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Why a paper square?

Brunno Jammes asks (25th May), why we mostly use square paper for origami. He says there has been a discussion on our sister list, Origami-francais.

This may have arisen indirectly from a brief exchange in Origami-L, at the beginning of this month which, however, didn't develop as much as I think the subject merited. (The whys and wherefores of subject generation and development in Origami-L are quite incomprehensible). Apparently Bruno didn't see my own contribution on the subject and I am brasenly having the temerity to repeat it here. As I said at the time, my thoughts were "off the cuff", and given as a series of disjointed ideas and not in any way as a reasoned discussion. Some points are more valid than others. James Sakoda criticised me for coming to a conclusion which was somehow in favour of square paper. Perhaps in my haste, I had expressed myself badly, but I was certainly not advocating square paper in preference to any other shape. I was merely trying to find out why, historically and in practice today, so much folding is done from a square. My own interests run the whole gamut of folding of every kind, known and remaining to be discovered.

Here is what I wrote, slightly edited for clarification.


1.A few months ago, I wrote in Origami-L about John Smith's ideas on "Origami Profiles" which analyses how each individual's preferences in folding fit into the general scheme of things. The theory accepts that everyone is entitled to adopt whatever rules for folding he or she chooses. A folder can use, or not use square or any other shape of paper at his or her absolute discretion.

2. Following from this a folder can choose to fold from a square or from a triangle or from A4 or a pentagon or a rectangle or a rhombus or a long ribbon of paper half and inch wide and ten yards long. He may even prefer silk ribbon or even string. Whether other people would include ribbon or string in their own concepts of Origami is another thing.

3. If you accept that cutting is legitimate (and cutting, too, fits into John Smith's Profiles of Origami), your can convert a square of paper into any shape you like. Or you can trim your dollar bill into a square. Or you can chop off any of those surplus bits of paper that get in the way. (No, I accept that most people who like to use scissors don't look at it in this extreme way, but I assert the possibility.)

4. Without even using scissors, you can convert most simple shapes of paper into most other shapes by folding alone. You can fold a square to make a triangle, or a hexagon or a 3 X 7 rectangle, even A4. So having done that, in theory, you can go on to fold anything that can be folded from a square equally from a Dollar Bill or anything that can be folded from a Dollar bill from a square. I write "in theory" advisedly, because all the preliminary folding to get the shape makes the model bulky and difficult or even impossible to fold in practice.

5. It seems to me obvious that the reason most people chose the square is that it has a greater symmetry than other shapes. [That is what I wrote, but the relevance of symmetry is a very difficult one and I have had further thoughts about it which I will mention in an addendum.] Other regular polygons also have their symmetry and triangles, hexagons and octagons have their merits and are useful to fold from, but a quick test will reveal that the four equal sides and for right angles of the square give a peculiar sort of symmetry that offers far greater possibilities for development than any of the other shapes. Once we leave regular polygons and come to rectangles, symmetry is, of course, very much reduced.

6. It may be suggested that a circle has more symmetry than a square!. Yet, curiously, while a circle CAN be folded (and may have hidden qualities which have yet to be explored by paperfolders), it is almost impossible to use for ordinary origami, because the circular outline is just not compatible with the straight crease lines that result when a piece of paper is folded.

7. It has always struck me as strange that while square paper is in practice by far the most popular for origami, square paper is rare in the everyday world. You just cannot go into an ordinary shop and ask for square paper. Of course, nowadays, a shop may stock "origami" paper, but by its definition, Origami paper is not ordinary paper. Writing paper is always rectangular, wrapping paper is always rectangular, banknotes are always rectangular. Even toilet paper comes in rectangular tear-off sheets! About the only common paper that I can think of that comes in squares are paper handkerchiefs and serviettes, obviously because proper cloth handkerchiefs and napkins are square. (Which makes me ask, why are handkerchiefs and napkins always square? I think a consideration of this question should throw considerable light on the problem of square versus rectangular paper.)

8. One of the greatest virtues of the square is that you can fold a bird base from it. I concede that you can fold quasi-bird bases from an equilateral triangle or from other simple polygons, but they do not have any of the virtues of the true bird base from a square. The bird base is a miracle of paper geometry and every paperfolder has enjoyed enormous pleasure form exploring its amazing properties, ( See John Skmith's BOS booklet "In Praise of the Bird Base"). In the 1960s the bird base dominated folding. It was almost impossible, for a time, to escape from its clutches. (Probably because modern folding derived from Yoshizawa's own discovery of the virtues of the bird base.) The domination of the bird base does not apply so much today when many other techniques of folding have been discovered. [And all praise for those new techniques in all their great variety.] But new techniques do not detract from the wonder of the bird base.

