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bos50 information

practical
Highlights from British Origami 140-149






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Raw Edge: February 1990
The Sincerity of Folding: Part 1. by Dave Mitchell

Since joining this Society not so very long ago it has seemed to me more and more that paperfolding stands in urgent need of a defence.

The pressures against which I feel paperfolding needs defending come both from within and without. From within we have a constant stream of new materials and techniques to assimilate and assess and from without the very success of paperfolding has created more and more interest in its commercial exploitation by specialist paper-manufacturers, publishers and advertisers, and as a result there are a small but steadily increasing number of people who depend to some extent upon folding paper for their living.

Advertisers in particular have not proved especially interested in whether the adverts are good, or even genuine paperfolding at all. If it looks right, it is right. Their interest is in the image, the final product, and external appearance of the model, regardless of the way in which it was made. Not all of this is necessarily bad, but there is a danger that paperfolding will lose its identity and become largely indistinguishable from other similar paper based techniques and crafts, in much the same way that poetry has largely lost its identity and is nowadays often indistinguishable from broken prose.

To defend anything it is first necessary to know what it is, to define what it is you will defend. And my definition is this, that paperfolding is, as its name implies, an indissoluble partnership between the medium, paper, and the activity; folding. Crucially it is a partnership and not a struggle. The paper and the folding ought not to be at war. In good paperfolding they work together.

In my first article I have already set out the first pillar of my defence, that a simple piece of paper has qualities that can be recognised and respected during the process of folding. That paper in fact possesses integrity, and the aim of good folding is to respect, maintain and if possible enhance this integrity in the creation of a finished model. The paper, like the potters clay, is not just a material that is passively worked upon. It itself inspires and directs the folding process. This recognition of integrity is crucial in setting paperfolding apart from and I believe above, other papercrafts such as quilling and papercutting, in which the majority of the qualities of the paper are both irrelevant and ignored.

The second pillar of my defence is this, that the kind of folding that respects this integrity also possesses a unique quality, and in order to discuss it I call this quality sincerity. Folding may be by definition sincere or insincere, depending on how sensitive the folder is to the qualities of the paper, and how much the folder understands the possibilities that the paper offers him or her at each stage of the models design. For if it is the qualities of the paper that set out the possibilities, then it is the quality of the folding that realises them.

It is important to realize that integrity and sincerity in this context are not two separate things. Rather they are the same thing viewed from two different angles. And therefore, as the more astute readers will no doubt already have realised, in plotting integrity in the form of a graph in my previous article I was inevitably plotting sincerity too. Given then, for the sake of this particular argument, that folding may be sincere or insincere how are we to tell the difference between them? What is a sincere fold and what is not?

At first this question seems to beg for a set of rules. Thou shalt not cut. Thou shalt only fold from squares. Thou shalt only use duo paper. But rules have a habit of becoming rapidly outdated and in any case I am concerned here with defending, not with setting limitations. In this context it is instructive to recall that one famous American folder was once incautious enough to publish a brief set of rules for paperfolding which included the precept that a fold once made must not be unmade. Which proves that the opposite is true of rules.

On the contrary what I am stretching towards is a set of concepts that are sufficiently flexible to be adapted to new circumstances and techniques, but sufficiently well drawn to provide a guide to what is and is not paperfolding, and more important a measure as to what good paperfolding ought to aim to be. It seems to me then that the sincerity of any particular fold can be assessed in two ways, firstly in relation to the qualities of the paper alone, and secondly, and with a much greater degree of theoretical, but not practical, difficulty, in relation to the whole sequence of folds of which it forms a part and the changing qualities of the paper within that sequence of folds.

The first approach is fairly simple, and not surprisingly leads at first down well-established pathways. Consider for instance that paper is foldable simply because it is flat and thin. It is an old trick to challenge a strong man to fold a sheet of newspaper in half more than eight times. It simply cannot be done. The paper quickly becomes too thick to fold. We are all familiar with more complex models that also demonstrate this particular principle, points that must be thinned to the extent that essential crimps become little more than inelegant bends, thick apexes that must be somehow developed into heads and beaks. The fault is the same. The folding quite clearly lacks sincerity.

