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practical
Highlights from British Origami 120-129







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Because It's There October 1986
Robert Lang discusses modularity.

Frankly, if someone had told me a year ago that I would have anything but contempt for modular folds, I would have laughed in their face. "Modular origami? That's an oxymoron!" I would have scoffed. "Modular folding is for those who would rather make 1 fold in 90 sheets of paper than make 90 folds in one sheet of paper. It requires no special ability - simply a dulled mental state that adapts itself to menial drudgery of the sort encountered when sorting screws, counting blades of grass or folding and gluing 24 bird bases together. It is a sort of pastime designed for those persons who need to fold something 144 times before they get it right."

Fortunately for all concerned, no one told me such a thing: consequently, I never made the preceding response and quite probably saved myself from a certain amount of embarrassment upon changing my tune, as well as a chance of bodily harm, depending upon who I responded to. Meanwhile, I have received several quite interesting modular folds in recent months (and invented one or two myself) which have convinced me that there may very well be some redeeming quality possessed by the genre.

Let's first look at what we gain by using many sheets of paper. We get edges. Cut a square into 4 smaller squares and you double the amount of "edge" available for the same amount of area. And, as readers of this column are by now doubtless aware, the more available edge there is, the more efficiently one can make appendage-like shapes. So, for example, 3-D stars (or to get technical, stellated polyhedra) are natural subjects which exploit this peculiar attribute of modular folds.

Is modular folding necessary for stellated polyhedra? Probably not. There are those (myself included) who derive a perverse delight from making stellated thingamagigs from single sheets of paper. The grandaddy of them all is the "Omega Star" which seems to have been invented by everyone at one time or another (and which, by the way, is generically called a stellated rhombic dodecahedron). Then ther is my own entry into the field of pointedness, a stellated cuboctahedron (a.k.a the Stellated Cubocta of BOS Spring Convention '85, sort of a truncated octahedron with points), Pat Crawford's stellated octahedron, the Jackstone (a stellated hexahedron) and more. These efforts at pointliness pale, however, beside the "sweetgum balls" of John Montroll. Conceptually, they are quite simple. You simply sink each of the points of the "Omega Star" twice, and then repeat the process ad infinitum to generate stars of 24, 48 and (God help us) 96 points. The process could be continued (John had only taken it to 96), I suppose, but the idea has been demonstrated; carrying it further seems rather, er, pointless. Heh heh.

In all of these stellar one-piece folds, however, as the pointliness increases, the model accumulates ever increasing amounts of excess paper that must be hidden away, excess paper that has a disturbing tendency to reappear at undesirable times giving one's star the appearance (to use another's words) of a burst concertina. Or in the case of the 96- pointed Omega Star, 96 burst concertinas. the larger the points, the smaller the amount of paper that contributes to the visible portion of the model. Whereas, if we had used lots of individual sheets, the design could be far more efficient - half (or more) of the paper can actually contribute to the visible surface of a modular fold, and this efficiency is what makes modularity appealing.

The whole idea of modularity rests upon some sort of basic unit that is repeated over and over again. And while many module designs must be fitted together in a specific way to make a single star, it is tempting to make a basic unit that can serve as the basic unit for several different stars - sort of a "configuration independent" module. For example, a unit (courtesy of Dave Brill) which can be assembled into a variety of shapes is shown in figure 1.

One of the beauties of origami is its ability to produce figures that belie their means of construction. Fo a multi-piece construction, one of the most desirable effects to acheive is monolithic solidity, or in less verbose terms, a rigid construction, without resorting to glue or tape. An example of a module which is remarkable for the rigidity of the finished star is shown in figure 2 (by Roy Roberts). It is also remarkable for the fiendish difficulty involved in assembling the last few units. The module is from an American dollar -- a number of them can be assembled to make any stellated uniform polyhedron, by replacing each edge of the polyhedron with the crease joining the two long triangles' bases together. (Definition: 'uniform polyhedra' have faces composed of one or more types of regular polygon. Actually, there are more possibilities for stars than the uniform polyhedra (e.g. Archimedean duals, prisms and antiprisms), but a complete enumeration will have to wait for another column). A stellated octahedron takes 12 dollars; a stellated dodecahedron takes 30 dollars; a stellated snub dodecahedron takes 150. At this rate one could easily go bankrupt.

