Virtual Origami

Virtual origami. Since I said already the title and my declare myself and abstract is too long, so I skip and you can. You can see, you can read it anyway on the net. So let me start from this page outline. I want to say four things. And the first is the. The system called E origami system and the language that is running on eos and geometrical constructions and verification by the oricult language and so on. So let me start what origami is. And now origami is international world. So I. I don't think I need an explanation. But sometimes people get confused and I do also because there are three ways of usage of origami. It's operation itself or sheet of paper, or the art of paper folding. So I abuse those differences. So let me allow this. So the use of origami did work quite early. I don't know exactly when. Nobody knows like not, not you don't know when. But origami is similar to that. But the unfortunate thing is that the use of origami is recorded in paper because you need paper to use origami. So before 10th century already in Japan, the people in the court aristocrats are using origami to symbolize their feelings and thoughts without explicitly saying what they want to say. So it's kind of protocols implicit, hidden in the. In the shape. So it's like it's you. You use emoji. And emoji has this tradition of origami modern virgin and as a paper production in the middle ages and onward, very in large scale and even common people can afford to use high quality papers. Then it's penetrating to daily life. It's like putting a symbol on a gift, et cetera, et cetera. But the change occurred in the 19th century when some people argued that origami can be used as a tool to teach mathematics. And so the formalizing origami started in the middle of 19th century until 20th century. And nowadays industries are also interested in origami because of. Well, just take an example. Imagine an example antenna have to be put in a rocket and on the space you have to expand it similar to tiny medical things that have to be put in your body and expand it in. In some ways so that it will work properly. So therefore origami is not only used in daily life, but also in high tech area. So after of this preface, I want to talk a little bit about origami geometry and Euclidian geometry. You see, construction tools of origami is just folding. Folding is what you can do, you cannot do RPG folding. You have to obey certain rules to superpose one object to the other or with some constraints. And there's a rule called Fujita Joshuin rules and this is used as a base for mathematical theory of origami. Whereas in Euclid you know everyone that maybe needless to say that storage and compass and how do I compare the power of two it is useless to compare the line and density of line density of points on the line or circle. The best way is to count or to check the intersections of those objects object. In the case of Euclidian geometry it's circle and lines. In the case of origami it's just fold lines and just and with. If you are careful enough to set up your. A theory, say we are trying to use a Galois theory for it and you start with a few rational numbers, then you expand it. In the case of Euclid you expanded square root of two and in the case of origami, in addition to square root of not square root of two, just square root, square root and two root. Therefore, it is easy to see that the origami point set of points of origami are strictly larger than the set of points that Euclidean geometry with Euclid tools can generate. And one example would be. Well, I enumerated five of them and one I'm going to show is that construction of both tri Sexton trisector of an arbitrary angle has been a long, long standing problem until 18th century. So people tried and tried, tried and some claim it has been solved, but turns out it's not. And in Euclidian geometry, which is negatively solved by the use of algebra, but origami you can do it. So I would like to show how to do it. It's just several lines of Oricoto programming language. But the Oricotto language has an implementation of the Fujita Justin moves which is a base of mathematical theory of origami. So apart from there are seven rules and one is very simple. If you look at PN2 which are given, then you are allowed to fold along the line that passes through PN2 and the other the superposition of the given object. So for instance, this one is that you are given two points. You are allowed to superpose two and by folding along the line that is defined by this. And if you unfold it, you will have in those cases fold what we call fold line or sometimes called crease. And if you add some more, O3 is the between superposition between two lines. M and N. Then you got this four line and see and 4, 5 and 6 are slightly more involved. I will skip 4 and 5 because well, time. It's not very occasion to speak about this the rules in Gitero. But I want to pay attention to this Rule 06 which says that given two lines and two points given two points and two lines fold to superpose 1.1 line and the other point the other line. So after a bit of maybe it's not trivial to do this, but you can do this by hand. You don't need to do a special computation. So for instance, you can get this one. Okay, and here's one more, but I will skip and Mathematica, our system start implementing this Fujita Justin rules. In the beginning it is very easy because it is seven rules. But by Mathematica is based on rewrite asset Mathematica you define a function using rewrite rules and with pattern matching with types, you can define these seven rules in one function. So we call HO later in my program. So let me. Explain the anguishai section. So in addition to this Fujita Justan rules, we need a lot of auxiliary functions to make origami. Possible. So