Consider again, for instance, that any fold of 180 degrees in this same flat thin sheet diminishes the surface area of the paper, and most subsequent folds will do likewise. As the complexity of the model increases the surface area grows smaller. The fact that this process is inevitable in two dimensional models immediately suggests that perhaps to allow the paper to occupy three dimensions would be more sincere. But even in two dimensions folding methods can vary in sincerity in relation to this basic quality of the paper. A model, for instance, that starts from a square, and immediately folds most of the paper away to form a hexagon, and never makes use of that paper again, seems to me to be basically flawed, however realistic the finished model may be.

It is a simple test for this aspect of sincerity to colour the visible surfaces of a model and then unfold the model and compare the proportions of the coloured and uncoloured areas. I would suggest that in a model where the vast majority of the papers original area has been lost the folding is prima facie (although not conclusively) in conflict with the paper.

Not conclusively, of course, because there are so many other factors to consider, but in general my own feeling is that as much of the paper as possible ought to be used in making one or other part of a model, or in the case of a working model of course in making the d .... d thing actually work. It is these 'other factors' that eventually grind this approach to a halt, for once the arguments move beyond simple qualities such as these the conclusions to be drawn are far from clear. This is because the qualities of the paper are being considered in isolation, as if they were set at the start and not subject to change, which in the folding process is certainly not true.

This aside, however, there is quite clearly a point at which paperfolding ceases to be paperfolding. In fact two points. One is where the medium ceases to be paper, and is for instance plastic or foil, and the other where folding ceases to be folding, and becomes for instance crushing or crumpling. In both cases the paperfolding partnership has been dissolved. In one integrity has been lost. In the other sincerity has been thrown away.

All this seems to me to drive towards a not altogether easily acceptable conclusion, which is that there are some potential models, or indeed, classes of models which are incapable of being folded in a sincere way, where the folding must, apparently inevitably, be in conflict with the paper, and the model be triumphantly wrestled from it. We may have to accept that some models cannot be folded without sacrificing the very essence of what paperfolding is all about, and that in fact further advances in techniques may only succeed in turning creative paperfolding into an obscure branch of the mathematical game of topology.

Personally I hope, sincerely, that this will not prove to be.

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Raw Edge: April 1990
The Sincerity of Folding: Part 2. by dave Mitchell

In the first part of this article I have maintained that to try to assess the quality of a fold in static terms is interesting but inadequate. Sincerity in folding is a dynamic quality and consequently requires a dynamic approach.

I would suggest that the universal starting point, a blank unfolded sheet, possesses its own integrity but since it is unfolded it has as yet no sincerity. Sincerity only occurs in the active process of folding itself. And here we come across what may become known to future, much cleverer, origamic philosophers, with much more time to waste, as 'the problem of the first fold., which is the best first fold to make? Which is the worst? Is the folding more sincere if the edges meet or if they don't? Need the first crease cross the paper from edge to edge at all?

The answer it seems to me is simple. Since no previous folds exist all first folds are equally acceptable. (But this is largely a matter of feeling and not sustainable in logic.) But there it stops. Once a single fold is in place the choice of a subsequent fold begins to become limited. Too thin an angle to the first fold is quite clearly unsincere. But most options are still open.

As the folding sequence proceeds, however, the options narrow and narrow, simply because the paper is becoming more limited in area, the movable flaps and points less free, and the folding structure more complex. And at this stage one of two things happens. Either the narrowing of the options means that folds must be forced into the model to provide the freedom that the folder needs, or the folds seem to naturally open up the opportunities to develop further folds, which open up still more opportunities, which....

So it becomes clear that the essence of what I have called sincerity lies not only in that each fold takes account of the properties of the paper at that stage in the models construction, but that each fold takes account of both the previous and the subsequent folds as well. In other words the secret of sincerity in folding lies in the sequence of folds rather than in the individual folds themselves, and the most sincere folding is that in which the folds in a model flower naturally from start to finish. This is I believe what gives the feeling to certain models that they have folded themselves.