What else does a high edge-to-area ratio net us? Colour changes! Quite a few modulars are made with rather mundane shapes but rely on detailed and intricate patterns formed from the colour reversals. Even the lowly cube offers several unique possibilities. The simple (6-crease) shape shown in figure 3 (from a 3x4 rectangle, by Lewis Simon) can be woven with 11 others to make the two-tone woven cube shown. The woven appearance is appealing in itself, but that twisted hole in each face also adds to the effect (and, as we have mentioned, there is something aesthetic about a hole in an apparently uncut piece of paper). This model is also remarkable for the sturdiness of the mechanism which locks it together. While in the stellated shapes, we used the same basic shape to make different polyhedra, we can take the opposite tack with this basic structure, and make differently patterned basic units which are woven together to make different patterns on the same cube-with-a-twisted-hole.

As an example of this, and as a final problem for the reader, observe the shape in figure 4 (by me). 12 of these get woven together in the same way as the Simon cube to produce the cube in figure 4 with the woven chain-link pattern running around its edges. I'm rather fond of it because of the large number of "islands" of one color in a sea of another. The problem of folding it from a 14x3 rectangle is left as an exercise for the reader.



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The Blintz, An Etymological Investigation, December 1986
Eric Kenneway on why the blintz is so called.

The following talk was given at the Cobden Hotel, Birmingham, during the Autumn Convention, October, 1986.

When translating Nakano's 'Easy Origami' (Viking Kestrel, 1985), one of the things I had to consider was what to do about the Japanese word 'zabuton-ori'. I could have translated it literally as 'cushion fold', but we know it as the blintz. By 'we' I mean paperfolders and not the rest of the British public, ninety-nine percent of whom are completely unfamiliar with the word. The remaining one percent consists of the Jewish community and, I shall argue, they don't understand it to have anything to do with folding corners to the centre of anything.

In the end I stayed with 'blintz' in my translation. It's become part of the British paperfolders vocabulary because we learnt it from American paperfolders; they use it because they eat them. (Blintzes are pancakes which contain a cheese or other filling. Blinchiki, the ancestors of blintzes, originated in the Ukraine and were introduced into other parts of Eastern Europe and ultimately North America by Jewish emigrants.) All the same, it seems a bit silly to substitute one foreign word for another foreign word (you won't find 'blintz' in English dictionaries), particularly if we use it in a way which misleads those people who are familiar with it.

I first questioned the way we use 'blintz' in an article entitled 'American Cuisine and Base Nomenclature', British Origami No.31 (1971). I quoted from a piece in 'The Observer' magazine (4 July, 1971) on American cooking. Therein it said that blintzes are rolled: there was no mention of corners being folded to the centre. Subsequently I had an opportunity to lay my hands on a lot of Jewish cookbooks. I now have a collection of blintz recipes from stodgy, traditional, fried blintzes to the new healthy, lightly baked ones. I was of course concerned not with the cooking, but to find out what the authors had to say about getting a blintz from a large pan on to a small plate. This is what I discovered:

Joan Nathan in 'The Jewish Holiday Kitche' (Schoken Books, NY, 1979) says, "Tuck in the opposite sides and roll up like a jelly roll." (Jelly roll is American English for Swiss roll.)

Jane Kinderlehrer, in 'Cooking Kosher the Natural Way' (Jonathan David, Middle Village, NY), distinguishes between the pastry and the filling. In her recipes for blintzes, she calls the pastry a 'bletl'. So as it's the pastry which is equivalent to origami paper, perhaps paperfolders should use 'bletl' instead of 'blintz'. That's assuming that the bletl is folded, but what does she say? She says, "Roll up ..."

Evelyn Rose is the leading Anglo-Jewish write and broadcaster on cooking at present. In her 'Complete International Jewish Cookbook' (Robson Books, reprinted 1984) she spells 'blintze' with an 'e' and she says, "Turn in to enclose and roll up." Blu Greenberg, in her 'How to Run a Traditional Jewish Household' (Simon & Schuster, NY, 1983), gives the clearest set of instructions so far. She says, "Place the cheese filling on one end of each leaf, and start rolling the dough over the cheese. Then fold over the sides, and continue rolling until you have a blintz, neatly sealed."

