Computational Origami

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My undergraduate Research on the Geometric and algebraic foundations of Computational Origami.

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<p> Discovering the Geometric and Algebraic Foundations Behind Computational Origami T h e U n i v e r s i t y o f A r k a n s a sD e p a r t m e n t o f M a t h e m a t i c sU n d e r t h e D i r e c t i o n o f D r . B . M a d i s o n 4 / 2 6 / 2 0 1 2Todd J . Thomas This paper provides the basic template for Research into the relativity new field of Computational Origami. In this paper we define Computational Origami based on the logic of the Euclidean Axioms and Algebraic Structures. We show that not only is the metric space in which this topology exists is well defined, but also that it is complete and efficient. But the truly most remarkable part is how we show that the real power behind Origami Folding is how it takes us out of the set of all Real Numbers and lets us fold our way in the set of all Complex Numbers. 1Contents1.0 I NTRODUCTI ON 22.0 PRACTI CAL APPLI CATI ONS 22.1 Practical Applications in the Space Program .....................................................32.2 Practical Applications to Biology and Medicine ...............................................42.4 Current Real World Applications ......................................................................53.0 FOUNDATI ONS I N EUCLI DEAN GEOMETRY 53.1 Axiomatic Systems ............................................................................................53.2 Euclidean Axiom Set for Geometry ...................................................................63.3 Huzita-Hutori Axiom Set for Origami ..............................................................83.4 Linking Euclidean Constructions to Origami Folding .....................................113.5 Linking Origami Folding to Euclidean Constructions .....................................133.6 The Fundamental Difference ...........................................................................144.0 ORI GAMI FOUNDATI ONS I N ALGEBRA 154.1 Definitions from Algebra .................................................................................154.2 The Origami Pair..............................................................................................164.3 Foundations of an Origami Constructible Set ..................................................174.3 Origami Roots of Polynomials .........................................................................205.0 BEYOND EUCLI D 215.1 Folding Cube Roots .........................................................................................215.2 Solving the Classical Problem of Trisecting Any Angle .................................216.0 CONCLUSI ON 23 2BI BLI OGRAPHY 24 1.0 I NTRODUCTI ON Origami is known to most as the ancient J apanese art of paper folding which was started by Buddhist Monks in the sixth century. The word origami is actualy a mash of two J apanese words ori for folding and kami which means paper. This art form has largely gone unnoticed by the sciences for nearly 2000 years; however, starting in the early part of the 21st century physicist, medical researchers, and of course mathematicians started finding solutions to real world problems hidden deep in the folds of this ancient art form. As scientist and mathematicians started probing the basic foundations of origami they have found a world of wonder just as intricate as the ancient art itself. 2.0 PRACTI CAL APPLI CATI ONS Although computational and mathematical origami may be interesting to some mathematicians, there is a practical aspect to this field of study as well. Real-world problems that require large surfaces to be compacted into small spaces for transport, then deployed reliably, are exactly the types of problems mathematical and computational origami solves. Some of the most promising areas where this discipline can be applied are the space program, medical sciences, and biology. Currently, the automobile industry uses techniques from computational origami to keep people safer on the roads. 