This concept of sincerity imposes no particular limitations on technique. There is no greater merit in a blintz than in a sink, except within the specific context in which a specific fold occurs. And this quite naturally points to another conclusion, and that is that it is quite possible for the same model to be folded in two different ways, the first sequence being sincere and the second not. The result is the same. The folding is not. Consequently, in paperfolding, as opposed to any other form of art with which I am familiar, the appearance of the finished model is necessarily secondary to the way in which it has been made. Secondary but not unimportant. But this is difficult territory and I quit it here.

Yet another perspective on sincerity can be gained by considering the contract between folding in cloth, for instance napkins, and in paper. Cloth, of course folds easily enough, but it will not naturally take a crease. A fold in cloth is a loose rounded form and if the cloth is unfolded the form generally disappears.

A fold in paper is something else entirely. Paper naturally takes a crease when folded. The fold is sharp and angular, and when unfolded paper retains a distinguishable crease. Wet paper on the other hand folds much like cloth, with the advantage so far as sculpting is concerned that it will retain the loosely folded form when dried, in a way that cloth will not. wet paper in fact can be moulded rather than folded into shape (although not all the techniques used to create a particular model are necessarily of this kind).

This of course raises an intriguing question. If in wet folding the paper is essentially a different medium treated in essentially a different way then is it really paperfolding at all? Or have the paperfolders who have followed this course in the interests of greater realism in fact abandoned the challenge of paperfolding in favour of the essentially easier and less disciplined art of papermoulding?

Perhaps the solution to this enigma lies in those folds which have used rigid fold lines in one part of a model to draw the paper into graceful curves in another, curves which only exist and are maintained by the overall rigorous fold structure of the model. So here is a challenge for the future. Can these techniques, pioneered, but not yet fully developed, in three-dimensional geometric folding, be applied to the representational form? Can we, for instance, produce a three-dimensional elephant where every fold is sharp and clean but which looks as though it is as soft and fluid as the best of the wet-folded models we have yet seen? Only time will tell.

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Models: June 1990
Spring into Action Model by Jeff Beynon

Remove a 2' wide strip from an A4 (113/4 x 8 1/4) Silver Rectangle. Colour side uppermost, mountain fold into sixths width ways. Then mountain fold into twelfths, lengthways. Lastly, valley fold the 'diagonals' of each small section. As befits twist folding, undertake this pre-creasing as accurately as possible. Consider the twist folding as six (widthways) sections, then begin, working from one end of the paper to the other. As you proceed, align the sections of twist folding so that the spring action (shown in final drawing) is present all the way through. When complete, check that length-way raw edges do not snag against one another, and that the spring action is free running by holding in the middle of the model, as shown, well away from the raw edges, and carefully pinching and releasing to produce concertina-like pump action.

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Clean Sheet : August 1990
10 Classic fold choices by Dave Brill

By now most of you will have seen Paul Jackson's new book. entitled 'Classic Origami'(published by The Apple Press at 7.95) which is a most interesting collection of traditional, Western, and Eastern designs. Two or three of us had copies of the book at a recent North West Mini Meeting and we folded several items from it during the afternoon. Inevitably, some we liked and some we didn't, and this promoted a short discussion about whether or not the subject matter was indeed 'classic'.

Of course, Paul Jackson's choice of the subject matter of the book is his own and I thought it might be interesting, therefore, to pose the hypothetical question: if you had to choose ten 'classic' designs which would form the basis of a book of a similar emphasis, what would they be? If this proposition interests you, perhaps you would like to write to the magazine listing your ten examples and giving three words for each design, justifying, at least in your eyes, your choice. Perhaps I could also add that Paul has limited the complexity of the designs which he has chosen for practical reasons: therefore we don't see any 'Cuckoo Clocks," or "Last Suppers,' and perhaps it's reasonable to suppose that a true classic design is not necessarily a highly complex one.

To start the ball rolling, therefore, I've given below a list of ten designs which I consider classic, additionally giving the source or book from which you may track down the folding instructions.