A new element is introduced by Anne London and Bertha K. Bishov in their 'Complete Jewish Cookbook' (W. H. Allen, reprinted 1983). The authors tell us to "fold edges over to form envelope" for a cheese blintz, but for a different recipe for use during Passover, the authors conclude, "... fold sides in and roll up." Does this mean that whether you fold or roll the blintz (or bletl) depends on whether you serve it on a feast or a fast day? I've checked: apparently not.

Florence Greenberg is the only author doesn't mention rolling up. She was of an earlier generation and died a few years ago at the age of ninety-nine. In her 'Jewish Cookbook' (Hamlyn, reprinted 1985), for a meat blintz, she says, "Fold in half and then in half again into three-corner pieces." Elsewhere in the book, for a blintz with another type of filling, she says, "Fold over into a triangle or ..." apparently as an afterthought, "... envelope shape." That's three different treatments from her.

So these sources contain one reference to a triangle shape; one to a doubled triangle shape; two to an envelope shape and six to rolling up. The envelope shape is outnumbered by 5 to 1 and the rolling up technique clearly comes out on top. This is confirmed by my researches among Jewish housewives: they all say they roll it.

The scene now moves to Grenoble and the FIPP exhibition in November, 1983. American expatriate Gershon Legman, who claims responsibility for introducing 'blintz' to the origami world in the 1950's was present. I told him of my doubts about its usage. He replied that, yes indeed, his own mother had said he'd got it wrong. She had once asked him. "Why didn't you call it a dolken?" I reported this in my addenda to the 'ABC of Origami', but when I returned to the above cookbooks (and Jewish housewives) I couldn't find the word 'dolken' anywhere. Nobody had heard of it. I began to feel that I wasprobably the victim of a Legman leg-pull.

The scene now moves to Israel where the search continued. Michael Asheri is both the author of a 400 page-plus book called 'Living Jewish: The Lore and Law of the Practicing Jew' (Dodd, Mead & Co, NY, 1978) and a keen paperfolder (He also rolls up his blintzes). I put the problem to him. Should what we call a blintz really be called a dolken? Had he heard of such a thing? I am very grateful to Mr. Asheri for entering into the spirit of the chase and for sending me such a full reply. This is what he wrote (2 March 1986):

"... I confess I shared your scepticism. The word DOLKEN was quite unknown to me and to a few other people I asked, but I felt I had better check. Weinreich's Yiddish-English: English-Yiddish dictionary failed to show it, but I had no real expectation that it would. Harkavy's Yiddish-English dictionary is far more exhaustive, but even there nothing appeared. I then made a careful check of Stuchkoff's monumental OITZER FUN DER YIDDISHER SHPRACH, generally conceded to be the most complete treatise on the vocabulary of Yiddish, but all to no avail: there was nothing to suggest that such a word as DOLKEN had ever graced our Mameloshen (mother tongue). And mind you, I looked not only under DOLKEN, but under DULKEN, DOLKIEN, TOLKEN, TULKEN, etc, etc.

"Finally I made a painstaking search of a book by Mordecai Kosover, one of the most qualified and thorough scholars of the Yiddish language. The book is called YIDDISHE MAYCHOLIM (Jewish Dishes). I had read it but it had been sitting on my shelves unopened for some years. On page 30, I found a long disquisition on a rare book MAKOR HABRACHA (Wellspring of Blessing), published in Munkacs some hundred years ago (1885). Kosover remarks concerning this book that it casts light on a corner of Easter European Jewry, passed by the mainstream of Jewish development in modern times. Among other things he observes that since Slovakia was part of the Austro-Hungarian empire, the list of dishes in the book reflects the relative economic wellbeing of the Jews in the region and 'is full of eatables and baked goods unknown elsewhere'.

"On page 18-b of MAKOR HABRACHA, appears the following heading: 'Knishes (Dolken) ... filled with cheese ..' (etc.) There it was! There can be no doubt that it is an extremely rare word, apparently a local term for the well-known 'knish', but that it is (or was) part of the Yiddish vocabulary, however regional and restricted, is not open to doubt. To my knowledge this is the only reference in print anywhere to the word.

"... And of course, Mrs. Legman is right: the knish is made by folding four corners of a square piece of dough to the center, over a filling. If we discard the word DOLKEN as archaic or at least extremely rare, the expression 'knish-fold' meaning the same thing, would be correct. Mr. Asheri goes on to say that he would be most interested to know if Gershon Legman's mother came from Munkacs or thereabouts.

