Origami By Andrew Ilyas The History of Origami The word Origami is is derived from the two japanese words: ori, meaning folding, and kami, meaning paper. Although from this most people think that Origami originated in Japan. If you think that, you might want to read this to know a little more about the history of Origami. Timeline: 0-200 AD: Origami originates in China 500-600 AD: The art reaches Japan, but at the time, the folds were used to summon spirits 794-1185 (Heian Period): Origami becomes part of ceremonial nobility. At this time, samurai, the Japanese warriors, exchanged gifts with noshis, which are luck tokens usually stuffed with meat. Shinto nobles at weddings drank out of cups with two butterflies folded out of paper on them: one male, and one female, for the bride and groom. Diplomas would be folded for secrecy, because once they were opened, they could not be refolded using the same creases. When paper became inexpensive: everyone could use it to fold Origami, so they used Origami for social stratification (classification of the community) 1338-1573 (Muromachi Period): Origami is used to determine classes between already high-class samurai 1603-1867 (Tokugawa Period): The Origami Bird Base comes up as in Senbazuru Orikato, meaning How to Fold 1000 Cranes, the oldest document on Origami 1845: Kan no Modo or Window on Midwinter is published Modern times: Origami is not only appreciated as a childish entertainment, but a tool to teach math and geometry Origami spread all over the world. The Arabs learned it from the Chinese prisoners they took in Samarkand in Central Asia. Paper making started in Europe in the 11th century. However, there is no proof of paper folding existence there before 1600 AD. Sadako Sasaki (January 7, 1943 - October 25, 1955) Sadako Sasaki was one of the many people who died because of the Hiroshima bombing. Sadako was two years old at that time. She was apparently unhurt, despite being about 1.5 km from the epicenter of the radiation. As the Japanese girl grew, she became a great athlete. One day, Sadako was practicing for a big race, but all of a sudden, she felt dizzy and fell to the floor. She was later diagnosed with leukemia. In the hospital, her friend folded her an Origami paper crane. This was a highly symbolic gift, as the crane was believed to live forever, and have the power to grant wishes. If you folded 1000 cranes, your wish would be granted. So Sadako Sasaki started folding every little bit of paper she could find. At first, all wishes were for health. Then, when she got weaker, she started wishing for world peace. By the time she died, Sadako Sasaki had folded 644 paper cranes. Her friends then folded the remaining 356 and buried her with all 1000. Later, they formed a club to raise money for a national monument. 3000 schools from Japan participate in donating money for this club, as well as nine other countries. Three years later, the Childrenʼs Peace Monument finally stood in Hiroshima peace park, and on Aug. 6 of every year, National Peace Day, people send in thousands of cranes to the park. The Childrenʼs Peace Monument. If you look closely, you will see that all their clothes are made of paper cranes Applications Origami is used in many ways Application Why Origami was used How Origami was used A space telescope At the Lawrence Livermore space center, there was a giant telescope with a diameter of 100 meters. This telescope needed to fit in a much smaller three meter rocket The Space Center contacted Robert Lang, and together with the scientists, they created a crease pattern that folded the telescope up into a cylinder with a 3m diameter. They have tried this pattern on a five meter prototype, and it works perfectly; going down to 1.5 meters A heart stent A folded origami piece to open Invented by Zhong You and Kaoru Kuribayashi blocked arteries. It starts small for inspired by the Water-bomb model (the Blowthe way there, then expands when up box) it reaches the clogged artery A Solar Array There was a realization that Studied and done by Koryo Miura through a crease pattern with a simple opening and closing, the array could be folded into a very compact shape Airbags Needed to be compact and Done by Robert Lang techniques developed expand quickly from Circle and River Packing to add detail to normal Origami designs Unconventional uses Artistic, less expensive, fun, more Gift wrapping, Advertising, recreational impressive purposes, etc. Basic folds Mountain fold: Fold away from you Valley fold: Fold towards you Squash fold: Open pocket and flatten* Preliminary Base (Square Base): First, start with the white side of the sheet towards you. Valley fold in half diagonally, then in half again. Perform the Squash fold. Also do the Squash fold on the other side. The final product should be a Square Base. Petal fold*: Valley fold and crease both edges to meet the center. Then fold the corners inside using a mountain fold. Lift the bottom of the flap to meet the top Inside reverse fold (Pocket fold)*: Fold and crease the tip at an angle. Then, partially open the layers up and push the tip down in between the layers Outside reverse fold (Hood fold)*: Open up the layers partially. This time push the tip outside to wrap around the paper Kite Base: This is the easiest of all bases it is as simple as creasing the square paper diagonally, then folding the