  1. CUBE by Shuzo Fujimoto 'Sequence, Surprise, Economy'. (British Origami Magazine No-94 p25/26).
  2. LAZY SUSAN (Traditional)'Round, Unique, Climax'. (Best of Origami by Sam Randlett).
  3. ASTRO TUBE by Stephen Weiss 'Simple, Flies, Spins'. (Wings and Things by S. Weiss).
  4. DISH by Philip Shen 'Beauty, Lock, Rigour'. (BOS Booklet No. 18 page 24).
  5. IRIS by Akira Yoshizawa 'Form, Rigidity, Volume'. (Origami Utsukushii) by A. Yoshizawa).
  6. LIZARD by Tomoko Fuse 'Obvious, Action, Simple'. (British Origami Magazine Number 126 p36-40).
  7. PINCH PUPPET by Eric Kenneway 'Action, Colour, Appeal'. (Folding Faces by E. Kenneway).
  8. PERFORMING SEAL by Fred Rohm 'Combines Traditionals, Engineering'. (Secrets of Origami by R. Harbin).
  9. CUBIC BOX by D Brill 'Sequence, Challenge, Satisfaction'. (Paper Folding for Fun by E. Kenneway).
  10. JACK IN THE BOX by Max Hulme 'Moves, Action-Toy'. (British Origami Magazine No.61 p23-26).


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Airmail: October 1990
Report from Hungary

A large and bulging parcel arrived from Susanna Kricskovics with lots of goodies inside; booklets, diagrams, photos, folds, news and a beautifully written six page epic letter. As you will have gathered, origami in Hungary is alive and well! Their group, Szorakatenusz Origami kor Keskemet, have an impressive membership of 266 (at last count) and receive four magazines a year. These consist largely of diagrams begged and cribbed from all over the world. They also offer a library service of 30 or so precious books, (if you have any spares, why not send them over to Kecskemet, dob. Krt 3.Viii.26, Hungary H6000).

This year they held an exhibition in Tatabanya at a children's Culture House. There were about 200 people at the exhibition and Susanna had to teach a group of 140 children using a microphone! The next exhibition was in Gyoti (North-West Hungary) and it then moved to Zalaegetszeg in the South West, this last involving a trip home that took seven hours. The hall was over 70 sq. metres with a high roof (imagine filling that!) so they hung folds from the walls, a 1000 crane exhibit and some kusudamas from the roof and the centre-piece was a selection of folds from other countries including a musical trio by Lang, folded by Laszlo Szemetey. Another feature of the event was a huge paper deer folded from a sheet 10x10m with a horn folded from a sheet 6x6m (see photo).

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Letters : December 1990
10 Classic fold choices from Rick Beech

  1. ROSETTE by Paul Jackson 'Simple, Climactic, Elegant' ('Classic Origami'by Paul Jackson, page 74)
  2. VALENTINE by Pat Crawford 'Ingenious, Satisfying, Impressive' ('Origami 3'by Robert Harbin, page 108)
  3. DAFFODIL by Ted Norminton 'Beautiful, Rigid, Realistic' (Classic Origami'by Paul Jackson, page 64)
  4. PANDA by Akira Yoshizawa 'Clever Contrast (Colour), Realistic' ('Sosaku Origami NHK'by Akira Yoshizawa, page 100)
  5. HANG-GLIDER by Akira Yoshizawa 'Glides Gracefully, Simple' ('Sosaku Origami NHK'by Akira Yoshizawa, page 144)
  6. ASHTRAY by Francis Ow 'Methodical Pre-creasing, Climactic' (Spring Convention BOS)
  7. DOUBLE-STAR FLEXICUBE by Dave Brill 'Flexible, Challenging Modular' (Unpublished)(Now 'Brilliant Origami' by David Brill, page 98)
  8. BUGATTI by Max Hulme 'Neat, Square, Pleating' ('Origami: A Step-by-Step Guide'by Paul Jackson, page 125; and BOS booklet number 15, Max Hulme: Selected Works, page 33)
  9. HEDGEHOG by John Richardson 'Pleating, Technical, Magnificent! ('Paperfolding for Fun'by Eric Kenneway, page 86)
  10. LION (FACE) by Kunihiko Kasahara 'Precise Contrast, Striking' ('Origami Omnibus'by Kunihiko Kasahara, page 52)