To sum up: 1) the way we use the word 'blintz' is misleading, to say the least. It would be more helpful if we reserved it for use in conjunction with the double-looped arrow to mean 'fold over and over'; 2) what we call a blintz could be more accurately called a 'knish-fold' and 3) my researches have shown me that there is a whole world of exotic, difficult-to-pronounce names in Jewish cooking which paperfolders could draw upon. Just one example: we could call a square folded diagonally in half a 'kreplich-fold'. And we haven't even started on Danish pastries and Italian pasta yet.

Since this article was written, blintz has been added to Chambers Twentieth Century Dictionary and the Oxford English Dictionary, at least, they were the ones that I checked. Both describe it as a pancake, no mention of paperfolding whatsoever!

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Double Diamond February 1987
A puzzle from Ted Norminton

This shape was the subject of a recent self-imposed challenge of Ted Norminton. Resembling somewhat an intermediate stage of Ed Sullivan's and Sam Randlett's Togetherness and Apartheid folds (see Randlett's The Best of Origami), the Double Diamond shape chosen by Ted has a few subtle differences. Each diamond has the same configuration as the classic diamond base (or a fish, bird or frog base for that matter) but the two diamonds must be self-coloured each using different sides of the paper. Naturally enough, the whole thing must come from a single square, no cuts or glue being allowed, and it must lock cleanly and effectively.

Ted's final solution filled all these requirements and came from a Blintz Bird Base. It was, I am sure he will admit, not the most straightforward or direct of his repertoire of creations! Maybe you feel you have your own interpretation of the shape... have a go at it: you'll find that it's not as easy as it looks! If you're successful do send your attempt in to us, drawings too if possible.

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Flights of Fancy of Planes in Creases, April 1987
David Lister on why a crease is a straight line

John Cunliffe asks why it is that when a crease is made in a sheet of paper the result is a straight line, rather than a curve (British Origami No.121, page 13). It is obvious what he means, but it only applies when we are considering plane sheets of paper. For a start, if you fold a sheet of paper in half the crease is only a straight line as long as the folded sheet is kept flat. Once you bend the folded sheet of paper, (so that it is no longer in a single plane) the crease becomes a curve!

Then again, you can fold a curved crease in a sheet of paper: it is a thing paper sculptors do as a matter of course. The simplest way is to score the paper with a semi sharp point to mark the line of the intended crease before you fold it: then it becomes easy. The paper can then be folded along the crease. BUT, the paper will not lie flat - in other words it will no longer lie in a single plane. Paul Jackson has demonstrated creases of this kind at his teach-ins at recent Conventions of the Society.

Three dimensional origami often produces curved fold, especially of the first sort (the "Susan" and "Super Susan" are simple examples). The second sort appear when you try to sharpen up creases in folds of things like dishes in the style of Philip Shen and they can be found on close examination in three-dimensional animal folds, usually formed towards the end of the folding process and often not pressed sharp. They are one of the tricks which Yoshizawa exploits to make his animals so lively: this is often why his models are so difficult to copy.

But this is all begging the question: why does the simple fold in a plane sheet of paper make a straight line? The answer is given in Martin Gardner's "More Mathematical Puzzles and Diversions" (1961) page 128 where he writes:

"The very act of folding raises an interesting mathematical question. Why is it that when we fold a sheet of paper, the crease is a straight line? High-school geometry texts sometimes cite this as an illustration of the fact that two planes intersect in a straight line, but this is clearly not correct because the parts of a folded sheet are parallel planes. Here is the proper explanation, as given by L. R. Chase in the American Mathematical Monthly for June-July 1940:

"Let p and p' be the two points of the paper that are brought in coincidence by the process of folding; then any point on the crease is equidistant from p and p', since the lines ap and ap' are pressed into coincidence. Hence the crease, being the the locus of such points a, is the perpendicular bisector of pp'".

So now you know! Yet... in some ways this begs the question. It all turns on the Euclidian proposition that the locus of all points equidistant from two other points is a straight line, and also the fact that Euclidian geometry confines itself (at this level at any rate) to the geometry of the simple plane. Go a little deeper and you may start asking why the locus of all points equidistant from two other points is a straight line. If you do, you may get into some very deep water or perhaps ascend to some very rarified planes.