visualization for instance, or manipulating originally. So we start with your session first and generate create origami of session size like this 10. Then I want to put a point any place. But since I defined this origami 10 by 10, I use I put E here by specifying the coordinates of it, right? So what I want to do is to trisect this part, trisect this angle. Okay? And in order to trisect we need two parallel lines. And how do I make two parallel lines? And also that parallel line should be parallel to this baseline, which is very Easy because of O2 rule simply says that if you bring say for A to. A to D, A to D and bring back this explanation Mark is a positive operator of unfold. So bring it, bring A to D and back. Then you get a crease fg, right? And similarly you get this hi by bringing A to F. And we are almost done for trisecting this angle. The secret is that we use this O6 rule to put point P on E and A on hi. But surprisingly, this doesn't give you a unique pose line. It gives you three possibilities. Here's one to three. We are at this time interested in trisecting this path. This and more we take that one. These others are not erroneously generated. It trisect other angles. The first one is supplementary angle and the second one is 2pi minus eab eab. But ignore it for a moment. So then fold it using number three. So we get this one right and we are almost done. Because the line that pass through here and there will be a trisector to make it nicer. I project those points and give a new name. And here we go. So. So you have K and J K A and K J, A are tricectors. Okay, so we are done. So this long standing problem is solved very eloquently by Japanese origamist somewhat long ago. So, okay, So the next thing is, are you sure that these indeed trisect? In order to do that, we need to tell the system what we should prove. So our verifier is based on equational reasoning, a system of equations. So we have to have equations among polynomials. So we have to have polynomials and to. We have to. Ideally we want to say that three angles are the same. But these angles are difficult to handle in equational settings. So we have to take those angle equalities into polynomial equalities. These are kind of transformations we do not explain here, but afterwards we make it a general one. So let ABCD be a unit square and E to be arbitrary G and prove it. And the system checks whether the given equations that are generated from the programs polynomial equalities. And if they do, they started to do to compute global basis. And then it's proven this time 11 o', clock, 1111 second. And if I click this, then quite a large proof document is generated and you can bridge it. And so I'm not going to talk about it more. So one interesting example of this trisection is Morris triangle. You may know that these three this little triangle. So given arbitrary triangle and you trisect each inner angle and pick up the intersecting point of neighboring trisectors and you get this tiny triangle. And this triangle in this case is equilateral triangle. This can be proven quite easily, but it takes more time. So this is one nice example of how trifraction is used in origami. Okay, so so far it's about geometry, origami geometry. And I'm going to turn to the traditional origami. Traditional origami has a lot more materials in it. Well, the base is very simple. You have valley fold and mountain fold, which is also O1R is a kind of valley fold as mountain fold. So the reason why it is mountain valley is. Well, if you look at this, it's quite obvious. If you make a fold like this, then this line looks like the path in the valley. So you have good imagination. And if you do opposite fold then this and long fold this become a mountain reach. So these two are the most basic. And this can be actually in computer implementation. It is one realization of fold operation which rotate the faces along rotating angle by rotating axis along this the first argument here you should know that this is still a function. So it's hurried function. So you need one more argument to make this valley fold work very fold produce something visible. Okay. And there are some more interesting ones. Inside the reverse fold is name B you. This is inside the reverse fold. If you look this way, then you will see that it's a fold. It looks like one fold, but one side, one part is mountain fold and the other side is valley fold. And we take it this as a single fold. And the other one is this one outside device. This the inside or the opposite the outside. So you get this this one. And you will see. Here certainly the mountain fold and valley fold are made outward. And there are one more which is quite often used as squash fold. I don't think I have time to talk about this, but I just show you a squash fold. Suppose you are given this and squash fold is to squash this part into here. Maybe. If you look this one, then it's quite obvious. So this part is squashed into here. All right, so this is squash fold a few fold. There are a number of. I have. I often use more so called classical fold. But for the. For the ones that flying crane I only use this G3 classical fold. So I'll show you this at the end. There are quite a long program and I don't think I will be able to expand it in detail. But just browse through all this. So we start with one side is yellow and we fold and the other side is this pattern. Okay, so we make one valley fold and then again valley fold. It will be lots of fold. And I. Therefore we have in the case of geometrical fold we have one plane. But traditional fold we have. Layers of fold. So we have to keep track of layers of code and the plain numbers. So here I really have to skip everything and I will show you the end of the. End of the picture. Yes. So. So at the end you will have this.