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Models : February 1991
Folding Method for an EXACT Heptagon by Humi Huzita



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Paper Folding in Ancient China : April 1991
Dr. Philip Shen discusses the origins of paper folding

Since British Origami No. 143 (p. 20) and No. 144 (pp. 12-13) referred to what I have or have not said about paper-folding in ancient China, perhaps I should respond to help "set the record straight".

  1. It is always my belief that paper-folding began in China (as the making of paper did in the 2nd century B.C.). The question is one of available historical evidence.
  2. Dr. Joseph Needham of Cambridge University, in his multi-volumed, monumental work on Science and Civilization in China, wrote that 'another widespread art, more related to topology like the Linked Rings, was that of the folding of paper (che chih shu), mentioned in a famous poem of Tu Fu" (Volume 3, on Mathematics and the Sciences of the Heavens and the Earth [Cambridge University press, 1959], p. 119). Some years ago, when Dr. Needham visited us at The Chinese University of Hong Kong, I told him I was unable to find any reference to paper-folding by this famous poet of the Tang Dynasty (618-906 A.D.). He wrote to me afterwards to say that his remark was based on a 1930 article by an Italian mathematician G. Vacca, who, as it turned out, misread or mistranslated the line of Tu Fu in question (more below). If paper-folding began in China, he said, the evidence has to be sought elsewhere.
  3. This appears to have been done in 1985, Vol. 5, Part 1 of Science & Civlization in China, on Paper and Printing, by Professor Tsien Tsuen-hsuin. The book provides the most comprehensive and authoritative discussion on the subject to date. Professor Tsien devotes a section (of 10 pages) to 'Papercraft and Recreational Use of Paper', including of course paper-folding. About its origin he writes as follows:

    Although paper-folding probably flourished in China for many centuries before it spread worldwise, there is no clear indication of how early it began. From all available evidence, its origin probably was not later than early in the Thang [Tang] dynasty, for several artificial flowers of folded and cut paper have been found in Tunhuang, and they show highly sophisticated techniques in paper-craft (pp. 126-127, with pictures of decorated paper flowers from Tunhuang, c. 10 cm. in diameter, kept at the British Museum).

    Professor Tsien mentions in a footnote the mistake in Vacca's reference cited in the earlier volume.

  4. Dr. James Sakoda in British Origami No. 145 (p. 15) cited I believe the same article of Vacca (1930). It was said that a chessboard was folded out of a piece of paper - done, I think, if it is for the Chinese chess, by folding 10 horizontal lines and 9 vertical lines, forming a grid of 72 squares. I can confirm that this is still quite common today. The lines are then traced out in ink, to make them more visible, but that is not always possible, or necessary. This way of folding a chessboard was done as far back as the time of Tu Fu (712-770), but according to Dr. Sakoda, Vacca gave no source for his statement. From what Dr. Needham wrote me, however, it was a line of Tu Fu's poem that was Vacca's source. The line actually reads (in literal translation) "drawing on paper for chess', which Vacca mistook to mean 'folding paper for chess'. But if the players first fold and then draw on the folded lines, Vacca's mistake may have been a point of inference, since one could draw the lines without necessarily folding them first. Dr. Sakoda noted that even if what Vacca said was true, it was 'no indication that origami as we know it was practised at that time". I agree. But what about the flowers in Tunhuang?


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Because it's There: Idiot Savant : June 1991
Robert Lang on persuading a computer to fold.