edges to meet your crease Rabbit ear fold: This fold creates a small flap that you will be able to move. First, start with a square piece of paper with a diagonal crease through it. Now crease one edge to meet the center. Then, flip the paper upside down and crease that edge to meet the center. Next, Crease the paper on the other diagonal. If you collapse the side with the creases, you get a rabbit ear fold. NOTE: Here they pinched the diagonal crease instead of folding the whole thing Water-bomb Base: Start with the paper colored side up for a colored Water-bomb Base. Valley fold and crease the top edge to the bottom and the right edge to the left one. Mountain fold and crease both diagonals. Collapse all folds. Windmill Base: Begin with two vertical folds through your square piece of paper. Valley fold the top and bottom edges to the center. Now again, valley fold the right and left edges to the center. Then, pull out the hidden corners. What you should get is a hexagonal shape. Fold the top-right corner to the right, and fold the bottom-left corner to the left. This is your finished Windmill Base Bird Base: Start with the Preliminary Base. Make sure that the point with all the layers of paper sicking out is at the bottom. Fold the right and left edges to the middle, and the top point down. Petal fold and repeat this on the other side Stretched Bird Base: As you may have guessed, this base starts with the Bird Base. Pull out the inner flaps until the top point pops inside. After you have flattened this, you have your finished Stretched Bird Base. Crimp: This fold is a combination of two reverse folds. There are three kinds of crimps: a step fold, an inside crimp and an outside crimp. A step fold is where there are two parallel crease lines: one mountain and one valley. If you fold them they look like a step. Another kind of crimp is an inside crimp. It looks like this: The last kind of crimp is the outside crimp. It’s the same as the inside one except it wraps around the paper, not in it Frog Base: Start with the Preliminary Base. Squash fold four times, one on each layer. Lastly, Petal fold on each layer. This is your finished Frog Base Sink Folds*: The three kinds of sink folds are the open sink, the closed sink, and the double sink. The open sink is simple, but hard: open layers, push point down, close layers. In the closed sink, just slide the point down. And in the double sink, perform an open sink, then push the point up. Fish Base*: to do this base, all you must know is how to fold a rabbit ear, as the Fish base is only two of them, one on each side of the paper These are but 17 of the many folds in Origami, so go out and explore * All folds with this mark need at least two layers to start. Pattern Grafting The point of grafting is to add pleats. This is so we can have more flaps to add detail to the design. A great example of grafting is the turtle. In the crease pattern, we realize that there is an unwanted rectangular strip of paper at the top Figure 1. The base can be folded without it. We also realize that the head, the shell, and the tail are all in one diamond. In addition, since the pattern has all 60° angles, to add details, we must force all new lines at 60°. Furthermore, we have eliminated the oval thatʼs going around, as we can just replace it by folding the edges of the shell down as shown in Figure 1. If you look closely, you will see that we are left with three hexagons and ten shell pleats. Since all pleats have to end at the edges of the paper, we will extend them until they do. We also have to give them a finite thickness. Because some of the pleats hit the edge at an angle, Figure 2, the paper is no longer square. We see that some pleats hit the edge at a flap (the hind legs), we can use those pleats to make a fancier hind leg, with toes, for example. Since we did this for the hind legs, we might as well do it for the front. So we will add a pleat at the forelegs, as shown in Figure 2. Now back to the problem of the paper not being square. If we want to fold a square, but the pleats turned the paper rectangular, what do we do? Well, remember the rectangular strip that could have been cut off? Here, when we cut it off, the paper becomes square again. Now once you pleat the pattern you have a much more developed Western Pond Turtle, by Robert Lang, as shown in Figure 3. Another good example of grafting is the Koi, a type of fish. Like the turtle, we want to represent texture to add more detail. First, letʼs look at what we want. We want scales that look like Cs. The closest things we can get are sideways Vs, as shown in Figure 4. Then, we realize that we can sink the tip to create a more blunt tip to the scale. To be sunk, the tip first needs to be freed. See Figure 5. Figure 4 The tip is now ready to be sunk. You can think of this next part as just duplication, as we are going to make an array of tiles as in Figure 6. Figure 6 An array of sunk pleat tiles Now for pleats, three things need to be chosen: the direction of the pleats, the width, and the spacing. We know the direction is 45°, but we can pick the spacing, and the width is given. If we unfold our single pleat, we get that the crease pattern is 60% larger (2.5 times larger) than the folded figure, as shown in Figure 7. Meaning, on average, there are two to three layers everywhere in the model. But thatʼs just an