Of course, when flattening.~_he model_valley folds must appear, say OMI, OM2, ... and we have (Figure 1) AIOM1 = M~IO~2 and so on. In the preliminary base, O and the verticesA~,A2 .... lie on a line, and the flaps like OA2MIA~O can be book- folded with that line as a hinge. Now, let us make a reverse fold on each flap, with the condition that the creases pass through the corresponding vertices, for instance through A1 and A2 for the flap OA2MIAIO. We shall call the result a bird base (Figure 3). See Figure 4 for the creases on the unfolded paper, and remark that the sides ,4./1i+~ of the polygon have been folded at points Ei that differ, generally, from the Mi’s (for ease of notation we consider that ,4n+1 = ,41 and so on). The bird base retains the above-mentioned property of the preliminary base, that O and the Ai’s lie on the same line A, which is an axis of rotation for the flaps like OPv4~A20. Now, let us say:
Figures 1-3 Definition: A bird base (Figure 5) is perfect if: (a) the triangular flap PIA2P2 can be bookfolded with P1P2 as a hinge, and so on for the other flaps. (b) when P1A2P 2 has been folded ’downward; A2 is in a new position A’ which lies on A and so on. (c) the flaps like P1A2P 2 consist everywhere of, exact(V, two layers of paper, that is the triangles A2P1E1 and A2P2E2 of Figure 4 are adjacent without gap or overlap. MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 155
Of course, such properties are among the good ones of the standard bird base, some others being let aside because they are too specific of the square.
\ /
\ \ . \ ’!
F~ures5~ 156 Z JUSTIN
3 PERFECT BIRD BASES
Figure 6 shows the hidden reverse folds. By property (c), the points h2, El, E2 lie on the same line. Let H2 be its intersection with PIP2. By property (a), E1 and E2 m~t be on...the side of P1P2 which do~ not con_tain A2. So we have: P1E1A2 <_ P1H2,,I2 (triangle P1E1H2), and P2E2A2 _< P2/-/2A2, hence (1) with equality only !f E1 and_~E are H2. Adding up the n inequalities like (1) and remarking that P.~.r,li + P.rE.t,4i+l = ~r, we get: n~r _< nzr. That inequality being a true equality, (1) must be also an equality. Hence E1 and E2 coincide with H2. Now, fold the flap P~A2P2 downward, giving PtA’P2. The sideA1A2 of the polygon lies now at the position A1H2Ao. But it is now completely unfolded, and so H2 lies on the (straight) line A1A’, that is on A by property (b). Of course, also, PIP2 is perpendicular to A2H2 at H2.
Figure 7 Now, let us look at Figure 7 which shows the above results on a part of the unfolded paper. We see that P1, for instance, is the incenter of the triangle OAIA2, that is the intersection of its inner bisectors. The circle inscribed in OA1A2 has P1 for center and touches the sides of the triangle at H1, El, H2. Actually that is exactly the consequence of folding OA1A2 by a rabbit ear procedure, but here we have the important fact that the circles inscribed in OAIA2 and OA2A3 touch 0.42 at the same point H2. We are now in a position to state: MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 157
Theorem: Given a polygon A1A2...An, (a) there exists a perfect bird base with origin 0 if and only if there exist positive numbers r and e1 ..... en such that ei+ei+l=AiAi+l for 1 OAi - OAi+2 = Ai+1 Ai - Ai+1 Ai+2 .for 1 _< i <_ n (3) (c) the polygon has at most one perfect bird base. Proof of (a): If the base with origin O is perfect, Figure 7 shows that the circle inscribed in OAr42 touches the sides at El, HI, H2. So we have OH1 = OH2 = OH3 = ... = r, andA1H1 = AIE1,A2E1 = A2/-/2 =A2E2. So, putting AiHi = ei for 1 -< i _< n, we have the relations (2). Reciprocally, if the relations (2) are satisfied, the three circles (O; r), (A1; el), (A2; e2) (where the first letter is the center and the second one is the radius) touch each other at points H"1 on OA1,/-/’2 on OA2, E’1 on AIA2. It is easy to see that these points are on the circle inscribed in OA1A2, and so, we have the configuration for a perfect bird base as given in Figure 7. Proof of (b): If the relations (2) hold, we have: OAi.- O.~/+2 = = ei - ~’+2 = (~ + ~+1) - (ei+2 + ei+l ) = Ai+lAi -Ai+I Ai+2, so relations (3) are satisfied. Reciprocally, suppose that (3) holds. Put (OA1 + 0.,42 - - A1A2)/’2 = rl, (OA2 + OA3 - A2A3 )/2 = ~, and so on. Then ~i = ~ .... = ~ is an immediate consequence of (3). Let r be the common value of the ri and put + ei = OAi - r. Then, for instance: e1 + e2 = OA1 + OA2 - 2r = OA 1 OA 2 - - (OA1 + OA2 - A1A2 ) = A1A2. So r and the ei’s satisfy (2). Proof of (c): Suppose that O and another point U are the origins of perfect bird bases. Let X be the perpendicular bisector of OU. As 0 and U are in the interior of the polygon A1A2..dln, there is at least one vertice that lies in each of the open (that is not containing X) half-planes (~2 ; O) and (~; U). On the other hand, the relations (3) in the above theorem give: OAi - O/1/+2 = Ai+IA/ -Ai+lAi+2 = = UAi - UAi+2, so we have: OAi - U,,~ = OAi+2 - UA~+2 . ThenA~ andA~+2 are on the same side ofX. So, A1,A3,A5 .... are in one of the half-planes (X; O), (X; U) and A2, A4, A6 .... are in the other one. So ~ crosses every side of the polygon. If this one is convex, this is impossible because S crosses the border at two points in this case. If the polygon is concave, it must have a re-entrant angle, likeAIA243 in Figure 8. But then, O and U must lie in the angle ~ because OAi, OA2, OA3, UA1, UA2, UA3, are in the interior of the polygon. ThenA2,A1, orA2,A3 are on the same side of X, a contradiction (for brevity’s sake, some details have been omitted). 158 .I. ,JUSTIN So there is at most one perfect bird base for any polygon. We propose, if it exists, to call its origin the ’point of Loiseau’ of the polygon (L’oiseau: French words for: the bird). Figures 8-9 Corollary 1: For any triangle there is exactty one perfect bird base. Its origin can be constructed with ruler and compasses. Proof: In Figure 9 we have e1 + e2 = A1A2, and so on. Then El, E2, E3 are the points where the circle inscribed in AIA2A3 touches the sides. Describe the circle (hl; el), that is the circle with center A1 and radius el, and in the same way, the circles (A2; e2), (A3; e3). Then by the relations ei + r = OAi, 0 must be the center of a circle that touches the three preceding ones. Therefore, O can be obtained by elementary geometry. Here is one possible construction (deduced from inversions with centers El). Let G1 on the line A1A2 be such that (A1; A2; El; G1) is harmonic. Describe the circle ((71; G1E1). It meets the circle (A3; e3) at H3 within A1A2A3. Similarly, construct HI, H2. Then O is the point of intersection of the lines A1H1, A2H2, A~/-/3. When the triangle is isosceles, AtA2 = A1A3 (Figure 10), it is easier to construct the point of Loiseau. Describe the circle with centerA1 and radius IA1A2 - A2A3 I. It meets the medianA~E2 at two points, X and Y. Trace the perpendicular bisector either ofA2XifA1A2 > A2A3, or of AYif not. It meetsA1E2 at O. Corollary 2: The quadrangle AIA2AsA4 has a perfect bird base if and only if A1A2 + A3A4 = A2A~ + A4A1. Then the origin 0 lies at the intersection of two MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 159 branches of hyperbolas. In the case where AIA3 is an axis of symmetry, 0 is the point where the circle inscribed in AIA2A3 touches A1A3. A, Figures 10-12 Proof: If O is the origin of a perfect bird base (Figure 11) we haveAzA1 -A2A3 = OAI -OA3 = A4A1 -A4A3. This impliesAIA2 +A3A4 =A2A3 +A4A1. Reciprocally, if that condition holds, then A2 and A4 lie on the same branch of some hyperbola with focuses A1, A3. Similarly A1 and A3 lie on a branch of hyperbola with focuses A2 and A4. These two curves intersect at some point O within the quadrangle. As O satisfies the conditions (3) of the theorem, it is the origin of a perfect bird base. In the case where hlh3 is an axis of symmetry (Figure 12), the hyperbola passing throughA1 andA3 becomes the lineA1A3, and then, OA1 - OA3 = A2A1 - A2A3 shows that O is the point of contact ofA~A3 with the circle inscribed inAIA2,43. Remark: If the quadrangle is convex the condition on the sides is equivalent to the condition that it is circumscribed to some circle. Corollary 3: (a) If the po~gon A1A2...An, with n even, has a perfect bird base, we must have: AIA2 - A2A3 + A3A4 -...+An_1 An - An A1 =0" (4) (b) If the po~ygon AIA2...An, with n odd, has a perfect bird base with origin O, then 0 can be constructed with rules and compasses. (c) Given a point 0 it is easy to construct irregular polygons with n sides having a perfect b~rd base with origin O. 160 1. JUSTIN Proof: (a) If n is even, the system of equations ei + ei+1 = AiAi+1 (1 _< i -< n) has solutions only if (4) is true. (b) If n is odd, we can find the Ei’s by solving the preceding system for the ei’s. After, the point O can be constructed almost as we did for the triangle. (c) Describe a circle C with center O (Figure 13), then a circle C1 externally tangent to C, then a circle C2 tangent to C and C1, then a circle C3 tangent to C and C2, always externally, and so on; at last Cn tangent to C, Cn_1 and C1. Join together the centers A~, A2 .... , An of the circles. We get a polygon having a perfect bird base with center O. Figure 13 4 FROG BASES LetA1A2...An be a polygon, and Bi a point on the side A.tAi+1 for 1 _< i _< n. We shall say that a bird base for the polygon AIB1A2..-4nBn is a frog base for the polygon A1A2...4n. A frog base is said perfect if the corresponding bird base is perfect. We have the following Theorem: If 0 is the origin of a perfect frog base for a polygon A1A2...An, then B1 is the point where the circle inscribed in AIOA2 touches A1A:~ and so on. Proof: (Figure 14). The relation (3) applied to the associated perfect bird base gives Br41 - BaA2 = OAx - 0.42, so that the circle inscribed in A~ OAz touches A IA2 at B~. MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 161 Figures Corollary: For a rhombus, the center of symmetry is the origin of a perfect frog base. Proof: (Figure 15). With O at the center, choose the Bi’s as above (for instance fold the perfect bird base of the rhombus, then the Bi’s are the Ei’s of Figure 7). Then by symmetry OB1 - OB2 = A2Ba -A2 B2 (= 0) so that O is the origin of a perfect bird base for the octagon AIBIA2..