One of the best ways to learn the fundamentals of origami is to try to teach it to a complete beginner. I have been doing that recently, and it is not an exaggeration to say that my pupil is a complete and utter idiot. Not only is he unaware of what it means to fold paper, he doesn't even know what paper is. "You stupid numbskull!" 1 rage. "How can you fail to understand such an obvious concept? Don't you have the brains God gave a cockroach?!" To which he blithely replies, "System Error ID=28," or some such thing. You see, my pupil indeed hasn't the brains God gave a cockroach. My pupil's brain is a 68000 micro-processor. He is a Macintosh SE microcomputer and, slowly and painfully, he is learning what it means to fold origami.

To say that a computer has no smarts is not exactly true; as everyone is aware, computers are capable of awesome feats of computation. Perhaps I should amend my statement: a computer may have smarts, but it hasn't the sense that God gave a cockroach. A computer cannot decide anything for itself; everything it does occurs because somebody told it what to do, and told it in very specific terms. What I decided to do several months ago was to tell my computer how to fold paper, maybe I could also tell it how to draw diagrams, thus relieving myself of that onerous chore. The process has been far more instructive for me than for my machine. In this article, I'm going to describe this project and some of the things I learned about clever origami and dumb machines.

If you are teaching an origami design to someone, you must describe the folded paper in terms that your pupil can understand. With an advanced folder, you can use concepts at a fairly high level: "Begin with a blintzed Bird Base. Sink the top corner of the model halfway." With an expert, you can describe ten-step sequences in a phrase or two: "Make this point into three; turn all the corners into Bird-Base points and symmetrize the model." To a beginner, though, you must be much more explicit: "Fold one corner of the square to the opposite corner." But what do you do if your pupil doesn't know what a "square" is, doesn't know what a "corner" is, and doesn't know what "fold" means?

My first step was to define the terms I would be using in terms the computer understood. That meant devising a mathematical description of a piece of paper and the various operations that could be performed upon it. That was easier said than done. I have to specify the shape of an arbitrarily folded paper in sonic sort of mathematical description that could ultimately be translated into the ones and zeros that are the language of computers.

The concept of a mathematical description of origami (as opposed to the pictorial one we are all familiar with) is not new. John Smith devised an algebraic description of paperfolding, "Origami Instruction Language" (O.I.L), described in his essay, 'The Nature of Paper Folding," and in The Origamian, vol 13, no. 2(1975). The language never caught on in the origami world, however, probably because it is very difficult to visualize the paper solely from the algebraic description. However, it contained some powerful ideas. Notable among them is the abstraction of the folded paper to a collection of geometric objects; in O.I.L., those objects are points (corners), edges (creases) and layers (of paper).

Try this experiment: Take a typical two-dimensional (i.e., flat) origami model - Montroll's Stegosaurus will do - and unfold it. If you did a neat job of folding, the surface of the square will be divided up into many polygons formed by the creases and their intersections. Now, if you number those polygons and retold the square, they will have moved around with respect to each other; some will end up on top of others, but each individual polygon the same size as it was in the unfolded paper. Each is, bounded by the same creases it was in the unfolded paper, and each touches the same set of polygons it touched in the unfolded paper. Only the position and orientation of each polygon have changed. A mathematician would say that the connectivity of the network of polygons, creases, and intersections is invariant, while their position and orientation are not. More importantly for our purposes, if you could write down the location and orientation of each polygon in the folded model, the location and orientation of each edge of each polygon, and the location and orientation of each vertex of each edge, you would have a complete and unique description of the folded model in the form of a few lists of numbers.

Actually, you'd probably need some more. Not only do you need to write down where each polygon is, but you also must keep track of which polygons are attached to which others-what I called connectivity-and it would help to know which edges make up which polygons, which vertices are attached to which edges, and vice-versa. And, if we're using two-toned paper, we need to note which side of each polygon is facing upward when the paper is folded, since in some cases the coloured side will be facing up and in others, the white side faces up.

That's an awful lot of writing (especially for something as complex as a stegosaurus) and a lot of numbers to keep track of. This sort of bookkeeping would be tedious for a person to write down and even more tedious to follow. O.I.L. simplified the process somewhat by only writing down what changed from step to step. O.I.L.'s main drawback was that it was impossible to pick up in the middle of a sequence. If you ever lost the first few steps, you would be completely lost. A more thorough mathematical model would include a complete description of the configuration of the paper at every step of the way. A complete description might entail hundreds of numbers and would be extra-ordinarily tedious for a person to keep track of. However, what is tedium to a person is mother's milk to a computer. A computer would have no trouble remembering the hundreds of numbers that represent a folded model.