average. Individual regions get much, much, thicker as you can see in Figure 8. Figure 7 Figure 8 Here, we realize that with a different sink fold, we are much more efficient. Between the two tiles in Figure 9, the crease pattern is 38% larger than the folded shape, meaning around 1.9 times bigger. The maximum number of layers here, seven, is almost half of the first tile. Figure 9 Now that we have the pleat that we are going to tile, we concentrate on where we want the pleats to go. We know that we want the pleats on the body, not the head, the tail or the fins. This is shown with color coding in Figure 10. Note that we want to put our pleats on the orange spots but not on the blue. First, what we come up with are the pleats in Figure 11. But this is not good enough. In this case, we can sacrifice a bit of the head to include more pleats as in Figure 12. We are now done the Origami Koi, with 400 tiles, each having to fit within one pleat length. Can you find an example of grafting? Look at Figure 13 for the Koi. Yours for the folding! And remember, each tile must fit into one pleat square!!! VS Compass and straight edge constructions are useful, but in my studies I have found that most of the time it is easier using origami to construct. Here, you will see two examples: constructing an equilateral triangle and trisecting an angle. 1. Equilateral Triangle Compass and Straight Edge Origami 2. Trisection of an angle What we are trying to do here is split one angle up into three smaller equal angles. It is impossible to do this with Compass and straight edge but see the figure below on how to do it with Origami. Software “Itʼs so simple that a computer can do it. You say, how simple is that?”, Robert Lang said about complexity of crease patterns. He then said: “With a computer, you have to be able to describe things in very basic terms. And with Origami, we could.” The application of Mathematics into Origami led to change in the art. With mathematical concepts applied, people could not only develop rules for things like Flat-Foldability, but they could also make powerful tools to help them make new designs. Treemaker This Origami designing software was made by Robert Lang. In 1989, he wrote a magazine article for Engineering and Science. What he wrote was: “Computing succumbed to the appeal of folded paper when, in 1971, Arthur Appel programmed an IBM System 360 computer to print out simple geometric configurations at the rate of more than one hundred a minute. Ninety percent were considered unsuccessful, but it raises an interesting question: could a computer someday design a model deemed superior to that designed by man? Since so much of the process of design is geometric, the prospect is not as outrageous as it may seem.” In the next decade, Robert Lang could see that his article was coming to reality. In the 1990s, with the idea of circle and river packing coming to many Origami folders, such as Robert Lang himself and Toshiyuki Meguro in Japan, Lang tried to make his article come true; he wanted to write up a computer program that could “design” a nontrivial Origami figure using circle and river-packing methods. So this is how Treemaker started. To use it, you start with a tree, it uses algorithms developed at the University of Maryland to pack circles and rivers so that it can come up with a crease pattern. Although Robert Langʼs first version of Treemaker didnʼt come up with a crease pattern at all, it got the idea of circle and river packing. The most current version is Treemaker 5. See Figures 1 and 2 to see how it works. Figure 1 Figure 2 ReferenceFinder This program was also designed by Robert Lang. Its goal is to find any point or line on a piece of paper based on x and y coordinates. It uses the Huzita axioms, axioms that describe any way to fold, so it can find thousands of points on a square. ReferenceFinder comes up with the five most accurate results. Usually, ReferenceFinder is as accurate as 1 thousandth away, the limit to human precision. The current version of ReferenceFinder is V4.0.1. Although itʼs a great program to use, you are using it at you own risk. It may crash your computer if you donʼt have enough RAM. See Figure 3 if you want to know how it works Figure 3 Flat Foldability LOCAL Flat Folding This section may be challenging to understand, but please do not get discouraged. In this section, we will also see the math behind Origami. Kawasaki Theorem The Kawasaki Theorem says that around any single vertex in the model, α1+α2+α3+α4...αN=2∏, α1-α2+α3-α4...αN = 0°, and that α1+α3+...αN-1 =∏, as well as α2+α4...