~B4 , that is a perfect frog base for the rhombus. 5 A PROBLEM Figures 16-17 Y. JUSTIN 162 The following property was found when trying to make irregular frog bases. Problem: Let OAB be a triangle and M be an arbitrary point onAB. Let I and J be the incenters of the trianglesAOM and BOM. Show that the circle with diameter IJ meets AB at M (of course) and at a second point, P, which does not depend of M (Figure 16). Hint for a geometrical proof. Project I and J orthogonally on AB at H and K, computeAH andAK and remark that HK and PM have the same middle. Origami solution of the problem. Mountain fold along OI an~d, OJ, then petal fold along IJ, which gives Figure 17. As PB and PA are adjacent, JPI = ~r/2. So P lies on the circle with diameter IJ. But PA + PB = AB, and also PA - PB = OA - OB, so the point P of AB does not depend of M. 6 WINDMILL BASES Figures 18-19 In current terminology of folders the windmill base made from a square is either the windmill itself, or the quadruple preliminary base obtained by squashing the four points of the windmill. Here we use the first meaning. Let .A1A2...An be a convex polygon (Figure 18), O a point in its interior and Mi a point of the side MA THEMA TICAL REMARKS ABOUT ORIGAMI BASES 163 A.tAi+I, for 1 _< i _< n. Let us valley fold the polygon in such a way that all the Mi’s come to O. If we pinch the comers and fold them fiat (Figure 19) we shall call the result a windmill base defined by O and the Mi’s. The main property is that the flap containingA1 can be bookfolded with OD1 as a hinge, and so on. Figure 18 shows that D1D2 is the perpendicular bisector of OM~. The creases D2M~ and D2M2 correspond to the hinge of the flap A2, as they coincide with D20 when the base is folded. Last, when we have flattened the flap A2, an extracrease has appeared, say D2X2, which is the perpendicular bisector ofM1M2. Now we shall say that the windmill base is perfect if, for 1 _< i _< n, the line D-~"i coincides with the line D.rAi. This amounts to saying that the flap AiXiD~O is a triangle, or consists everywhere of two layers. We have the following Theorem: Given a convex po~ygon A1A2...An, the points 0 inside and Mi on the sides AiAi+l,for I <_ i <_ n, de.t-me a perfect windmill base if and on~ if." (1) there exist positive ei’s such that ei + el+1 = AiAi+1 and AiMi.1 = AiMi = ei; (2) 0 lies outside every circle (Ai; ei); (3) 0 lies inside every circle (MiMi+IMj), for I <_ i, j _< n; j ;~ i; j ;~ i+l, where (PQR) denotes the circle circumscribed to the triangle PQR. oO Figure 20 Proof: (a) The conditions are necessary. In the unfolded perfect windmill base (Figure 20), D;yl2 is the perpendicular bisector of M~M2, so A2M~ and A2M2 have the same length, say e2, and the ei’s satisfy (1). NowA2 and O must lie on opposite Y. YUS TIN 164 sides ofDiD2, soA20 > A2M1 = e2, and then conditions (2) are satisfied. Last, the polygon DID2..J)n must be convex with O inside it. So O and Di lie on the same side of D1D2 for i # 1, i # 2. So M~/~ > Di O. But Di is the center of the circle (Mi.lMiO). So M1 lies outside this circle. But if we remark that O and M 1 lie on the same side of Mi.IMi (opposite to Ai) this is equivalent to the fact that O lies inside the circle (Mi_IMiM1). So conditions (3) are satisfied. (b) Reciprocally, if the conditions (1), (2) and (3) are satisfied, then, by reversing the arguments above, we see that the perpendicular bisectors of OM1 and OM2 intersect at some D2 on the inner bisector of A1 ~"~2A3, and that the polygon D1D2..~Dn is convex and contains O. So, by valley folding along D1Dz..J)n we obtain a perfect windmill base. Corollary: lf a convex po~ygon is circumscribed to a circle, then it has infinitely many perfect windmill bases. (Proof left to the reader). Remarks: (a) the conditions (1) can be satisfied if and only if the lengths A.~li+1 satisfy some conditions easy to state (solve ei + ei+1 = Aiai+1 and write that the ei’s are positive). It is also equivalent to say that the polygon A1A2...4n can be deformed by modifying its angles but not its sides, so that it become circumscribed to some circle. Coordinates: A1 (0, 14), A2 (-16, 8) A3 (-8, 0), A4 (8, 0), A5 (16, 8), O (12, 7) Figure (b) when conditions (1) are satisfied, the region corresponding to conditions (2), that is the intersection of the inside ofAy42...4n with the outside of all the circles MATHEMATICAL REMARKS ABOUT ORIGAMI BASES 165 (hi; ei) is not empty. Though rather intuitive, this fact is difficult to prove. Maybe it could be proved by a continuous deformation of the polygon in the way said just above. In any case, it follows easily from the proposition hereafter which can be proved by methods of Graph Theory. Proposition: Consider a convex polygon AIA2...An and real numbers ai associated with its vertices, such that AiA~+1 >_ ai + ai+1 for I < i <_ n (with the convention that an+/ = al). Then there exists at least one vertice Ai such that AiA~ >_ ai + ajfor all i # j. (c) However it is not always possible to satisfy both (2) and (3), or even (3) alone. Try for instance to fold a windmill base defined by the Mi’s and O of Figure 21. You will obtain interesting results, but not exactly as wanted. 7 CONCLUSION Many geometrical problems arise when one tries to generalize the traditional origami bases. We have studied here some natural generalizations. Bases with irregular polygons or with polygons with many sides are probably of little use in Origami. However some may be amusing. For instance fold a kite base from a square, then fold the kite into a perfect bird base, then pull out the two corners of the square that had been folded first. With this ’kited bird base’ you can fold a flapping bird with long neck, short tail, medium-sized wings and small legs. Symmetry: Culture and Science Vot 5, No. 2, 167-177, 1994 EVOLUTION OF ORIGAMI ORGANISMS Jun Maekawa Address: 1-7-5-103 Higashi-Izumi, Komae, Tokyo 201, Japan E-mail: [email protected] Abstract: This article (based my book Viva! Origami (Maekawa, 1983) will show origami design as the tiling work. First, the rules of origami and some of the notions of origami taxonomy will be considered. Furthermore, traditional models will be analyzed; the meaning of ’basic form’ will be discussed; and an introduction to my original designs will be explained. 1 ORIGIN OF THE SYSTEMATIC STRUCTURE In origami, an organism is regarded as a transformed flexible sheet. This sheet is able to be split and fused keeping the distinction between its surface and its reverse side (Kasahara and Takahama). Origami folding begins with a sheet of paper, which is transformed by folds. Of course, there are exceptions in that sometimes several sheets of paper are used, and, on occasions, scissors. However, for the moment, we shall consider the strict and traditional rules of origami. Most of paper-folders have implicit rules. The following are "the five commandments" by Husimi, arranged by Kasahara (1989). These are typical rules of origami. 1. Start with a sheet of square paper. 2. Cutting and gluing are forbidden. 3. Fold model fiat just before its completion. 4. Straight folding is only permitted. 5. When constructing a model, bear in mind the physical quali~ies of paper. I think Husimi made his rules from the character of traditional origami. Most clas- sical origami models (in documents) violate all of the above rules. Models that adhere to the rules have been handed down by tradition, and are not described in any extant document. These models are the result of historical selection over a 1000 years. (I describe ’over a 1000 years’, but there is no established view when origami had its beginnings. This is my guess by the introduction of paper.) It can be compared to natural selection. In this analogy, the most important question to be considered is: What is selection pressure? 168 I.. MAEKAWA Fish (tradition) Crane (tradition) Frog (tradition) ~ Uttle bird (tradition) / Wind mill (tradition) Ff~ure t Figure 1 shows some traditional models and their creases. Their primary character reveals a sense of ’easiness’. An easy model stands the strongest chance of survival. However, it is very difficult to define the meaning of ’easiness’. It has, at least, two meanings: 1. Easy to make. 2. Easy to learn. The former concept relates to technique; the latter to process. Another keyword in traditional origami is ’natural’. Bearing these two key words in mind, we can now explore the rules of origami in more detail. Most of the rules (one sheet, no cutting, no gluing, and fiat folding) encapsulated in the word ’fold’. These are related to the physical qualities of the paper. We trans- form a sheet of paper by rolling up, crumpling, folding along curve and folding along straight line. Why fiat folding is important in the rule of origami? Miura’s studies will give us a hint to solve this problem. He has shown us the peculiar fiat folding as a solution of the strength of materials (Miura, 1989). We can find bits of this peculiar fiat folding in the traditional models. EVOLUTION OF ORIGAMI ORGANISMS 169 As for the characters of traditional origami, we shouldn’t ignore the distinction between the surface of paper and the reverse side of it, though it isn’t included in Husimi’s rules. On most of traditional models, surface and reverse side become outer inner sides as a result that edges of paper are fitted to another edges. It is natural and easy process of origami. In biological terms, it correspond to blastula. It is a wonderful coincidence for me. However, ’origami sheet biology’ is very simple. At present, origami creatures are belonging to a kind of Coelentelata like a jelly fish. I may have overlooked the rules of origami. I think the rules of origami and its sys- tematic structures are originate from both physical qualities of paper and ’easiness’ as selection pressure. These systematic structures lead me to the taxonomy, and my ’tiling method’ is based on the taxonomy. 