And indeed, mine didn't. For my program I chose to use a language called Object Pascal, which is a highly structured programming language that permits one to define complicated objects such as "vertices," "edges,' and "polygons," in terms of their constituent parts. A "vertex," for example, was made up of its coordinates in real space, its coordinates on the unfolded square, and a list of any edges that were connected to it. An "edge" contained a list of its vertices and a list of any polygons that were bounded by it; a "polygon" contained a list of its edges and a note as to which side (coloured or white) faces upward, among other things. An important part of the definition of each object was the list of what it was connected to; the vertex is connected to the (pause) edge bone, the edge bone's connected to the (pause) polygon bone, and so forth.

(Those readers with visions of telling the computer to design new models should note, however, that such a description says nothing about how you would go about actually folding the thing. A mathematical description of a folded piece ofpaper tells where you are, not how you got there.)

(I should also take this opportunity to express my indebtedness to Jack Fastag, a Cornell University student who independently took on the same project, and when we eventually met, gave me many tips and ideas and introduced me to object-oriented origami programming.)

Now, at this point (skipping over many months of programming and debugging), I could represent any fold mathematically, and with a bit more work, I could convince the computer to display an image of the folded paper on the screen. However, my goal of a dedicated slave faithfully converting my designs to diagrams was still distant. The problem was that I had to fold a paper version of the model; label each vertex, edge, and polygon with a number; calculate its coordinates, both in real space and on the square; and then individually enter all the numbers. What I really wanted to do was to begin with a square on screen and interactively 'fold" the square into a finished model.

Up to this point, I had been teaching the computer about nouns: creases, folds, paper, edges, vertices, and polygons. It was now time to begin with verbs. Just what, to a computer, is meant by "Make a valley fold"?

When we make a valley fold, we pick up a flap, move it to another location, and flatten the paper, creating a new crease. We specify the location of the fold by (usually) telling the folder to bring one point to another, as in, "Fold one corner of the square to the opposite corner and flatten the paper." Obviously, one cannot physically reach into the screen to pick up one corner of the square displayed there. It was necessary to first create paradigms for the actions of "picking up a corner," "moving it to another corner," and "flattening the paper," that would be applicable to a computer screen. Second, I had to devise an algorithm to implement the action.

The paradigm was straightforward. The Macintosh, like many graphically oriented computers, uses a mouse (or other device) to move a pointer around on the screen, and a button, which you press to indicate some action. For my paradigm, the "pointer' for a valley fold looked like an open hand. You position it over the flap to be moved, press the button (the hand closes on the point, thus indicating that you've picked it up), and move the pointer (and flap) to its new location, where you release the button (and flap). New creases are to follow.

Ah, but where do I put the new creases and how do I add them to my model? The first part was easy. A little experimentation with a piece of paper will show that when you bring one point to another and flatten the paper, the new creases lie on the perpendicular bisector of the line connecting the two points. By constructing that line, the computer could find its intersections with all the existing polygons and split any that the line crossed. By this means, it created new edges wherever the fold would occur. Then, any polygons that lay between the line and the moving point were reflected around the crease line to simulate the action of folding.

(Actually,there was a bit more to it than that, because not all polygons that the bisector crossed needed to be split, and not all that are split needed to be reflected. Suffice it to say that the hardest part of this entire project was developing an algorithm that could figure out what to split and what not, and what to reflect and what not, and that could handle arbitrarily complicated paper configurations.)

This algorithm was sufficient to persuade the computer to do Pureland Origami. Pureland Origami (coined by John Smith, who seems to have been remarkably prescient about computerized folding) is a version of origami that restricts itself to a very small subset of folding techniques; Pureland Origami is that which can be done solely with valley folds performed one at a time. It is very easy to teach, and is a good starting point for instructing beginning folders, be they animate or silicon.