3 2.1 Practical Applications in the Space Program The Lawrence Livermore National Laboratory in Livermore, California has plans to put a telescope into deep space, however; this is not just any typical telescope. These forward-thinking researchers are in the design phase of engineering a deep space telescope with a 100-meter aperture for deployment at some point in the future. J ust for reference, the Hubble Space Telescope has an aperture of a paltry 2.4 meters by comparison. According to Dr. Robert Lang, an expert in Origami Sekkei or Technical folding, the major problem these researchers face is how to fit a lens that measures 100-meters across into a shuttle or rocket transport whose cargo bay is only a few meters wide (Wertheim, 2005). This is accomplished by treating the lens like a laminar surface using hinges that follow along the crease lines. As shown in Figure 1, the prototype of the lens was composed of 72 segments or panels made up of sixteen rectangles, thirty-two right triangles, and twenty-four isosceles triangles. These panels are then subdivided into eight petals each sweeping and area of 45 degrees. Each petal consists of three isosceles triangles, four right triangles, and two rectangles (Heller, 2003). Likewise, Lang notes the solution to this problem and other similar problems such as a 500-meter solar sail will require some type of folding on an unprecedented scale. A first success using origamis came in 1995 when the Space Flight Unit, a J apanese satellite, was launched into low orbit. Its solar arrays deployed utilizing an Figure 1: The prototype lens is composed of 72 segments which are then divided into eight petals. Each petal, one of which is highlighted, sweeps 45 degrees or one-eighth of the structure. 4origami technique known as Muira-ori which has also been found to occur naturally in leaves. 2.2 Practical Applications to Biology and Medicine Medicine has its own set of unique problems, and researchers at Oxford University, U.K. wanted to develop an artificial stent that could easily travel through the circulatory system of a patient, but be large enough to hold open the collapsed artery once it was put into place (World Science, 2007) Using an origami pattern call the Waterbomb Base, they were able to develop a stent that when folded was a mere 12mm and would easily pass through small capillaries for long distances without damaging the stent or the patient. Upon reaching its destination it would then expand to 23 mm when unfolded to hold open a collapsed artery and restore blood flow. Aside from the naturally occurring Muira-ori technique as discussed above, researchers at the Dana-Farber institute have combined origami with nanotechnology to fold sheets of DNA into different shapes, such as octahedrons, smaller than the thickness of a human hair (The Scripps Research Institute, 2004). In the near future these folded objects could be used to transport medicines directly into a cell. According to Dr. William Shih, the lead researcher on this project and assistant professor in the Biological Chemistry Department, his group was able to fold DNA to make several different shapes to include a genie bottle, two kinds of crosses, a square nut, and a railed bridge (Shih, 2008) Figure 2. An artificial stent that uses the origami patter Waterbomb bases to save human lives. 52.4 Current Real World Applications At first thought the automobile industry may be a strange place to find the likes of Dr. Robert Lang, an expert in Origami Sekkei, however; it seems that anyone who has survived an automobile crash due to the deployment of the cars airbags is deeply indebted Dr. Lang. Working with researchers at EASi Engineering in Germany, Dr. Lang developed a crease pattern that allows for a three-dimensional polyhedron (the airbag) to be folded into a flat surface that deploys easily, completely, and at the proper pressure within microseconds. Using an algorithm dubbed the Universal Molecule that was created by Dr. Lang, it shows that there is little difference between folding a three-dimensional polyhedron onto a flat plane and origami folding a flat sheet into another flat polygon (Lang, 2004-2012). EASi was able incorporate this research into their airbag design system which has been used by automotive manufacturers all across the globe. 3.0 FOUNDATI ONS I N EUCLI DEAN GEOMETRY 3.1 Axiomatic Systems In order to ensure that our origami system is consistent, we need to develop it as an axiomatic system. Similar to the approach Euclid of Alexandria took with Geometry, an axiomatic system is logical and will possess a set of axioms from which we can derive other statements (Weisstein, 1999-2012). Furthermore, we shall strive for economy and Figure 3. Using the Universal Molecule algorithm, Lang was able to identify a crease pattern that is highly efficient for deploying airbags. 6efficiency with our axioms, we want them to be independent, which means we do not want to assume any axiom can be proven from the others. Lastly we want to make sure that we can in-fact prove or disprove any statement about our system from the axioms alone, that is we want to be able to say that our system is complete. The advantage to having an axiomatic system is that once a set of fundamental axioms are determined, we can then start deducing other properties (e.g., as lemmas and theorems) from within our system, and thus construct a wholly consistent, independent, and complete origami mathematical system. 