2 THE ORGANIZATION OF TAXONOMY There have been some attempts at producing a taxonomy for origami. There are four viewpoints as follows. 1. Process. 2. Symmetry of the complete model. 3. Technique. 4. Structure. The most famous study on origami taxonomy is that known as ’origami tree’. The pioneer of this study was probably Ohashi (1977). The origami tree was a system- atic method used in learning how to construct origami models. Its main concept is the notion of ’basic form’. Basic forms are simple and geometrical forms which can be applied to many different kinds of origami designs. In fact, the crease patterns in Figure 1 aren’t actual complete figures, but basic forms of them. There are about 10 basic forms. Returning to the biological analogy, the origami tree is a kind of a genealogical tree, for example, the frog’s legs and the Iris’s petals are homologous. It is an interesting viewpoint, but it has a rigid aspect because of its adherence to the folding processes. The symmetrical analysis of complete models is the second type of taxonomy. This view is mentioned by Kihara in his book (Kihara, 1979). He classifies traditional models by point group (a term of crystallography). A new type of work in this field is that by Kawasaki who wasn’t aware of Kihara’s book. Kawasaki’s work is called ’isoarea folding’ which is a design of 4-times rotatory inversional symmetry (Kasahara and Takahama). The third viewpoint considers the origami design to be an assembly of techniques. This view classifies folding techniques. For example, tsumami ori (pinch fold), nejiri 170 £ MAEKAWA or/(twist fold), sizume or/(push fold) ... At present, this study is only an idea, but there are many designs which can be explained by their peculiar techniques. Taxonomy by structure will be described in Sections 3 to 5. 3 ORDERLINESS OF TRADITIONAL MODELS In Figure 1, there is a systematic pattern in the fish-crane-frog lines. On the other hand, the pattern in the windmill is different from the others. The minimum angle of it is 45 degrees. This pattern and its extension have been given the fitting name of "box pleating" (Lang, 1988; Lang and Weiss, 1990). It has great potential in making new structures. Figure 2 is such an example. However, this type of folding is not always structural because relations between each crease are weak. In short, they are agglutinative. Rhinoceros (Maekawa) A fundamental shape of the fish-crane-frog system is the right-angle isoscale triaL- gle which is half of the fish base (Fig. 3). I call it the crane unit. The crane base is assembled by 4 units, and the frog base by 8 units. The 8-unit form is not the limit of this system; the other form can be arranged by the same number of units. Figure 4 shows these examples. The flower dish is assembled by the same number of units as the frog units, whereas the bug is assembled by 16 units. This system has been known for a long time. Uchiyama shows the spider base (4 frog bases = 32 crane units) in his book (Uchiyama, 1979), and it can be traced back more. Figure 5 shows origami designs with cutting from the Edo period (1600- 1860’s). The upper figure was introduced in Kayaragusa (Adachi, 1845). In a slightly different sense, the lower figure is an example of renkaku (chained cranes) in Senbazuru Orikata (Rokoan, 1797). I have designed other forms using the crane unit assembly. Figure 6 shows an example. EVOLUTION OF ORIGAMI ORGANISMS 171 Flower dish (tradition) Bug Figure 4 Crab (tradition) Seigaiha (blue wave) (tradition) 172 ~ MAEKAWA Crocodile (Maekawa) The crane unit is a right-angle isoscale triangle. We can extend this unit to arbitrary triangles. Husimi first explained its geometrical meaning in his book (Husimi and Husimi, 1984). Husimi’s innerpoint theorem states: Any triangle is folded into a form in which all sides are gathered in a straight line. This theorem has been extended to include any quadrilaterals with inscribed circles, and has been extended to quite arbitrary quadrilaterals (Fig. 7). The inner point theorem (Husimi) The Husimi-Maekawa Folding The quadrilateral molecule (Meguro) We can make crooked cranes using Husimi’s theorem. Since I know the arbitrary triangle unit, I have a tendency to use the right-angle isoscale triangle unit. Using the peculiar triangle, we can design new models easily. 4 ORIENTATION OF THE ELEMENTARY UNITS The ’crane unit’ is not an elementary unit (like an atom) of the crane. It is subdi- vided here into two types of triangles (Fig. 8). They are elementary units of crane type origami; the little bird base is also assembled by those units. EVOLUTION OF ORIGAMI ORGANISMS We can regard the basic forms as the results of conditional tiling work using the elementary units. The following 22.5 two theorems corre- spond to conditions of ~ the tiling work, though 67.5~,~ they aren’t sufficient conditions (Kawasaki, The elementary units 1989). Figure 8 The Maekawa theorem states: At any node of a flat folding, except of those on the edge of the plane, the difference of the number of mountain creases and the num- ber of valley creases is equal to two (Maekawa, 1983). The Kawasaki theorem states: At any node of a flat folding, except of those on the edge of the plane, the alternate total of angles between the creases is equal to 180 degrees (Kasahara, 1989; Kawasald, 1989). Unit (type I) Unit (type 2) Unlt (type3) In designing the new model, I do not use the elementary units themselves, but sec- ondary units which are assembled by several elementary units. In short, this tiling work is hierarchical. Figure 9 shows examples of these secondary units. Of course, the crane unit belongs to the group of the secondary units. Recently, a well arranged work using those units was made by Meguro (1991-92). He emphasizes the ’univalency" of the secondary units. In origami, ’univalency’ means the character shown in the Husimi’s innerpoint theorem, that is to say, that all sides of the figure are gathered in a straight lihe by flat folding. ’Univalency’ is a concept to increase efficiency of the tiling work. 174 L MAEKAWA Devil (Maekawa) Deer (Maekawa) Uzard (Maekawa) Beast (Maekawa) 1.S ------) 1 New pattern Adaptation Already Known pattern Rotational view Adaptation Already known pattern Addition Slide view EVOLUTION OF ORIGAMI ORGANISMS 175 5 THE ORIGINAL DESIGNS I have created new designs using the tiling work. Figure 10 shows some of these designs. In designing them, I have used various methods - among them: ’adaptation’, ’addition’, ’rotational view’ and ’slide view’. These are illustrated in Figure 11. ’Adaptation’ is a re-introduction of arbitrary triangle folding. ’Addition’ and others are extended methods of already known forms. 2 times self similar figures Figure It is interesting that two peculiar figures appear in those forms. One is a rectangle which is square root 2 wide per other side, and the other is a right-angle isoscale triangle. They are figures that can be divided into two self-similar figures as in Fig- ure 12. The crane system is a good example of this pattern. I have achieved some interesting results by starting with a sheet of this peculiar-ratio rectangle paper instead of a sheet of square paper (Fig. 13). (This ratio is very common.) Square root 2 is the magic number of origami, because we find this ratio everywhere. Giraffe (Maekawa) Fish (Maekawa) Figure 13 This magic number is significant under the conditions that folding angles are restricted within multiples of 22.5 degrees. If we use other angles, we will find other magic numbers or will not see it. 176 1.. MAEKAWA Unit anqle Tilin.q Triancjles 45~ 45 (l~)x90degrees 75 (l~)x90degrees 18 ’,l/4)x90degrees (lf3)x90degrees Table I Table 1 shows an extension of the elementaq¢ units. The hexasection of the right angle is productive. I have tiled the hexasectional units on a square field as in Fig- ure 14. The trisection has a possibility on regular hexagons and rectangles: the ratio between the width and the length is an integer division or a multiple of the square root of 3. The pentasection can be used on regular pentagons and the Penrose tiles, but I have not accomplished presentable designs to date. EVOLUTION OF ORIGAMI ORGANISMS 177 REFERENCES Adachi, K. (1845) Kayaragusa [The Collection, in Japanese], (not published), A part of copy in: Kasahara, K., Origami 5, Tokyo: Yuki-Shobo, 1976. Husimi, K. and Husimi, M. (1984) Origami no Kikagaku, [Geometry of Origami, in Japanese], enlarged ed., Tokyo: Nihon-Hyoron-Sha. Kasahara, K. (1989) Origami Shin-sekai [The New World of Origami, in Japanese], Tokyo: Sanrio. Kasahara, K. and Takahama, T. (1987) Origami for the Connoisseur, Tokyo: Chuo-Kouron-Sha. Kawasaki, T. (1989) On relation between mountain-creases and valley-creases of a flat origami, In : Huzita, H., ed., Origami Science and Technology, Proceedings of the International Meeting of Origami Science and Technology, Ferrara, pp. 229-237. Mantis (Maekawa) Kihara, T. (1979) Bunshi to Uchuu [Molecule and Cosmos, in Japanese], Tokyo: lwanami. Lang, R. (1988) The Complete Book of Origami, New York: Dover. Lang, R. and Weiss, S. (1990) Origami Zoo, New York: St. Martin’s Press. Maekawa, J. (1983) Viva ! Origami, Kasahara, K., ed., Tokyo: Sanrio. Meguro, T. (1991-1992) Jitsuyou origami sekkeihou [Practical methods of origami designs, in Japanese], Origami Tanteidan Shinbun [The Origami