From Flapping Birds to Space Telescopes: The Modern Science of Origami

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1 From Flapping Birds to Space Telescopes: The Modern Science of Origami Robert J. Lang Notes by Radoslav Vuchkov and Samantha Fairchild Abstract This is a summary of the presentation given by Robert Lang at the MASS colloquium on September 25, 2013 at the Pennsylvania State University. We would like to thank Dr. Lang for providing the slides for information and images used in this paper. 1 Introduction Origami is the Japanese art of folding paper. The complexity ranges from creating the iconic crane to the intricate cuckoo clock designed by Robert Lang himself and shown in Figure 1. For many years origami artist thought is was impossible to create complex figures like Lang s cuckoo clock, but using mathematics and the help of computers, the recent decades have given birth to a new level of complexity in origami. Figure 1: The Black Forest Cuckoo Clock 1

2 Figure 2: Akira Yoshizawa 2 History There is no documented evidence on when origami began, but there is evidence from as early as 1797 of relatively complex origami such as Senbazura Orikata, which is the folding of multiple cranes from a single piece of paper. Even earlier, there was documented evidence in a Japanese newspaper from 1734 including designs such as the crane, boat, table, and box, revealing that origami was thoroughly developed before the 1700 s. Starting in the 1900 s however, Origami began to evolve into a much more complicated art form due to Akira Yoshizawa (Figure 2). Akira Yoshizawa made new creations and also debuted a common language that allowed the origami artist to communicate the instructions on folding their designs. The change that has come to origami allowing for complex designs like the cuckoo clock in Figure 1 is that origami artists such as Lang have developed a new way of viewing origami from a mathematical perspective. The power of the mathematical view of origami is that it has allowed for complexity in origami that had never been envisioned before. 3 The Math Behind it all No matter how simple or complex the structures, origami can be reduced to looking at the crease pattern on the paper. Looking at the paper from the above, the crease pattern will be a collection of in creases, called valley folds, and out creases, called mountain folds, represented by dotted lines and solid lines, respectively. There are four rules that must be satisfied in order for a 2

3 Figure 3: The two possibilities for creases, and they must be NAE crease pattern to be made into origami. They are as follows: 2 colorability: any two adjacent shapes (separated by a mountain or a valley fold) must be colored a different color, and this can be completed on the crease pattern with only two colors. Mountain-Valley Counting: if M stands for the number of mountain folds, and V stands for the number of valley folds, then it must always be the case that M V = ±2. Alternate angles around a vertex must sum to a straight line, so in the language of 2-colorability, the angles of the first color must sum to a straight line, and the angles of the other color must also sum to a straight line. Layer Ordering: There cannot be any self-intersections at overlaps. Making a crease pattern that satisfies the first 3 rules is relatively simple, but it is the rule of layer ordering that can be challenging. To understand this last rule, for any layering or pleat, there must be a valley on one side, and a mountain fold on the other side and they are assigned a value true or false based on Figure 3. Then, when three pleats come together to make a layered fold, in order to satisfy the fourth rule, they must be Not-All-Equal as shown in Figure 3. In fact, any NAE-3-SAT problem can be encoded as a crease assignment problem. The NAE-3-SAT problems are NP-complete, so no fast or easy solution is known, showing that origami is indeed challenging to create. The basic question faced when designing origami is how to fold a square to produce the desired form. The main algorithmic approach in creating origami uses circle packing. The first step in design is to choose a subject, and then create a tree or stick figure out of that subject as seen in Figure 5. Then from 3

4 Figure 4: The Use of Circle Packing to Create a Crease Pattern that tree, create a flat folded piece of paper called a base that has flaps corresponding to all the limbs in the tree. From there, it is up to the artist to make the base into the artistic model they desire. The mathematically challenging step in the process is transforming the tree into a base, which is where the circle packing is used. The idea behind circle packing is that to make a flap on an corner, a quarter of a circle is required. To make a flap on an edge, a half circle is required, and to make a flap from the center of the paper, a full circle is required. So, to optimize the flaps and the placement of flaps, circle packing is used as in Figure 4. Note that the lines between the centers of touching circles are always creases. Those creases divide the square into distinct polygons that correspond to pieces of the original tree which holds the intended shape of the sculpture. The circle packing gives a division of the paper, then from that division the crease pattern must collapse so that its edges form a stick figure. These crease patterns are called bun-shi, or molecules. A triangle has only one possible molecule, but as the sides of the polygons increase, the number of possible molecules grows rapidly, and it is the artists decision as to what molecule to use. This completes one approach for creating origami: Take the subject and create a stick figure from the subject. Use circle packing to optimize the placement of the flaps for the limbs of the tree or stick figure. Fill in the resulting polygons with molecular crease patterns. Fold! This process can be completed by hand, or also by computers, which involves using optimization, linear objective function, linear and quadratic constraints, and nonconvex feasible region. 4

5 Figure 5: Lang uses Computers to transform ideas like a stick figure into an origami deer. 4 Applications Lang himself has used origami for projects with famous companies such as Google, an advertisement for a car company, and also to fold the lens for a new space telescope. By using his computer programs and algorithms, Lang can create these new designs on demand. Origami is still an art, which Lang showed is documented by the bug wars : a friendly competition between fellow origami artists to make the most intricate bug possible. The computers are an essential part to creating these pieces (Figure 5), but the final folding design still cannot easily become a masterpiece without the steady and precise hands of an origami artist. Origami is also used geometrically, where tessellations are used to create elaborate geometric origami. In fact Lang went as far as to show the audience that origami is not bounded to Euclidean geometry only, but can be produced by using hyperbolic paper as well. Thus the idea of hyperbolic origami was introduced. (Figure 6) 5 Conclusion Origami is an ancient Japenese art that has now merged with mathematics and the modern computer age by using cicle packing and other methods to create intricate masterpieces that had never been dreamed of before. 5

6 Figure 6: The result when folding the crane with hyperbolic paper 6