9. I sometimes wonder whether the first paperfolding started not with a square, but with a rectangle. In the days when people began to fiddle with trimmings of paper left over from the primitive paper making process, it is unlikely that any of those trimmings would be square. As paper was cut to the requisite sizes, rectangular offcuts would be more likely. Some of our simplest paper folds are in fact from rectangular paper. Like the simple hat that turns into a boat. Or the paper dart,. Even the swallow or glider starts with a rectangular sheet of paper, from which a square is cut or torn to leave a rectangle of paper which will form the tail of the swallow. Did these models originate in time, before the multiform series, windmill and pajarita?

10. A square is a simple shape. Every square has the same proportions and can therefore be standardised. On the other hand rectangles are infinitely varied in shape, so that it is hard to standardise any particular proportions. (As evidenced by our continued jnablility to standardise the sizes of typing paper throughout the word, even today.) So it is easy to produce square origami paper, but impossible for manufacturers and shopkeepers to produce a rectangular paper that will command general acceptance.

11. Arising out of this, historical inertia (or, if you like, historical momentum - it is the same thing) takes over and square origami paper is square because it always has been square; square paper IS origami paper and origami paper doesn't come in any other shapes otherwise (for the manufacturer or shop-keeper) it isn't origami paper.

12. Despite the infinite possibilities of folding from square paper, the universal availability (I'm speaking relatively here, of course: they aren't very available to most people) of banknotes has meant that they have inevitably been folded, so that in banknote folding we have for long a strong tradition of folding from rectangles. In particular, the Dollar Bill with its curious, but immutable proportions of 3X7 has had a particularly powerful influence in the United States.

13 Again, despite the power of the square, and despite the fact that most folders of animals and other living creatures start from a square, when it comes to inanimate objects, folders often use a rectangle. From Neal Elias, through Max Hulme and David Brill and on to Yoshihide Momotani and Robert Lang, folders of railway trains, jack-in-a-boxes and cuckoo clocks have all favoured rectangles, often of extreme proportions. One of the reasons for this is that rectangles can be folded longitudinally and laterally to form the grid from which so many inanimate objects can be folded.

14. Origami is not only an art: it is a puzzle and a game. We like to pit our wits against the paper and its inherent geometry. A game must have rules and the most obvious rule, ready to hand and accepted by a tacit general consent, is that we should begin with a simple, easily definable shape of paper, and that is the square.

15. As has so often been said in Origami-L, creative art is arises where the constraints are greatest. The accepted constraint in Origami has traditionally been that the paper should be a simple square.

16. To summarise, the square has become the classical shape of paper for folding. [by "classical", I meant the universally accepted shape for what we ordinarily call "origami paper"].


As I re-read this after three weeks, there is not much I would want to change, provided it is accepted as a basis for discussion and not an attempt to pronounce on the laws of paperfolding.

However, the point about symmetry and the square does make me think again (as indeed it did when I wrote the above. But since I had already written enough, I decided to skate round the question. In any case, I'm not a mathematician.)

A square has many summetries. It has reflective symmetry from each of its four sides, and from each of its four corners. It has rotational symmetry, too: 90, 180, and 270 degrees in each direction.

Yet other figures have even more symmetry. An octagon, for instance has even more reflective and rotational symmetry. Yet it does not give the freedom for folding that get with the square. You cannot, for instance fold a true bird base from an octagon.

A circle has infinite reflective and rotational symmetry, yet, as my short comment on circular paperabove, indicated, it is almost useless for folding using currently known techniques. (As an aside, I personally think that there is enormous scope for developlkment of techniques fror folding circular paper, but such techniques have not yeyt been discovered).

So, it is not entirely symmetry that makes the square so appropriate for paperfolding. I can only think that it is the peculiar geometry of the square, and in particular, the fact that 45 degree angles can be generated in it, and indeed that it gives rise to the bird base, that gives it its great advantages. Perhaps the more mathematical among the subscribers to Origami-L will be able to home in on the essentials of the problem.

Apologies to everyone who thinks I have already had my say on this subject and shouldn't be repeating myself or taking up so much of everybody's mailbox space.

David Lister Grimsby, England.

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