Detectives Newsletter], 5-36-7 Hakusan Bunkyou-ku Tokyo: Gallery Origami House, Nos. 7-12, 14. Miura, K. (1989) Map fold a la Miura style, its physical character and application to the space science, In: HuTJta, H., ed., Origami Science and Technology, Proceedings of the International Meeting of Origami Rail (Maekawa) Science and Technology, Ferrara, pp. 39-49. Ohashi, K. (1977) Sousaku Origami [Creative Origami, in Fibre 14 Japanese], Tokyo: Bijutsu-Shuppan-Sha. Rokoan, (1797) Senbazuru Orikata [Folding Forms of 1000 Cranes, in Japanese], Republished: Kasahara K., Origami 2 -- Senbazuru Orikata, Tokyo: Subaru-Shobo, 1976. Uchiyama, K. (1979) Junad Origami [The Pure Origami, in Japanese], Tokyo: Kokudo-Sha. Syr, vnetry: CuRu,’e and Science Vo~ 5, No. 2, 179-188, 1994 PAPER SCULPTURE Didier Boursin 17 rue Sainte Croix de la Bretonnerie, F-75004 Paris, France Origami, or paper folding, is always practised with both hands. This reflects the symmetry implicated in paper folding. This symmetry is accompanied by a further aspect; that of the way the folding is performed: the folds made on the paper introduce successively different points of reference; points around which the symmetry turns, and folded lines which act as the axes of symmetry. The marks into the space are also dualistic operations: left - right, above - under, up - down, right-side - wrong-side, and so on... Symmetry is ever present though its form is dependent on which model do you wish to construct. For instance, in origami, animals generally have an axis of symmetry along a single axis; plants and flowers, on the other hand, have their symmetry organised around a central point. All the traditional bases and techniques are also objects of analysis from the point of view of symmetry: the water-bomb base, the preliminary base, the windmill base, the bird base, the fish base; the reverse fold, the squash fold, the rabbit’s ear. The paper shape used is often the square, which has many axes of symmetry, like polygons for example. Symmetry is the foundation of any equilibrium. It puts the mind ease. It is the origin of aesthetic. It allows the glance not to be lost. The symmetry is a glance on a human being, who is symmetrical too. In the sense of origami, a human being is symmetrical and is, moreover, represented in a closed surface. We start from a piece of square paper, and, through successive symmetrical folds, reduce it while we get a closed, completed origami model. The most important feature of origami is not the final object created, but the operations that are performed to obtain that final product. That final product may become elegant and simple, for it to be seen as interesting and aesthetically pleasing. If not the best techniques are applied or too much paper is used in the construction of a model, it will be seen as inelegant and not very artistic. The application of paper in successive layers is considered unaesthetic and spoiling its artistic and symmetrical charm. Ideally, origami should involve delicate folds, each of which should be reflected upon, like the moving of chesspieces on a chessboard. 180 D. BOURSIN 2 Fold in front and behind 4 Fold and unfold 5 Fold twice A and behind join the dots 6 Unfold and return I///) ~ / / ~XX N7 Join the dots 8 Join the dots ( / [ ~ and return folding in half 9 Fold angles in half and unfold ~ ~~ 10 Open;fold in half Connect t/" / 5 pieces~ 14 Lock by this last fold inside Figure !: Star by D. Boursin (1992). PAPER SCULPTURE 181 t’,’ I,/ Figure 2: Star in 3 dimensions by D. Boursin (1992). 182 D. BOURSIN 1 Take a A4 paper and fold it in quarters and cut it in half 2 Fold all the quarters except two parts and cut until the middle 3 Fold each triangle then fold it in 3 Dimensions 5 The cube where appears V’ and Vz 4 Fold the both pieces of A4 and connect roger her. Figure 3: Cube structure by D. Boursin (1990). PAPER SCULPTURE 183 1 pher les 4 pointes d’un carte jusqu’au centre pu~s marquer les phs au t~ers dans les 2 sens ouvrir completement 2 et retourner :~ plier les 2 triangles derriere et marquer derniers ptis comme indiquOs puis retourner 4 ghsser les cbtes I’un darts I’autre 5 rabattre les po=ntes ~. I’~nt~r~eur puls remonter le tout ~ la verticale 6 embo~ter I’un dans I’autre "~ le cube terrnln~ Figure 4: Cube by D. Boursin (1990). 184 D. BOURSIN The model’s symmetry should explain the essence itself of the paper and symmetry: minimum of matter for a maximum of expression should be one of the fundamental rules to follow in creating a new aesthetic of symmetry and generally origami. To practice the folding, it’s to sculpt the matter trying to explain tae best expression. Personally, I prefer simple origami; I try to be simple in my origami works, however it’s very difficult. For me, origami is most of the time, a wink of life, generally very ephemeral. Now the task is to fold different irregular sizes, or polygon shapes of paper from the usual square-shaped paper. A4 size paper owns interesting proportions: 21 x V~ = 29.7 (~/~-is the diagonal of a square, ~ is the diagonal of an A4 size paper and also the diagonal of the cube). The relation between 2 and 3 dimensions is also interesting to explore and can be used for folding different sorts of cubes and polyhedrons with this paper size. Summing up, symmetry cannot be dissociated from a sense of equilibrium, proportion, elegantness and aesthetic beauty. PAPER SCULPTURE 185 2 Hold the paper with both hands and fold dot to dot 1 A4 paper : fold and unfold as indicates 4 Fold inside the little triangles (4) then fold the right side down to 3 Make the creases then makes j the dot. ’ reverse folds ~ Repeat on the other side. 5\\// Put the left triangle inside the central one 6 Fold and unfold the middle on the both sides.For making in 3 dimensions,pull out the opposite corners. 7 The octahedron finished Figure 5: Octahedron by D. Boursin (1992) 186 D. BOURSIN Fisure 6: Folding frame: The structure (D. Boursin). PAPER SCULPTURE 187 Figure 7: Folding Frame: Details (D. Boursin). 188 D. BOURSIN THE AUTHOR’S PUBLICATIONS ON ORIGAMI: Bout-sin, D. (1983) Le ticketplid. Boursin, D. (1988) Manuelpractique d’origami, ed. Celiv. Boursin, D. (1989) Papiers plids, des idees plein les mains, ed. J’ai lu. Bout’sin, D. (1990) Pliages en mouvernent, ed. Dessain et Tolra. Boursin, D. (1991) Pliage des serviettes, ed. Dessain et Tolra. Boursin, D. (1992) Pliages en libertY, ed. Dessain et Tolra. Boursin, D. (1994) Pliagespremierspas, ed. Dessain et Tolra. Symmetry: Culture and Science VoL 5, No. ~ 189-210, 1994 SYMMETRIC GALLERY ORIGAMI 190 D. BOURSIN Symmetry: Culture and Science VoL 5, No. 2, 190-196, 1994 Shooting Stars, one piece paper with curve, (Didier Boursin). PAPER SCULPTURE -- GALLERY 191 3 Horns, it is a work on curving paper then continue and discontinue, (Didier Boursin). 192 D. BOURSIN Tetrahedron, from A4 sheet of paper, (Didier Boursin, photo: Fabrice Besse). Cubes, from A5 sheet of paper, (Didier Boursin, photo: Fabrice Besse). PAPER SCULPTURE -- GALLERY 193 Octahedron, (Didier Boursin, Photo: Fabrice Besse). Polihedra, (Didier Boursin, photo: Fabrice Besse). 194 D. BOURSIN 5Points Star, (Didier Boursin, photo: Fabrice Besse). Star in 3 Dimettsions, (Didier Boursin, photo: Fabrice Besse). PAPER SCULPTURE -- GALLERY 195 Growing Plant, it contains 9 meters o[ paper in one piece, (Didier Boursin). Folding Frame. It is impossible to fold directly by hands. The artist has different flames like this with a minimum of foldings; this is one of the best solutions, (Didier Boursin, photo: Fabrice Besse). 196 D. BOURSIN Mobile, (Didier Boursin, photo: Fabrice Besse). Symmetry: Culttwe and Science Vo£ 5, No. 2, 197-210, 1994 Robert J. Lang ¯ Stag Beetle2 ¯ 1. Fold and unfold 2. Fold and unfold along and angle along the diagonals. b~sector Make the crease sharp only where ~t h,ts the edge Fold and unfold 4. Fold the corners m to 5. Fold the comers out to 6. Fold the comers m to overlap he along the new edges he on ex/stlng creases. 7. Partially unfold 8, Repeat steps 4-7 on 9. Fold and unfold. 10. Fold and unfold 11. Fold and unfold 12. Fold mbb~t ears from 13. Unfold the two mbb:t 14. Pleat the paper Note that the two comers mountain folds hits the colored raw edges where two existing creases (made in step 12) hlt the edges. 198 R..L LANG 15. Fold the paper in half 16, Fold and unfold 17. Reverse-fold the edge along the diagonal and rotate 1/8 turn counterclockwise 18. Fold and unfold 19. Open-sink the corner. 20. Spread-sink the edge symmetrically 21. Fold the edge upward 22. Fold the upper edge down 23. Pull the paper out from under to align with a hidden edge, the pleat and sink a square region crease, and unfold, of the paper outlined by the creases you made m step 21 24. Flatten the model 25. Repeat steps 23--24 b~hmd 26. Asymmemcally squash-fold the right side Note that the two circled crease intersections come together STAG BEETLE 2 -- FOLDING INSTRUCTIONS 199 27. Reverse-fold the edge 28. Bring two layers of paper 29. Reverse-fold the far edge upward to the front. 