+αN=∏ where αi is the ith angle between two crease lines around a vertex. Proof: Fold the model at the left in Figure 1, what you should get is the model on the right. Remember, dot-dot-dash for mountain folds and dash-dash for valley Figure 1 Left:Crease pattern of a local vertex Right: Folded model If we travel along the creases, switching directions at every crease, and assume that going right is positive and left negative, we get that positive α1 (because it is going to the right) minus α2 (because it is going to the left) plus α3 minus α4 =0°, because we are back where we started. If we unfold, we see that our path was a full circle, or 2∏. So we have these two equations: α1+α2+α3+α4=2∏ α1-α2+α3-α4= 0° If we add both equations we get 2*α1+0+ 2*α3+0= 2∏ /2 α1+α3=∏ the same can apply with α2 and α4 if you assume left is positive and right is negative Maekawaʼs Theorem This states that the number of mountain folds minus the number of valley folds around a vertex = ±2 Proof: 1.Fold the same thing you folded in the first theorem; realize that all the valley folds are 2∏ folds and the mountain folds 0° 2.Cut off the top part 3.A polygon should appear when unfolded, refer to Figure 2; notice that the number of creases is the same as the number of points on the polygon which is also the same as the number of sides 4.Let n be the number of creases and also the number of sides Figure 2 The cut polygon 5.We know that the summation of angles in a polygon is (n-2) ∏ 6.Since n=the number of creases, then n=the number of mountain folds + the number of valley folds. We will express this with n=M+V 7.Therefore (M+V-2) ∏ = ∑ of interior angles=angle of mountain fold * M + angle of valley fold * V (M+V-2) ∏ = 0M +2∏V ÷∏ M+V-2= 0+2V -2V +2 M-V-2=0 M-V=2 When you turn the model inside out M-V=-2, because V=0 and M=2∏ Corollary The number of creases around one vertex are even Proof: If n is the number of creases then n=M+V (1) M-V = 2, then M = 2 + V substitute in the equation (1) n=2+V+V n = 2 + 2*V n = 2 (V + 1), which is even because any thing times two is even. The same thing works if V-M=2 GLOBAL Flat Folding Global Flat Folding is Flat Folding with more than one vertex. M-V= ±2 ?? Unfortunately, local flat folding theorems do not always agree with global flat folding theorems. For example, in local flat folding, M-V= ±2. In the Origami crane, M-V=15. We know that when a vertex is pointing up, MV=2, and when a vertex is pointing down M-V=-2, so how might we fix this problem? One of the ways to do this is saying M-V=2(#of up vertices)-2(#of down vertices), but this is still incorrect, because we may have counted some creases twice. So, we will define an interior crease as a crease with its two endpoints inside the paper. After we do this, we get our proposition: Proposition: M-V= 2(#of up vertices)-2(#of down vertices)-(#of interior mountain creases) + (#of interior valley creases) The only thing that can thwart this proposition is that there are creases that do not intersect any other ones. For example, if we have one mountain fold going through the paper, the proposition tells us to do this equation: M-V=2 x 0 - 2 x 0 - 1 + 0=-1 This is not right, because M-V=1-0=1. So, for these creases we have to assume a vertex and make the crease into two. This agrees with both the Proposition and the Theorem. These are some of the formulae used to generate flat Origami, but there are many more, so go out and search!! Famous Origami Artists Here you will learn about some of the most famous people who folded origami. They helped the art in different ways. Akira Yoshizawa Akira Yoshizawa was one of the best- and the most important origami folders of his time. Not only did he create more origami designs, but he invented a way for all origami artists to communicate, through symbols. When we see arrows that tell us to mountain fold, valley fold, squash fold or anything else, we do not really think about where it came from, we just assume it was there. But we can do this thanks to Akira Yoshizawa. Akira also invented “wet folding”, where you dampen the paper so it creates a three dimensional effect when it is dry. He was born son of a dairy farmer, but was in a factory job in Tokyo, Japan. In the evening, Akira would continue his education being a draftsman, using origami to help him understand and solve geometric problems. He was also fascinated with nature and wildlife, so after he was introduced to origami when he was given a paper boat at the age of three, he liked to design the animals he saw outside. For two years, the artist actually wanted to be a priest of buddha. He didnʼt complete his studies, but he was still very devout to his religion. One time in his life, Akira Yoshizawa was asked to fold an origami zodiac to go with an article in a newspaper, and that was all the recognition he needed. In two years, he published his first in twelve books on origami. He also helped make the Origami Center in Tokyo. The artist died March 14, 2005, the same date that he was born 94 years ago, in 1911. Robert Lang As of 2010, Robert Lang has been doing origami for around 40 years. He has had lots of experience and excelled in different jobs. Before he worked on origami full time, he worked as an engineer, a physicist, and an R&D manager. During this time, he had 80 technical authored or coauthored publications and 50 patents awarded on semiconductor lasers, optics, and integrated optoelectronics. In 1992, he was the first western to be invited to the NOA (Nippon Origami Association). He is a fellow of the OSA (Optics Society of America) and a member of the IEEE Photonics Society, and Editor-in-Chief of the IEEE Journal of Quantum Electronics. In 2009, he received the highest award of Caltech, The Distinguished Alumni Reward. He also contributed to creating software and using origami to solve problems (see “Applications”). Born in Ohio, and raised in Atlanta, Robert Lang now lives in Alamo, California. He has authored or coauthored 9 books on the subject of origami. As you can see these two folders helped origami a lot. Notation, software, and much more.
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