Well, although I had hopes of typing in, 'Fold a bullfighter, cape, and, bull from a single sheet" and have the diagrams for same come flying out of the laser printer, by the time I finished the program, I was somewhat less ambitious (and, to be honest, I was getting tired of messages notifying me that I had erased some essential portion of the computer's memory and must restart the machine). The computer can now begin with a square on screen and, using the mouse, fold it into various configurations, turn it over, rotate it, and pan around over the image, but that's about all. Still, it's enough to fold some simple shapes. Unfortunately, there are still quite a few bugs and the program still crashes spectacularly on occasion. However, it was, as we say, a learning experience - for me, if not for the computer. I have waxed lyrical on many occasions about how a computer makes a superior draftsman for origami diagrams, but for actual folding, I am happy to report that there is still no substitute for a square of real, as opposed to virtual, paper, and a pair of thoroughly human hands.

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Japanese Origami in Denmark : August 1991
Claire Chamberlain watches the Japanese watching the Danish folding paper.

Last night on a commercial TV programme, "World Adventure", which introduces Japanese culture to various countries each week, origami was the subject. The programme opened with the show's hosts explaining that origami originated in Japan's Heian era (794 - 1185), but became a popular craft in the Edo era (19th century). It was also explained that the word 'kami'(paper) derives from 'kami' meaning God.

We were then taken to Denmark, where it was explained that 'kirigami', paper cutting, is the popular paper craft, mainly thanks to the influence of Hans Anderson.

Mr Yamaguchi, whom I had the pleasure of meeting at the last NOA Symposium, was shown teaching Danish primary school children, who were quickly able to fold on their own from books. When one girl created her own model pig, this amazed the 'hosts' as individuality is not exactly encouraged in Japanese schools or the arts. Yamaguchi then accompanied two of the children home, who promptly began to teach their family.

The next scene is a restaurant, where the manager and Yamaguchi exchanged napkin folds. Yamaguchi's traditional crane with pleated wings went down very well with customers. Then on to a pub where cups folded from paper really impressed the clientele. The wide range of kirigami available for sale was then shown together with an expert who seemed somewhat disdainful of what he considered the limited possibilities of origami. (I'm afraid I don't understand Danish, and couldn't read the Japanese subtitles!)

Back to origami, and a newspaper advertisement for an exhibition by Thoki Yen. Thoki was introduced (including his age), and it was explained he got into origami through his joint interests in kirigami and magic. Displayed models included a flexagon and his famous rabbit. Move on to a Senior's Day Care Centre, where Yamaguchi, assisted by his new Danish friends teaches module flowers to the aged. The Occupational Therapist was very enthusiastic, praising origami for it's maintenance of hand flexibility and stating her intention to keep on teaching it.

Finally, the climax, folding a dinosaur from a 5m square - it only took 4 hours! Both Thoki and Yamaguchi looked exhausted - and I always thought hobbies were for relaxing! The model was then placed outside the door of the exhibition hall. We then saw Thoki with a huge folded box which turned out to hold a stunning bunch of folded flowers. Other exhibits included Correia's iguana ("Is that really origami?" queries the host, a pig's face, and loads of children's works as well as the inevitable kirigami. It is reported that over 200 attended. The aforementioned "cutter" is impressed, but still doesn't seem sufficiently won over to throw away his scissors!

The programme ended in the studio with an impressive array of works by Akira Yoshizawa, with the comment that some models took over a month to fold (very encouraging for budding folders), and the commentators still "oohing" and "aahing" over what is still really stuff for kids in Japan. When will origami be regarded in the same light as "ikebana" or the tea ceremony ..... Well, at least the programme did close with the suggestion that viewers take a new look at origami and try some for themselves!

Sorry I didn't get a copy of the programme, but I'm one of those antediluvians that still don't have a video!

On a different note, although it is hard to believe, I've now been in Japan for a year and it's time to get back down under, where I hope I can maintain my enthusiasm, and find the time, to spread the work and recruit a few more origolics.

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