30. Fold and unfold along a 31. Fold and unfold along a 32. Pleat the nght side, horizontal crease, which lines crease perpendicular to the plVOtlng the bottom corner so up with a hidden edge adjacent folded edge that its edges align with the raw edges on the far layers The model will not lie fiat 33. Reverse-fold the corner 34. Full out a single layer of 35. Close up the model and paper (open unsmk) to make flatten completely the pleat symmetric 36. Mountain-fold the narrow 37. Simultaneously squash-fold 38. Lift up one edge and bring flap behind to make the model the left edge and the bottom corner up so that symmetric from front to back mountain-fold the bottom stands out away from the underneath, tucking tt into the model pocket Repeat behind 2OO R]. LANG 39. Pull a colored comer 40. Repeat steps 38-39 41. Reverse-fold the edges 42. Fold and unfold out from lnstde the pocket, behtnd Repeat behLnd squash the top down, pull the sides out, and flatten 43. Fold and unfold, 44. Fold the comer down 45. Fold the comer back 46. Pull out a raw edge. Repeat behind whde squash-folding the up but keep the squash edge underneath fold in place 47. Reverse-fold the edge 48. Reverse-fold the edge 49. Reverse-fold the edge 50. Repeat steps 42-49 back out again behind 5;1. Fold one layer up, 52. Fold the raw edge over 53. Reverse-fold the white 54. Wrap one layer from repeat behind to the right and ptnch the point to the left the mstde to the front. excess paper upward Repeat behind. Repeat behind STAG BEETLE 2 -- FOLDING INSTRUCTIONS 201 55. Fold and unfold, repeat 56. Open-sink the corner, 57. Ptnch the narrow point behind repeat behind m half and swing it downward Repeat behind 59. Stretch and narrow the 60. Now we’ll work on one 61. Fold and unfold point and swing tt side at a nme for a while downward Repeat behind Fold and unfold 62. Reverse-fold 63. Reverse-fold on the 64. Reverse-fold again crease you made in step 61 65. Reverse-fold the 66. Pull out a raw edge 67, Reverse-fold the edge remaining edge halfway between existing creases 202 R.£ LANG 68. Reverse-fold on an 69. Reverse-fold the edge 70. Repeat steps 66-69 existing crease back down to hne up with behtnd the other edges 71. Fold two edges down 72. Lift up one layer 73. Reverse-fold the ms,de to the lower left and shghtly edge spread-squash ~e top horizontal edge 74. Close tt up 75. Fold a mbb*t ear from a 76. Fold two layers 77. Fold a rabb*t ear from single layer upward the remaining layer 78. Reverse-fold the point 79. Sink the edge 80. Mountain-fold two upward; the reverse goes edges together, forming a between the topmost edge small swivel fold at the and the remaining edges base of the point STAG BEETLE 2 -- FOLDING INSTRUCTIONS 2O3 81. Repeat steps 60~0 82. Fold and unfold, 83. Fold and unfold behind 84. Reverse-fold two 85. Fold and unfold The 86. Closed-sink the corner points coming step will be easter tf Repeat behind you mountain-fold the paper through the thick layers as well Repeat behind 87. Fold the left edge to he 88. Pull out some loose 89. Open out the flap along the diagonal crease paper and squash the excess Repeat behind upward Repeat behind 90. Pleat the flap upward 91. Pull out some loose 92. Squash-fold the edge and paper swing the excess paper to the right 2O4 93- Pull out some more 94. Detml of the left half of 95. Petal-fold the edge excess paper the model Squash-fold the flap 96. Pull one of the corners of 97. Squash-fold the corner 98. Reverse-fold the edges the square all the way out symmetrically from ms,de 99. Fold and unfold 100. Fold the point up to the 101. Pull out some paper left 102. Squash-fold the paper 103. Pull out some paper 104. Outside-reverse-fold d~e over to the right corner STAG BEETLE 2 -- FOLDING INSTRUCTIONS 205 105. Fold and unfold 106. Fold one flap over to 107. Fold and unfold the left 108. Reverse-fold the corner 109. Enlarged view 110. Squash-fold the point Mountatn-fold the white corner underneath and swing the two potnts down 111. Inside petal-fold the 112. Fold two points up 113. Open-sink both corners 114. Fold two potnts down edge 115. Fold two points upward 116. Close up the model, 117. Reverse-fold two edges incorporating a reverse fold 206 118. Pull out some loose 119. Valley fold the mw edge paper down Repeat behind 120, Fold half of the top layers forward and ball of the bottom layers behind 121, Carefully spread-stnk the corner, spreading all of the layers symmetrically 122. Closed un-smk the pocket 123. Grasp two edges of the point you lUSt pulled out and stretch them to the left, the point disappears in the process 124. In progress 125. Fold and unfold STAG BEETLE 2 -- FOLDING INSTRUCTIONS 207 126. Closed-sink the edge on the crease 127. Spread-sink the edge symmetrically you lUSt made It helps to open the model out from the underside while you do this 128. Sink the bottom two corners and 129. Bring one layer m front squeeze the excess paper upwnrd Be careful to keep the paper from filbng up tbe vertxc:al sht between the sidest Repeat on the top 130. Mountain-fold one layer behind on 131. valley-fold a double edge to the top and bottom center hne of the model above and below 132. Undo two reverse folds 133. Turn the model over 2O8 /~J. LANG 134. Pleat the abdomen 135. Mountain-fold the edges of the pleated par~ 136. Crimp the thtck (tuner) points so that 137. Mountain-fold the edges and tuck they stand straight out from the body them into pockets Repeat behind 138. Next wews am a close-up of the 139. Fold and unfold about 2/5 140. Fold and unfold Note that right side of the model of the pomt the vemcal crease h~ts the top edge at the same place the last crease d~d 141. Camp the point downward 142. Pull out two layers on the 143. Wrap one layer to the on the existing creases top and bottom front Repeat behind STAG BEETLE 2 -- FOLDING INSTRUCTIONS 209 144. Pull out some loose paper 145. Reverse-fold the point 146. Reverse-fold the corner Repeat behind 147. Reverse-fold the h~dden 148. Repeat steps 138-147 on 149. Pinch the two flaps in half corner the lower flap through all layers 150. Turn the model over 151. Mountain-fold the blunt 152. Pinch the two small points corner inside m half 153. Ptnch the two small points 154. Pull out two points 155. Turn the model back over in half 210 R.Z LANG 156. Crimp the antennae to 157. L~ke th~s 158. This shows the entire model Crimp the ngbt {he forelegs (on the rJght) Reverse-fold the h*nd legs 160. Turn tbe model over 159. Reverse-fold the hind feet Reverse-gold each m~d leg twice, 161. Round the abdomen and pleat *t down 162. F*mshed Stag Beetle {he m(ddle Make the body 3-D Syrametry: Culture and Science VoL 5, No. 2, 211-212, 1994 RESEARCH PROBLEMS ON SYMMETRY RESEARCH PROBLEM 1 Two-sided wallpaper groups of periodic origami patterns Origami is useful to make infinite periodic patterns, more precisely, to represent a finite part of them. These periodic patterns are not 2-dimensional in a strict sense, because they are not totally fiat, but rather occupy a narrow layer. We may call them 2.5-dimensional patterns, continuing Coxeter’s joking remark that the 7 frieze-groups (strip-groups) are 1.5-dimensional. The 2.5-dimensional periodic patterns can be analyzed with two-sided wallpaper groups (layer groups). These were firstly enumerated by C. Hermann (1929): there are exactly 80 types. In the same year L. Weber (1929) illustrated all of them by black-and-white patterns where the colors refer to the front and the back sides, respectively (see, e.g., Shubnikov and Koptsik, 1972, where Weber’s figures are reprinted). From these 80 types only 46 ones are really interesting, while the remaining 34 types are degenerate cases (i.e., the 17 strictly 2-dimensional wallpaper patterns, as those cases where only one side is used; and those further 17 cases where the above mentioned 17 patterns are mirrored to the back side). These 46 patterns are also known as two-colored (black-and-white) patterns. This interpretation with colors is useful in both fields: crystallography (where the colors refer to different physical properties, e.g., positive and negative magnetism) and design. Indeed, these patterns were discovered and studied not only by the mentioned crystallographers, but also by the textile engineer H. J. Woods (1936). His patterns are reprinted in a more recent monograph on pattern analysis of ornamental arts (Crowe and Washburn, 1988). Question: How many of these 46 types can be represented by origami without using cuts? REFERENCES Hermann, C. (1929) Zur systematischen Strukturtheorie: 3. Ketten und Netzgruppen, [About a systematic structure-theory: Chain- and net-groups, in German], Zeitschriflfiir Kristallographie, 69, 25O-270. 212 RESEARCH PROBLEMS ON SYMMETRY Shubnikov, A. V. and Koptsik, V. A. (1972) Simmetriya v nauke i islo~sstve, [in Russian], Moskva: Nauka, 339 pp.; English trans., Syrmnctry in Science andArt, New York: Plenum Press, 1974, xxv + 420 pp. [See Chap. 8]. Washburn, D. K. and Crowe, D. W. (1988) Symmetn’es of Culture: Theory and Practice of Plane Pattern Analysis, Seattle, Wash.: University of Washington Press, x + 299 pp.; Paperback ed., ibid., 1991. [See Chap. 3]. Weber, L. (1929) Die Symmetrie homogener ebener Punktsysteme, [Symmetry o1~ homogeneous planary point-systems, in German], ZeitschtiftffirKristallographie, 70, 309-327. Woods, H. J. (1936) The geometrical basis of pattern design: Part 4, Counterchange symmetry in plane patterns, Journal of the Tepaile Institute, Transactions, 27, T305-T320. D6nes Nagy Symm~ay: Culture and Science Vot 5, No. 2, 213-218, 1994 SYMMETR O-GRAPHY Section Editor: Ddnes Nagy, Institute of Applied Physics, University of Tsulazba, Tsukuba Science City 305, Japan; Fax: 81-298-53-5205; E-mail: [email protected] BOOK REVIEW Boursln, Didier, Pliages en mouvement, [Folding in Movement, in French], Paris: Dessain et Tolra, 1990, 79 pp.; Reprint, ibm, 1991. The author of this book is the President of the Mouvement Franfais des Plieurs de Papier (French Association of Paper Folders). The work starts with a one-page introduction with an historic survey, including some interesting data about the spread of paper-folding in France. A group of Japanese touring in France actually demonstrated the ’real’ origami in 1860. The author emphasizes, however, that there was in Europe an independent tradition based on folding table napkins, which originated in the courts of Henry III and Louis XIV. The book has five main parts: Animals, Magic, Gravity, Sounds (i.e., instruments producing sounds), Cards. The contents has an important feature: the items are classified into three categories of very simple (*), simple (**), and difficult (***). This is followed by the detailed explanation of the visual symbols used in paper folding. In each case of the presented ’paper compositions’ we find double illustrations: a color photograph of the completed work and the detailed explanation how to make it, with very well made drawings. The quality of the photographs, made by Fabrice Besse, represent a high level of artistic skill. In many cases the installation was made by the Japanese wife of the author, Setsuko, a fashion designer. Although this book is addressed to a broad public, it is also useful for specialists. Thus Boursin gives credit to the inventor of each individual piece, including such details that, for example, he developed a frog (p. 16) using the idea of K. Kasahara. Note that many of the pieces were invented by Boursin himself. One of the items should be made of cloth (pp. 24-25). We also have some symmetry-related remarks. Boursin gives credit to Luisa Canovi of Italy for inventing a closed chain of tetrahedra that provide an exciting flexible object (pp. 36-37). This item was, however, discovered earlier outside of the world of origami. An early discovery of this idea is due to Paul Schatz in 1929, which was probably independently rediscovered by Schattschneider and Walker in the 1970s, see their book M. C. Escher Kaleidocycles (New York, 1977). Another related object is discussed by Boursin under the name ofFleragon (p. 39). Indeed, it is a very expressive name, because the flexible chain of units form a hexagonal SYMMETRO-GRAPHY 214 shape. Boursin remarks that we can make with this object kaleidoscopic images. From a didactical point of view, it would be better to reverse their order: to explain first the Flexagon and then to turn to the more complex chains of tetrahedra. The piece Equilibrium (pp. 50-51), designed by the author, has an interesting connection with the topic of symmetry. Balancing toys were popular in earlier ages: each of these objects, usually dominated by a U shape turned upside-down, has a supporting point higher than its center of gravity. Thus, these toys can balance on the top of various sharp objects. It is exciting to see the origami version of this mechanical toy and the photograph where it balances on the top of a bottle. Another item, a beautiful periodic pattern (pp. 44-45), inspired us to formulate a research problem about symmetry groups of this type of structure. Note that Robert Harbin’s name is misspelled on p. 38 as "Herbin". Illustrations: it is basically a picture book with very many drawings and photographs. Address: Cr~ati0n Setsuko, 17, rue Sainte Croix de la Bretonnerie, F-75004 Paris, France. D6nes Nagy Harglttai, Istv~in and Hargittai, Magdolna, Symmetry, A Unifying Concept, [Symmetry in general], Bolinas, Calif.: Shelter Publications, Inc., 1994, 222 pp. Istv~in and Magdolna Hargittai have published several books, partly as editors, partly as author, on the topic of symmetry. They are chemists, who started their investigations from the symmetry of molecules to discover further symmetries of the invisible and visible world. As scientists, they explain the regularities, patterns, and laws of hidden symmetries, while as fans of photography, they present the beauty of the visible ones’ to the reader. One meets symmetry in sciences, arts, and even in everyday life, nevertheless to write a book on the topic conceals traps: to aim at a broad public, from mathematicians to biologists, from physicists to linguists, from chemists to artists, from students to professors limits the scientific level; to keep high scientific standards limits the widths of the audience. The Hargittais manoeuvre skilfully between these Scylla and Charybdis, well known for any interdisciplinary author. They ferry the reader from the point-group symmetries through the examples for antisymmetry to the space-group symmetries. They do not sacrifice the wide understandability for deep scientific exactness. The basic symmetry concepts are introduced mainly by a visual way, special terminology is mainly avoided, in other cases simple definitions are given. Clear drawings help the understanding. Layman readers can discover new connections, new patterns in the world they had thought they knew. One explores, step by step, the way to the most recent symmetry- inspired discoveries like the quasicrystals showing fivefold symmetry which has been not usual in the inanimate nature. The book is well illustrated with hundreds of black-and-white photos, reproductions, and drawings. Gy0rgy Darvas SYMMETRO-GRAPHY 216 SYMMETRIC REVIEWS 5.2 The "Symmetric Reviews" (SR), as a regular subsection, publish brief notes about books and papers. These are not conventional reviews; their main goal is to emphasize the connections with symmetry and, in same cases, the required backgrourwL Correspondence should preferab~v be sent to both the section editor (for reviewing) and the Symmetrion in Budapest (for the data bank). SR 5.2 - 1 (Origami: popular) Boursin, Didier, Pliages en mouvement, [Folding in Movement, in French], Paris: Dessain et Tolra, 1990, 79 pp.; Reprint, ibid., 1991. SR 5.2 - 2 (Interdisciplinary book: popular) Hargittai, Istv~in and Hargittai, Magdoina, Symmetry: A Unifying Concept, Bolinas, Calif.: Shelter Publications, Inc., 1994, 222 pp. The authors of this books are Hungarian chemists, husband and wife. Istv~in Hargittai is a Honorary Member of ISIS-Symmetry; see the list of his symmetry- related books in this quarterly, Vol. 3, No. 3, p. 324. The new book of the Hargittais covers various aspects of symmetry in art and science. It is addressed to a very broad public using the advantage of large number of illustrations. This book is reviewed in more details; see the section "Book review". Illustrations: 850. References: to books 29, to illustrations 177. Address: E0tv0s Lorhnd University, P.O. Box 117, Budapest, H-1431 Hungary. SR 5.2 - 3 (Origami: popular) Pataki, Tibor, Hajtogatni j6, [Folding is Exciting, in Hungarian], Budapest: Gyorsjelent~s Kiad6 Kft., 1993, 79 pp. This book belongs to the rare category of Western books on origami where the majority of the pieces are designed by the author himself, and there are only very few adaptations. In those cases the author refers to the sources, which are either traditional origami items or the works of modern inventors. The book starts with a two-page preface by a noted Hungarian author, G~ibor N6gr~di. Pataki himself is even shorter: his introduction is just a half-page. The rest of the book is dominated by figures with brief notes. First of all Pataki explains the notations and presents the four basic folding operations. After these, he creates almost an entire origami zoo with very many animals. There are a few other objects: a sailing boat, a piano, a Santa Claus, a decoration for Christmas-tree, a steam boat, a fighter, an envelope, three masks, a geisha, a flying machine, a clown, and a rocket. Altogether there are 40 pieces in the book. Some of the animals feature in more than one work. For example, the book presents a running rabbit, then a sitting one. It is followed by a fox, and we think that children may immediately create stories with these animals. In Hungary origami paper is not widely available, therefore the book includes at the back 12 sheets. The front side of each these square-shaped origami papers is SYMMETRO-GRAPHY 217 yellow, while the back side is white. These colors help the author to better explain the process of folding. Specifically, the drawings clearly indicate when we should work with the yellow front side of the paper, and when with the white back side. In the beginning the author consistently uses this helpful ’color coding’, but later he frequently violates it: thus, the back side is sometimes gray, while in other cases the background (the table) is yellow and the two sides are distinguished by white and gray colors. In a possible new edition of the book, we suggest being more consistent in coloring the figures, which is, indeed, a very useful idea. The book includes color photographs of the objects: these are collected in a section at the middle. The captions of these photographs clearly refer to the page numbers where the folding processes are explained. One may claim that it would be better to have the photographs of the pieces and the corresponding drawings of the folding processes side by side, but that arrangement would make the book much more expensive. We suggest supplementing any possible new edition of the book with a table of contents. Illustrations: about 800 drawings and 30 color photographs. Address: Budapest, Kir~ily u. 67. V.em. 3., H-1077 Hungary. D6nes Nagy Peacock by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 88-89.) 