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A Simple Story June 1987
Humi Huzita ponders the meaning of the word simple

(Japanese born Italian Humi Huzita has a remarkable way with words - in several languages. At the London Convention 1987, he related the following history of the word simple.)

SIMPLE can be divided into it's two Latin roots: SIM and PLE. SIM means 'one time' (as can be seen in such words as 'single', 'symbol', etc.). PLE means 'fold' (as can be seen in 'multiple', 'plywood', 'apply', etc.). To 'fold-one-time' is really very simple and the first step in any origami process. However SIMPLE is nort really so simple, according to the intellectuals. They claim another interpretation, i.e. that SIM means 'together'as in 'sympathy', 'ensemble', 'symphony' etc. We know what PLE means.

So SIMPLE can also mean 'fold-together'. That sounds right for us; participants at the origami conventions naturally fold together. If you think about it, these interpretations are not contradictory but express the same meaning: All is one (not 'hole in one'!)

MORAL: SIMPLE is the keyword for paperfolders. Those who keep things simple will not have any complexes.

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Because it's There: August 1987
Robert Lang on Wet Folding (Part 1)

Origami has had a long-standing tradition of precision folding; razor-sharp creases and needle-like corners have been the norm, and origami models have been collections of planar geometric triangles. That all changed ten or so years ago when Yoshizawa began visiting the West and promoting the idea of "folding softly." The technique he used was wet-folding, folding origami from a damp paper, which could retain gentle curves and incomplete creases when dried.

The idea caught on in England, but not so much in America. It was quickly realized that foil-backed paper possessed many of the advantages of wet-folding without the mess. Foil was certainly easier to work with, but the shiny surfaced proved offensive to many. The advent of Spray-Mount solved that problem; by affixing tissue over the shiny side, one could make paper in any desirable color. Carrying the idea one step further, one could band tissue directly to aluminum foil (both sides) to make a paper simultaneously thin, strong, pretty, and almost infinitely moldable.

Through all these developments, wet-folding has persisted with two unique characteristics. First, the look of a thick paper, particularly one with texture, cannot be duplicated by a thin foil-based paper. Second, and more important, when dry, wet-folded models are durable. The same metal that makes foil-based papers so malleable makes them fragile. It is common at conventions to see an origami artist pull his foil model out of a box and laboriously smooth out the points that have been crumpled in transit. By contrast, a properly wet-folded model is quite rigid and can take the shall knocks of travel.

This aspect of folding has significance beyond simply getting a model to and from a show in once piece. Most origami models are ephemeral. Over time, the paper spreads; the creases relax; the color fades. Over months or years, acids in the paper can attack it, yellowing and embrittling the fibers. most folders accept that origami is folded one minute, crumpled and tossed away the next. They send models to conventions, not expecting them back, and why should they, when they can make another in half an hour?

I think this attitude has hurt us in the eyes of the general public, most of whom see origami as either a child's toy or a clever trick - but a trick only. The tricks are there, in the design of a fold, but artistry lies in the performance, the craftsmanship and spirit of the finished model. But how much spirit can there be in a model that has to be reshaped every time someone picks it up? If origami is to be art it musts not be disposable. Each piece must be unique, and it must endure.

Endurance; that is the strength of wet-folding. However the method has its drawbacks. It's messy; the paper is difficult to work with and easy to rip. Precise folding is difficult what with the swelling and shrinkage of the paper, and complex multilayered models are almost out of the question because of the thickness. The soft creases and gentle shaping can be matched by foil-based papers; for complicated folds, other materials prevail; but for the 'Combination of beauty and permanence, there is nothing better.



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Because It's There: October 1987
Robert Lang on Wet Folding (Part 2)

Many people have developed wet-folding techniques. These are some that I've found useful.

The paper is the first consideration. Most paper contains a material called sizing, a water-soluble adhesive that binds the fibers of the paper together and adds stiffness. Sizing is the secret of wet-folding. When the paper is wet, the sizing dissolves and releases the fibers, making the paper soft and floppy. When it dries, the sizing holds the fibers in the shape they were in when they dried.