3.2 Euclidean Axiom Set for Geometry Over 2000 years ago Euclid of Alexandria approached geometry from an axiomatic stand point as mentioned earlier. By using five independent postulates, Euclid constructed a logical and consistent geometry from which all other geometric lemmas, and theorems would later be derived. In this section we will recall and briefly discuss these first five postulates, and then in Section 3.3 we will compare them to the postulates of Mathematical Origami. A quick search of the internet reveals thousands of websites devoted to Euclids Elements; however, we refer to the website of Dr. David E. J oyce of Clark University, and his translation for the remainder of our discussion. Note that the designations E1 to E5 are not part of Dr. J oyces work, but will be used in future sections as references to the specific postulate: (E1) Postulate 1. Let it have been postulated to draw a straight-line fromany point to any point. (E2) Postulate 2: And to produce a finite straight-line continuously in a straight-line. (E3) Postulate 3: And to draw a circle with any center and radius. 7(E4) Postulate 4: And that all right-angles are equal to one another. (E5) Postulate 5: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side which are the angles less than the two right angles. (J oyce, 1996) Postulates three and four are self-explanatory so we will not go into detail about these, however for the other postulates we shall give a brief interpretation. Postulate 1 (E1) gives us our first construction using a straight edge. In this postulate, Euclid is simply telling us that for any two points in a plane we can use a straight edge to construct a straight line (AB) between the two points. What he does not explicitly say, but it is implied, is that line is also unique. In Postulate (E2), given line segment AB, we can construct and extend the segment AB to CD. It is interesting to note that Euclid does not tell us how far CD can be extended. Postulate 5 (E5) is also known as the parallel postulate. Thus if we have two lines l and m and then a third line t intersects both l and m, the two lines will intersect on the side where the angle each line makes with the traversal is less than ninety degrees (J oyce, 1996). Starting exclusively with one of these five postulates, a system is considered geometrically constructible if we can show that by starting at a given point/line/circle we end up at another point/line/circle that makes up whatever geometric object we are seeking. Figure 4: I f two lines a and b are parallel, then angle theta is equal to angle beta. 83.3 Huzita-Hutori Axiom Set for Origami Origami is the art of paper folding, and as most of us know from our elementary school days, there are certain folds that just seem fundamental. For instance, we can fold a straight line quite easily, however trying to fold a curve, although possible, is quite a difficult undertaking, and nearly impossible to control. Since we are trying to establish a link between Euclids postulates and origami, we will consider the folding of a curve to be non-fundamental and exclude it for our purposes (Geretschlager, 1995). Now consider an origami construction such as a crane. We start with a flat sheet of paper, a plane if you will analogous to the Euclidean plane in geometry constructions. Whereas in geometric constructions we start with a point, in an origami construction we start with a fold, and as we develop our folds, a more complex object is formed. Of course when we fold an origami object, we are going from the two dimensional plane to a three dimensional object; however, after we have created an origami object we can then unfold object and return the paper back into a plane. What we will then consider are the creases that are left behind from the unfolded origami object. Thus since we wish to find a relation between Euclidean constructions and origami constructions, we must first define a set of allowed operations, similar to five postulates that Euclid defined for plane geometry. The first six postulates were presented by Humiaki Huzita at the First International Meeting of Origami Science and Technology in 1991 and a seventh one was found by Koshiro Hutori in 2002. Although Huzita and Hutori were the first to present their axioms that bear their names, it is important to note that in 1989 J acques J ustin published a paper entitled Resolution par le pliage de lequation du troisieme degre et applications geometriques, in which he accounted for seven combinations of alignments 9(Lang, Huzita-J ustin Axioms, 2004). We will enumerate these postulates (O1),(O7) to distinguish them from the Euclidean Axioms (E1),(E5). (O1) Given two points p1 and p2 we can fold a line connecting them. Figure 5 (O2) Given two points p1 anu p2 we can fold p1 onto p2....</p>

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