218 SYMMETRO-GRAPHY Sparrow by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 94-95.) Tree-frog by T. Pataki. (See the folding instructions in Symmetry: Culture and Science, 5 (1994), 1, 92-93.) Syraraetry: Culture and Science VoL 5, No. 2, 219-223, 1994 SFS: SYMMETRIC FORUM OF THE SOCIETY (BULLETIN BOARD) All correspondence should be addressed to the editors: G~yOrgy Darvas or Ddnes Nagy. ANNOUNCEMENTS SECOND INTERNATIONAL MEETING OF ORIGAMI SCIENCE AND SCIENTIFIC ORIGAMI November 29 - December 2, 1994 Otsu, Shiga, Japan SCOPE AND TOPICS Origami no longer means only orizuru, a folded paper crane, the most representative example of a classic style work, but is the proper subject of mathematics and involves the field of science and technology, which have developed the new idea of origami art. The first International Meeting of Origami Science and Technology, held at Ferrara, Italy, in 1989, was the result of this involvement. Because of this good first step, it was desired to have the second meeting in Japan, the home of Origami. Therefore, it has been decided to have this event in the City Otsu, near Kyoto, from November 29 to December 2, 1994. The aim of this meeting is to bring together specialists in origami science and scientific origami. It is intended to provide the participants with information about the important topics. The contributed papers may cover origami science and scientific origami. However, ’science’ includes wider fields, in particular, mathematical foundations of origami, natural forms and origami, classical vs. modern origami, origami art, origami and industrial design, Euclidean geometry and origami axiom, origami and algebra, education and therapy by origami, mathematical design of origami, paper materials, curve folds modular origami, definition of origami, etc. 220 SFS Participating Organizations Ars+, Asociacion Espanola de Papiroflexia, British Origami Society, Centro Diffusione Origami (C.D.O.), Dansk Origami Center, ISIS-Symmetry, Korea Jongie Jupgi Association, Mouvement Francais des Plieurs de Papier (M.F.P.P.), Nippon Origami Association, Origami Deutschland (O.D.), The Board of Education, City of Otsu, Shiga Prefecture, The Form and Culture Society, The Friends of the Origami Center of America, The Society for the Science of Form. CALL FOR PAPERS SYMPOSIUM: Origami: East and West will be organized in the framework of the following congress and exhibition: S YMME TRY." NATURAL AND ARTIFICIAL Third Interdisciplinary Symmetry Congress and Exhibition of the INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) August 14- 20, 1995 Old Town Alexandria (near Washington, D.C.) U.S~4. CALL FOR PAPERS, WORKSHOP TOPICS, AND EXHIBITION ITEMS FIELDS OF INTEREST SYMMETRY: NATURAL AND ARTIFICIAL The congress and exhibition present a broad interdisciplinary forum where the rep- resentatives of various fields in art, science, and technology may discuss and enrich their experiences. The concept symmetry, having roots in both art and science, helps to provide a ’common language’ for this purpose. The new ’bridges’ between disci- plines could inspire further ideas in the original fields of participants, as well as facilitate the adaptation of existing ideas and methods from one fieM to another. The title of the congress emphasizes the presence of symmetry (dissymmetry, bro- ken symmetry) both in nature and in the objects created by artists, scientists, and engi- neers. SYMMETRIC FORUM OF THE SOCIETY 221 Exhibition: Ars Scientifica There have been several exhibitions representing the specific impact of certain fields of science and technology on art, but ISIS-Symmetry has initiated a regular forum for a broader interface of art and science. The exhibition will consist of two parts: a professional exhibition and an informal one, based on the objects illustrat- ing the lectures given by the participants. Some workshops will be conducted in the exhibition rooms. SPECIAL INTERESTS ofArs Scientifica are, among others: kalei- doscopes, polyhedra, model designs, new media. CALL FOR PAPERS A lecture proposal should include a maximum 4-page extended abstract in a camera ready version. Keeping in mind the interdisciplinary goals of the congress and the composition of the participants, please try to help the readers outside of your main discipline e.g., by explaining some special concepts, using intuitive approaches, or giving comprehensive tables and illustrations. The extended abstracts should either (a) describe concrete interdisciplinary ’bridges’ between different fields of art, sci- ence, and technology using the concept of symmetry; or (b) survey the importance of symmetry in a concrete field with an emphasis on possible ’bridges’ to other fields. Note, please, that the central topic of the present congress Symmetry: Natural and Artificial opens a wider door towards technological applications. Papers discussing links between any form of symmetry-asymmetry phenomenon or law in nature on the one side, and artistic, technical achievements on the other, are preferred. Please consider tlxat the meetings of ISIS-Symmetry are informal and do not substitute for the disciplinary conferences, only supplement them with a broader perspective. The extended abstracts should be submitted in 2 copies, mailed 1 each to G. Darvas and D. Nagy, on A4 or letter size pages, printed on one side of each sheet, with at least 2.5 cm (1 inch) margins both sides, top and bottom, double spaced, 12-point characters. Sample: TITLE WITH CAPITAL LE’Iq"ERS [two line-spaces] Joe Symmetrist and Josephine Asymmetrist Department of Dissymmetry, Fibonacci University San Symmetrino, SY 12358, Symmetryland E-mail: symmetrist @ fibonacci.edu [two line-spaces] The text should be printed in one column. Figures (black-and-white only) may interrupt the text. Please avoid using any other heading (e.g., ’Extended Abstract’, ’submitted to ...’). Page numbers should be marked with pencil. References [at the end]: Alphabetical order, full bibliographic description. 222 SFS For more details refer to the "Instructions for contributors" on pp. 110-111. CALL FOR EXHIBITION ITEMS Items for the exhibitions should be introduced in the same form as lecture-abstracts on A4 or letter size sheets, in black-and-white camera ready, reproducible form. Please mark with pencil at the top of the sheet: (EXHIBITION). A short descrip- tion and/or explanation of the items, as well as the connection to the main theme of the congress and exhibition, is preferred. Please give the dimensions of each item. Art works, models, demonstration materials, etc. are welcome, e.g., in the following sections: Kaleidoscopes, Polyhedral symmetry, Origami, The beauty of molecules, Aesthetics of man-made constructions, Mechanical structures inspired by nature: Artificial and natural structures, Design principles, New Media. Proposals for further sections are encouraged. CALL FOR WORKSHOP TOPICS Please give the approximate title, short description (how do you plan to organize the workshop), other expected/proposed contributors, etc. Proposals emphasizing interactions, mediated by symmetry, between different disciplines; science, art, and technology; cultural origins, and relying upon the interest of participants with dif- ferent backgrounds, are preferred. CALL FOR PROPOSALS FOR EVENING ACTMTIES OR PERFORMANCES Music, dance, video, laser, paper folding, etc. programs are welcome. Please submit your proposals, similar to the lecture abstracts, with descriptions of the feasibility and the technical requirements. Please mark with pencil at the top of the sheet: (PERFORMANCE), (VIDEO), etc., respectively. (Formal requirements are the same as above for papers.) DEADLINES for application and short description of contribution and other proposals: December 15, 1994; for submitting final (camera ready) versions of the emended abstracts: March 31, 1995. SYMMETRIC FORUM OF THE SOCIETY 223 THE FORMAT OF THE CONGRESS AND EXHIBITION The tradition, initiated by ISIS-Symmetry, to facilitate interdisciplinary dialogues among scientists, engineers, and artists will be continued. There will be no parallel sections (which would lead to disciplinary separation of the participants), but each morning there will be plenary sessions, while the main ideas will be discussed and developed in afternoon workshops. For the evenings there are scheduled perfor- mances and informal meetings, including recreational, and ars scientifica programs. The working language of the congress is English. The Scientific Advisory Committee of the Congress and Exhibition is the Board of ISIS-Symmetry (see inside front and back covers). CONTACT PERSONS (for the Congress and Exhibition) Martha Pardavi-Horvath, Site Coordinator George Washington University Department of Electrical Engineering and Computer Science Washington, D.C. 20052, U.S.A. Phone: 1-202-994-5516; Fax: 1-202-994-5296; E-mail: [email protected] Gy0rgy Darvas, Executive Secretary, ISIS-Symmetry 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 Na~, President, ISIS-Symmetry Institute of Applied Physics, University of Tsukuba Tsukuba Science City, Ibaraki-ken 305, Japan Phone: 81-298-53-6786; Fax: 81-298-53-5205; E-mail: [email protected] APPLICATION FORM Name: ...... Affiliation: ...... Mailing Address: ...... City: ...... State/country: ...... Fax: ...... Phone: ...... E-mail: ...... I intend to: © attend the Congress © submit a paper © exhibit Tentative title of my contribution: ...... 224 AIMS AND SCOPE There are many disciplinary _periodicals and symposia in various fields of art, science, and technology, but broad iaterdisciplioary forums for the connections tmtween distant fields are ve~, rare. Consequently, tl~e interdisci- plinary papers are dispersed in very different j.ournals and proceedings. Th~s fact makes the cooperation of the authors dilficult, and even affects the ability to locate their papers. In our ’split culture’, there is an obvious need for interdisciplinary journals that have the basic goal of building bridges (’symmetries’) between various fields of the arts and sciences. Because o[ the variety of topics available, the concrete, but general, concept of symmetry was selected as the focus of tbe journal, since it has roots in both science and art. _SYl~t~Rlq,_ CULTURF.. AIgD SCIENCE is the quarterly .of the It¢’r~_I~C,~TIOI¢~_SO~.I~. ]~oR ~ INTERDISCI~PLIN~_ y ¯ ~’TUDYOFb’YJO~TRY (abbreviation: ISIS-Symmetry, shorter name: ~ymmemy ~’o¢iety). ISIS=s]t, mmetry was rounded during the symposium Symmetry of Smacture .