Thus, a good wet-folding paper contains a lot of sizing. One of the best is "calligraphy parchment," which contains so much that your hands become sticky while folding it. Almost any paper will work, however, and watercolor paper, because of the range of colors and weight and its availability, is a good all-around paper.

The paper must not be glazed, however, or water will stand on the surface and/or soak in unevenly, resulting in rippling. It also must not be too thin or there won't be enough fiber for the paper to support itself. A good paper to use for larger models (e.g., 50 cm square) is 160 gm/m2 watercolor paper; thinner papers work better for smaller models.

You should dampen the paper before you cut your square from it. It is easy to do this evenly with an atomizer that can spray a fine mist, but simply wiping the paper with a damp cloth will work too. Both sides should be dampened equally. The trickiest part of wet-folding is getting (and keeping) the paper at the right consistency; it should be neither stiff nor soaked, but of a leathery texture. if there are shiny patches of wetness it is too wet and you should leave it to dry a bit. The same applies if the creases begin to get "fuzzy".

With the paper thoroughly and evenly damp, you may cut your square and begin folding. Naturally, the paper will begin to dry almost immediately. You deal with this by (1) folding swiftly, and (2) periodically wiping the paper with a damp cloth to re-moisten it, paying particular attention to the corners, which dry faster.

As you begin to fold, you will find that damp paper is a recalcitrant medium reluctant to hold its shape. You should avoid making unnecessary creases, as they will show on the finished model and weaken the paper, making it susceptible to tearing. At this point, drafting tape (which looks like masking tape but peels off more easily) can be invaluable. You can use it to reinforce weak areas of paper and to hold layers together until they dry. You can also use it to reinforce areas that several creases run through, like the the center of the paper, but you should put the tape there before you make all the creases. Otherwise, the tape may stick to the fuzz on the crease.

The beauty of wet-folded models lies in their three-dimensionality, so you should not press everything flat. In fact, major features of the body (like the back) look good if you just round the folds, rather than creasing them. Again, you can use strips of drafting tape to hold the shape until the model dries. If the model is to stand on its own, you should tape the feet to a flat surface to ensure that the legs are all the right length.

As you are folding, don't be afraid to crimp, stretch, or bodge. The thicker paper and softer creases are much less likely to form a jumbled mass of wrinkles where thinner papers might. On the other hand, complicated models with many layers should be avoided because if you try to squeeze too many layers too much, the model will burst.

When you are finished, set the model aside to dry, When it is thoroughly dry you can easily strip off the drafting tape. Minor changes can still be made by bending (but not creasing). Finally the model, beautiful and sturdy, is ready for long-term display.



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Because It's There: December 1987
Robert Lang on 'The Corner Sacrifice'

It was called "the chess game of the century." In 1956, at the Lessing J. Rosenwald Trophy Tournament, Donald Byrne, one of the finest players in America, faced his opponent, a thirteen-year-old boy. As the game progressed, Byrne sensed a weakness. Carefully, he isolated the boy's queen - the most powerful piece on the board - and, springing his trap, closed in and captured it. The boy did not even blink. He made his move; Byrne countered; and as the game progressed, he realized that a trap had been sprung, but he, Byrne was in it. With the superior position his game had bought him, the young Bobby Fisher drove deep into Byrne's territory to win. Fischer went on to become the world champion of chess while Byrne remained one of the better American players, but the queen sacrifice" became one of the classic scenarios of chess.

So what has chess to do with origami, besides an unusual fascination with squares? They both have fixed rules: bishops move diagonally; no cuts allowed. There are also rules of thumb. "Thou shalt not tamper with king-side pawns;" "Thou shalt protect thy queen;" "Thou shalt make points of thy square's corners." Yes, in origami, we are taught to conserve our corners (and to a lesser extent, edges) almost as much as a chess player protects his king and queen. The reason is clear. To make most subjects, we need several points for appendages. The difference between using a corner or edge to make a point, and using the interior of the paper, can make a four fold difference in the amount of paper required and the thickness of the point. For that reason, the notion quickly becomes ingrained in every inventor that for efficiency's sake, all four corners must eventually wind up as major features of the model.
lizard thang
Corners are precious, and so we must ration them out as heads, tails, arms, legs and wings. Alas, a square possesses but four corners. That allows us to have a head and two front legs, but only one back leg. The annals of the three- legged animals are richer for this restriction. For all its pleasant symmetry, the square is a pretty lousy shape from which to make an animal.

Consider the symmetry of a lizard, in figure 1. Besides head and tail, it has four appendages, two fore and two aft. Now, consider the square in figure 2. It has two "appendages," centered along its length. It is more suited to a two-legged lizard like that in figure 3.How sad it is, that such a creature doesn't exist. It if did, it would be a more popular subject than the elephant.

Obviously, to make a four-legged animal with the same amount of "stuff" fore and aft, we need a hexagon. Let us begin with the requirement of six points. How can we get six equal points from a square in the most efficient manner? Mathematically, the problem boils down to putting six points on the edges of a square as far apart as possible. It is not hard to find the solution, which is shown in figure 4. (The diagonal lines are angle bisectors.) The interesting thing is that two of the corners - the shaded regions - go completely unused; yet this is the most efficient solution for getting six equal points from the edge of a square. The corners might as well be cut off.

Herein lies the beauty of the sacrificial corner; by not using the corner, we get longer points (and thus, a more efficient structure) than if we had required four of the points to come from corners. Now, the conceptual leap from the crease pattern of figure 4 to a finished model is large, but the efficiency of the sacrificial corner can be illustrated by a fold from the literature.

Montroll's Dog Base (Animal Origami for the Enthusiast, Dover, 1985) is shown with its crease pattern in figure 57 Again, I have shaded unused regions of the paper. The useful area is unmistakably hexagonal, albeit with some distortion, which shifts extra paper to the head. Like the hexagon of figure 3, the Dog Base is a maximally efficient structure for the number and proportions of its points. Therefore, the loss of the two corners was inevitable and could be predicted even without knowing exactly how the Dog Base goes together. The corners have been sacrificed, and while the result is not nearly as unexpected as Fisher's it is at least as effective.



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Because It's There: February 1988
Robert Lang on Creativity

For the October 187 Convention, Richard Hallman, convention organiser asked for a few words on the subject of "Creativity." Forthwith, I set pen to paper, and forthwith, drew an instant blank. After about ten pages of false starts' I came to the conclusion that I didn't really have anything to say about the general topic of creativity, but I did have a few thoughts on the topic of creation -- namely, how a folder comes up with a design for a new model.

Different folders approach it different ways. Some talk about their folding "inspiration." Some merely play with a base, twisting it this way and that until it begins to resemble something. Others follow their intuition. An American colleague says he dreams new designs. And some, of course, invent totally new symmetries and techniques to accomplish their goal. Alas, my own method is quite mundane. Most of my models are the result of plugging away at a subject, using systematic folding and geometric techniques. Where I part company with the giants of geometric folding is that most of "my" techniques are not even home-grown, but are either lifted from other models or continuations of published techniques.

The secret, I think, to adapting other people's techniques without finding yourself simply duplicating their work is to look beyond the finished model and sequence, and to figure out why each step was done the way it was. I try to figure out the thought process the inventor used to develop his or her model; if I am successful, I can draw not only on his folding technique, but on his line of thinking as well. Often this necessitates pulling a model apart to discover the symmetries that went into it. The symmetries and dimensions are not always clear from the folding sequence, which is usually developed with the goal of simplicity and sequentiality in mind rather than as a guide to the structure of the model. In fact, a good simple folding sequence often obscures the instrinsic structure more than a more complicated one ("bring these 20 creases together now") might, since a complicated step usually gives a more global view of the model. It's often not until I've completed a model and unfolded it that 1 can say, "ah ha,so that's why you made that precrease in Step 2!" I get a pleasurable feeling from inventing a new model of my own, but it's also a heady feeling to know exactly how someone else invented their model -- and it beats inventing something yourself and then finding out you've reinvested the wheel.

The other thing I do when I appropriate other techniques is to generalize them. As an example: in the recent Kasahara/Maekawa book Viva Origami, Jun Maekawa developed a technique to get five little points from a single large flap. His secret was to generate a lot of excess paper in the flap that could then be reverse-folded until the five tiny points appeared. It occurred to me that one could get the necessary excess paper from any sort of bird-base-like flap by crimping it and pulling out the excess paper, as in John Montroll's "wing-fold." I thought this technique could be used so I got an old hawk I had rattling around in my drawer of rejects, and crimped and reversed the wing flaps. Ta-da! Instant feathers. However, five feathers seemed to be a bit sparse. Then I saw that there was no reason you had to restrict it to five points; you could make six, seven, or any number of points you wanted (I settled on nine). Irregularities prompted some changes in the folding method, but the final result was successful (it was displayed at the Spring 1987 convention, and will appear in Origami Zoo by Robert Lang and Stephen Weiss, published by St. Martin's Press in late 1988. End advertisement).