(First lnterdiscipFauzry Symme_ tty Sympos~ura and Exhibition), Budapest, August 1:3-19, 1989. The focus of ISIS=Symmetry is not only on the concept of symmetry, but also its associates (asymmetry, dissymmetry, antisymmetry, etc.) and related concepts (proportion, rhythm, invariance, etc.) in an ~nterdisciplinary and intercultural co.ntext. We may refer to this broad approach to the concept as syrn~trolog),. The suffix -1o~ can be associated not only with knowledge of concrete fields (cf., biology, geol- ogy, philologyj psychology, sociology, etc.) and discourse or treatise (cf., methodology, chronology, etc.), but also with the Greek terminology of proportion (cf., logos, analogia, and their Latin translations ratio,i~rol~o~io). The basic goals of the Soci¢o~ are (1) to bnng together artists and scientists, educators and students devoted to, or interested in, the research and understanding of the concept and application of symmetry (asymmeh’y, dissymmetry); (2) to provide regular information to the general public about events in symmetrology;_, (3.) to ensure a regular fornm (including tr~e organization of symposia, congresses, and the publication of a periodical) for all those interested in symmetrology. The Society organizes the triennial Interdisciplinary S_ym/~.Fy Congress and Exhibition Cstarting with the sym- posium of 1989) and other workshop, meetings, and exhibitmns. The forums of the Society are informal ones, which do not su~titute for the disciplinary conferences, only supplement them with a broader perspective. The Quarterly - a non-commercial scholarly journal, as well as the forum of ISIS-Symmetry - publishes original papers on symmetry and related questions wliich present nov results or nov connections between known results. The papers are addressed to a broad non-speciahst public, without becoming too general, and have an interdis- ciplinary character in one of the following senses: O) they describe concrete interdisciplinary ’bridges’ between different fields of art, science, and technology using the concept of symmetry;, (2) they survey the importance of symmetry in a concrete field with an emphasis on possible ’bridges’ to other fields. The Quarterly also has a special interest in historic and educational questions, as well as in symmetry-related recreations, games, and computer programs. The regular sections of the Quarterly: ¯ Symmetry: Culture & Science (papers classified as humanities, but also connected with scientific questions) ¯ Symmetry: Science & Culture (papers classified as science, but also connected with the humanities) ¯ Symraetry In Education (articles on the theory and practice of education reports on interdisciplinary pro’ects) ¯ SF~: Symmetric Forum of the Society (calendar of events, announcements o[ ISIS-Symmetry, news from members, announcements of projects and publications) ¯ Symmetro-graphy (biblio/disco/sogtware/ludo/historio-graphies, re, cloys of books and papers, notes on anniversaries) Additional non-regular sections: ¯ Symmetrospective: A Historic View (survey articles, recollections, reprints or English translations of basic papers) ¯ Symmetry: A Special I~o~us on ... (round table discussions or survey articles with comments on topics of special interest) ¯ Symmetric Gallery (works of art) ¯ Mosaic of Symmetry (short papers within a discipline, but appealing to broader interest) ¯ Research Problems o/~ Symmetry (brief descriptions of opei~’problems) ¯ Recreational Symmetry (problems, puzzles, games, computer programs, descriptions of scientific toys; for example, filings, polyhedra, and origami) ¯ Reflections: Letters ~o the Editors (comments on papers, letters of general interest) Both the lack of seasonal references and the centrosymmetric spine design emphasize the international charac- ter of the Society; to accept one or another convention would be a ’symmetry ~iolation’. In the first part of the abbreviation ISIS-Symmetry all the letters are capitalized, while the centrosymmetric image iSIS! on the spine is flanked by ’Symmetry’ from both directions. This convention emphasizes that ISIS=symmetry and its quarterly have no direct connection with other organizations or journals which also use the word Isis or ISIS. There are more than twenty identical acronyms andmore than ten such periodicals, many of which have already ceased to exist, representing various fields, including the history of science, mythology, natural philosophy, and oriental studies. ISIS-Symmetry has, however, some interest in the symmetry-related questions of many of these fields. German). FR Andreas Dress, Fakuhat ~ur Mathemauk, Cognmve Science Douglas R. Hofstadter, Center for Research Umvers=tat Blelefeld, on Concepts and Cognition. Ind~ara Unlvers*ty, D-33615 Btelefeld I, Fosffach 8640, F R Germany Bloomington, lndmnz 47408, U S,A [Geome{ry, Mathemauzatmn of Sc,ence] Then Hahn, lasutut fur Kristallographle, Computmg and Apphed Mathemattcs Sergei P. Kurdyumov. Rhem~sch-WestF;ihsche Technische Hochschule, Insutut pr~kladnot matematlk~ ~m. M V Keldysha RAN D-W-5110 Aachen, FR Germany (M V Keldysh lastilute of Apphed Mathematrcs. Russmn [Mineralogy, Crystallography] Academy of Sciences). 125047 Mosk’~a, Miusskaya pl. 4, Russm Hungary. M~h~tly Szoboszlai, l~pit6szm~rnok~ Kar, Educatton. Peter Klein, FB E~ehungsw~sseaschafi. Budapesu M,3szak~ Eg2/etem Umvers~tat Hamburg, Von-Melle-Park 8, (Faculty of Arch*tecture, Techmcal Umvers*ty of Budapest), D-20146 Hamburg 13. F R. Germany Budapest, P.O Box 91, H-1521 Hungary [Architecture, Geometry, Computer Aided Arch.ectural Des*gn] Hzstory and Phdosophy of Science" Klaus Mainzer, Lehrstuhl ~ur Philosoph~e, Umversltat Augsburg, Italy Giuseppe Caglioli, Ist.uto d~ Ingegneria Nucleate - Umvers~mLsstr. 10, D-W-8900 Augsburg, F R Germany CESNEE Pohtecmco d* Mdan, Vm Ponzm 34/3. 1-20133 Mllano. Italy ProJect Chatrpersons [Nuclear Pb.3,sics, Visual Psychology] Poland" Janusz R~;bielak, Wydzml Arch*tektury, Archttecture and Mustc Emanuel Dimas de Melo Pimento, Pohtechmka Wmc~aw~ka Run T~eruo Galvan, Lute 5B - 2 °C, P-1200 Lisbon, Fonugal (Department of Architecture, Techmcal Umvers*ty of Wrc.ct’aw), ul B. Pmsa 53155. PL 50-317 Wrocl’aw, Poland Art and Biology Werner Hahn, Waldweg 8, D-35075 [Architecture, Morphology or" Space StructuresJ Gladenbach, FR Germany Portugal Josd Lima--de-Faria. Centro de Crtstalografia Evolutton of the Umverse: Jan Mnzrzymas, [nstytut Flzyk~. e Mmeralogla, Iasntuto de Invest~ga¢:~o Cientifica Tropical, Umwersytet Wroct’awski Alameda D Afonso HennqOes 41.4.°Esq , P-1000 Lisbon, (Insmute of Theoreucal Physics, Umvers~ty of Wroctaw), Portugal ul Cybulskiego 36, PL 50-205 Wrocl’aw, Pc, land [Crystallography, Mineralogy. H,story of Sc*ence] H~gher-D~menstonal Graphics Koji Miyazak~, Romama" Solomon Marcus, Facultatea de Matematica, Department of Graphms, College of L~beral Arts. Unlvegsihatea dm Bucure~tl Kyoto Umvers~ty, YosMda, Sakyo-ku, Kyoto 606. Japan (Faculty of Mathematms, Umversity of Bucharest), Str Academ*el 14, R-70109 Bucure~it~ (Bucharest), Romama Knovdedge Representatton by Meta.strucmres Ted Goranson, [Mathematical Analysm. Mathematical Lmgmstms and Poetics. S~nus Incorporated. 1976 Munden Point, V~rgmm Beach, Mathemahcal Scm~oucs of Natural and Sccla} Sciences} VA 23457-1227. U.S A. Russia" Vladimir A. Koptsik, F*z*chesku fakultet, Pattern Mathemattcs Berl Zaslow, Moskovskll gosudarstvenny, umversltel Department of Chemistry, Arizona State Umverstty. Tempe, (Physmal Faculty, Moscow State Umver~,b,) AZ 85287-1604, U S A. 117234 Moskva. Russ*a I Crystalphysms] Polyhedral Transformations Haresh Lalvani, School of Arch.ecture, Pratt Insutute, 200 Wdloughby Avenue, Scandinavia. Tuft 9,’~ster, Sksvelaborat~,net, Brooklyn, NY 11205, U S A Baerende Konstruktioner, Kongelige Daaske Kunstakadem, - Ark~tektskole Proportion and Harmony m Arts S. K, Hemnger, Jr. (Laboratory for Plate Structures, Department of Structural Department of Enghsh, Umvers~ty of Norlh Carolina at Chapel Smence, Royal Damsh Academy - School of Arch.ecture), Hdl, Chapel Hill. NC 27599-3520, U S A. Peder Skramsgade 1, DK-1054 Kobenhavn K (Copenhagen), Denmark [Polyhedral Structures. Biomechamcs] Shape Grammar" George Stiny, Graduate School of Architecture and Urban Planmng, Umvers~ty of Cahforma Los Angeles, Swttzerland. Caspar Schwah¢. Ars Geometnca Los Angeles, CA 90024-1467, U.S.A. Rfim,strasse 5, CH-8024 Zhnch, Sw~taerland IArs Geometrical Space Structures. Koryo M~ura, 3-9-7 Tsurukawa, Mach~da, Tokyo 195, Japan U K " Mary Harris, Maths ra Work Project, Tibor Tarnai, Techmcal Umvers~ty of Budapest, Insutute of Educauon, University of London, Depanmem of C~vtl Engineering Mechamcs. 20 Bedford Way. London WCIH 0AL. England [Geometry, Ethnomathematies, Textile Design] Budapest, Mfegyetem rkp. 3, H-IIII Hungary Anthony Hill. 24 Charlotte Street. London WI. England [Visual Arts. Mathematms and Art] Liaison Persons Yugoslavm: Sla’vik V. Jablan, Matemau~.ki mstnut Andra Akers (Internattonal Synergy Institute) (Mathemattcal Institute), Knez Mlhaflova 35, pp 367, Stephen G. Davies (Journal Tetrahedran Assyrnmetry) YU-II001 Beograd (Belgrade), Yugoslavia Bruno Gruber (Symposta Symmetries m Science) [Geometry, Ornamental Art, Anthropology] Alajos K~lm~in (lnternattonal Umon of Crystallography) Cha~rperso~ of Roger F. Malina (Journal Leonardo and lnternauonal Society for the Arts, Sciences, and Technology) Art and Sctence F_xhtbittons L.’iszl6 Beke, Magyar Nemzett Gal6rta (Hungarran Nauonal Gallery), Tohru Ogawa and Ryuji Takaki (Journal Forma and Socmty [’or Scmnce on Form) Budapest, Buda~ri Palota, H-1014 Hungary Itsuo Sak~ne, Faculty of Env*ronmental Dennis Sharp (Curule6 lnternauonal des Cn.ques Information, Keio Umversity at Shoran Fujtsawa Campus, d’A rch~tecture) 5322 Endoh, Fuj,sawa 252, Japan Erz.~bet Tusa (INTART Socmty) ISiS-Symmetry Budapest, P.O. Box 4 H-1361 Hungary CALL FOR PAPERS AND WORKSHOP TOPICS SYMMETRY: NATURAL AND ARTIFICIAL aq~ .~o uo!qq!qx~[ Laomm£S Third Interdisciplinary Ln;u!ld!os!p~muI P~!qz Symmetry Congress of the INTERNATIONAL SOCIETY FOR :A~LLCJIAIINAS THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) S V~J~I NOLLKtlHX~ ~oa ~qvD August 14-20, 1995 Old Town Alexandria (near Washington, D.C.) ~6unH U.S.A. f, xo8 "O’d ’lsedepn~] fu~,em LuAS-S~S~