Now, I doubt that when Mr. Jun sees that model in print he's going to say "hey-- this bloke nipped my five-fingered reverse fold!" any more than John Montroll will accuse me of stealing his wing-fold. For one thing, the finished sequence bears practically no resemblance to the original structures. Nevertheless, the connection is there, at least in my mind. It is with a long series of small, progressive changes that I (or anyone) can go from an existing technique to one that, to the casual observer, appears to be entirely new.

Fortunately, there has never been a better time than the present to engage in this process of analysis and adaptation, as the origami literature is full of new ideas just waiting to be borrowed. The two Montroll books alone are treasure troves of clever techniques, as are the two Kasahara books Viva Origami and Top Origami (Ed. Top Origami is now called 'Origami for the Connoiseur'). Plus, John has a third book That will be out by the time this appears (Origami Sculpture); Peter Engel has one that will published by Simon and Schuster in 1988; Eric Kenneway's (Complete Origami, in America from St. Martin's Press) is now in press, and you might even find a nugget in my own solo book, The Complete Paperfolder, from Dover (Autumn of 1988). This is not to say that recent books are the only place to look by any means. Ingenuity with a bird Base is unmatched in the Harbin/Randlett books. The point of this all is that folding techniques are readily available. One need not create new ones to be creative.



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Profile: April 1988
Robin Macey

Although Robin Macey has played many roles in the BOS, including organising facilities at two Nottingham Conventions, nevertheless he will probably always be associated in people's minds with the group photograph. After all, he has been herding members into the open twice a year for the past ten years in order to take their picture, come rain, come shine, come snow once at Cambridge.

Yet he is not a photographer by profession even though photos taken by him have appeared in a great many publications aside from the BOS Magazine. In fact he earns his living as a computer programmer for the Nottingham City Treasury where he went following a degree in mathematics at Nottingham University. In the little time that's left after photography and origami he goes hot-air ballooning, plays table tennis and works out on the trampoline. He does community work, too, such as taking a weekly class in paperfolding at a detention centre for 14-16 years olds. Clearly a very active individual, so what does he find in origami, a sedentary, solitary occupation much of the time? Chiefly the challenge of a difficult fold, which explains his admiration for the work of Max Hulme, Neal Elias, Paul Jackson and Martin Wall. His all-time favourite fold is Max's Jack-in-the-Box. As to creating his own originals, this he rarely does, preferring to highlight the skills of others through the camera's eye.

Robin's introduction to origami came in 1971 at the age of ten via a book by A. Van Breda called Paper Folding and Modelling. First folds he ever did were the traditional Salt Cellar and Flapping Bird. He was an avid viewer of Robert Harbin's origami programmes on TV in the seventies. All the while he was also doing photography, having begun at the age of eight with a box camera graduating to a 35 mm Agfa-Silette in 1974. These days he uses a variety of Canon equipment with standard 50 mm lens. For group photographs he uses a Bronica ETRS large format camera and in addition owns a very old 5x4 in. plate camera which may be slow but gives superb quality pictures. Needless to say he does his own black and white processing. (Editors note - Robin's equipment has probably changed since then!)

Best origami pictures? Possibly those of the A.Yoshizawa folds at the '83 convention. Most spectacular? Undoubtedly the cover of BOS mag.109 which showed a large folded bird being launched from a top floor hotel balcony. To get the angle right he had to climb over the rail and lie on a nearby roof. A dangerous occupation sometimes this photogami.

Photographs clockwise starting from top right: Building Society balloon taken during a flight over Nottingham; John French launching his "bird" - winner of novelty competition, Cobden Hotel, Birmingham,1984; Bull and Matador (Neal Elias); Chicks (A.Yoshizawa); Best dressed paper-folder, Joan Homewood, autumn convention, Cobden Hotel, 1980.



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