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<div>First up is the tree method, whose software implementation TreeMaker I demoed at the end of Lecture 3. I'll describe how it lets us fold an optimum stick-figure (tree) origami base, although computing that optimum is actually NP-complete (as we'll see in Lecture 5). This algorithm is used throughout modern complex origami design; I'll show some examples by Robert Lang and our own Jason Ku. Second we'll look at a simple, fully understood case: the smallest square to fold a cube. Third we'll look at a classic problem that we made progress on recently: folding an n n checkerboard from the smallest bicolor square. Finally we'll look at the latest and most general method, Origamizer, for folding any polyhedron reasonably efficiently. Here we don't have a nice theoretical guarantee on optimality, but the method works well in practice, provably always works, and has other nice features such as watertightness. No support for video detected. Please use an HTML5 browser.</div><div></div><div></div><div>The second part is somewhat true, in that the development of complex origami design techniques evolved roughly in parallel with the development of origami-specific design software. But the design algorithms can still be applied by individuals working by hand (and there are good reasons for doing so), and for the type of complex origami highlighted above, even today, most designs do not require specialized origami software.</div><div></div><div></div><div></div><div></div><div></div><div>treemaker origami download</div><div></div><div>Download Zip: https://t.co/i8CNXlcRZe </div><div></div><div></div><div>TreeMaker allows one to set up quite elaborate relationships between flaps, their lengths, and their angles: far more complex relationships than are possible using pencil-and-paper origami design. Which meant that it was now possible, with TreeMaker, to solve for origami bases that truly were more complicated than anything a person could design by hand.</div><div></div><div></div><div>The laser cutter was growling away, scoring one of Lang's Hanji sheets. He twiddled with his computer. On the screen was a lacy geometric pattern. Lang had designed it with software he started writing in 1990 called TreeMaker, which is well known in origami circles; it was the first software that would translate "tree" forms -- that is, anything that sort of resembles a stick figure, such as people or bugs -- into crease patterns. Another program he wrote, ReferenceFinder, converts the patterns into step-by-step folding instructions. This secured his position as the most technologically ambitious of the origami masters. In 2004, he was an artist-in-residence at M.I.T., and gave a now famous lecture about origami and its relationship to mathematical notions, like circle packing and tree theory.</div><div></div><div></div><div>To the non-origami person, the sequence that transforms a sheet of paper into a beautiful folded object can seem miraculous. Even to the origami aficionado, however, the idea that a single drawing of the creases conveys the full folding sequence can seem equally miraculous. But in fact, a crease pattern can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. And thus, it can actually give the folder insight into the thought processes of the origami composer in a way that a step-by-step folding sequence cannot.</div><div></div><div></div><div>For centuries, origami patterns had at most thirty steps; now they could have hundreds. And as origami became more complex it also became more practical. Scientists began applying these folding techniques to anything -- medical, electrical, optical, or nanotechnical devices, and even to strands of DNA -- that had a fixed size and shape but needed to be packed tightly and in an orderly way.</div><div></div><div></div><div>Anyway, early in the show was a demonstration by Robert Lang, noted origami expert, where he designed crease patterns on computer then used his laser engraver/cutter to engrave the patterns into his large sheet of folding paper. He was able to get extremely complex models this way.</div><div></div><div></div><div>The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve up-to cubic mathematical equations.[1]</div><div></div><div></div><div>Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases.[2] Computational origami results either address origami design or origami foldability.[3] In origami design problems, the goal is to design an object that can be folded out of paper given a specific target configuration. In origami foldability problems, the goal is to fold something using the creases of an initial configuration. Results in origami design problems have been more accessible than in origami foldability problems.[3]</div><div></div><div></div><div></div><div></div><div></div><div></div><div>In 1893, Indian civil servant T. Sundara Row published Geometric Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the kindergarten system. Row demonstrated an approximate trisection of angles and implied construction of a cube root was impossible.[4]</div><div></div><div></div><div>The first complete statement of the seven axioms of origami by French folder and mathematician Jacques Justin was written in 1986, but were overlooked until the first six were rediscovered by Humiaki Huzita in 1989.[15] The first International Meeting of Origami Science and Technology (now known as the International Conference on Origami in Science, Math, and Education) was held in 1989 in Ferrara, Italy. At this meeting, a construction was given by Scimemi for the regular heptagon.[16]</div><div></div><div></div><div>In 1996, Marshall Bern and Barry Hayes showed that the problem of assigning a crease pattern of mountain and valley folds in order to produce a flat origami structure starting from a flat sheet of paper is NP-complete.[18]</div><div></div><div></div><div>In 2002, belcastro and Hull brought to the theoretical origami the language of affine transformations, with an extension from R \displaystyle R 2 to R \displaystyle R 3 in only the case of single-vertex construction.[23]</div><div></div><div></div><div>In 2003, Jeremy Gibbons, a researcher from the University of Oxford, described a style of functional programming in terms of origami. He coined this paradigm as "origami programming." He characterizes fold and unfolds as natural patterns of computation over recursive datatypes that can be framed in the context of origami.[27]</div><div></div><div></div><div>In 2005, principles and concepts from mathematical and computational origami were applied to solve Countdown, a game popularized in British television in which competitors used a list of source numbers to build an arithmetic expression as close to the target number as possible.[28]</div><div></div><div></div><div>In 2009, Alperin and Lang extended the theoretical origami to rational equations of arbitrary degree, with the concept of manifold creases.[29][30] This work was a formal extension of Lang's unpublished 2004 demonstration of angle quintisection.[30][31]</div><div></div><div></div><div>The construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is an NP-complete problem.[32] Related problems when the creases are orthogonal are called map folding problems. There are three mathematical rules for producing flat-foldable origami crease patterns:[33]</div><div></div><div></div><div>As a result of origami study through the application of geometric principles, methods such as Haga's theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths. Other theorems and methods have allowed paperfolders to get other shapes from a square, such as equilateral triangles, pentagons, hexagons, and special rectangles such as the golden rectangle and the silver rectangle. Methods for folding most regular polygons up to and including the regular 19-gon have been developed.[36] A regular n-gon can be constructed by paper folding if and only if n is a product of distinct Pierpont primes, powers of two, and powers of three.</div><div></div><div></div><div>The classical problem of doubling the cube can be solved using origami. This construction is due to Peter Messer:[38] A square of paper is first creased into three equal strips as shown in the diagram. Then the bottom edge is positioned so the corner point P is on the top edge and the crease mark on the edge meets the other crease mark Q. The length PB will then be the cube root of 2 times the length of AP.[14]</div><div></div><div></div><div>Angle trisection is another of the classical problems that cannot be solved using a compass and unmarked ruler but can be solved using origami.[39] This construction, which was reported in 1980, is due to Hisashi Abe.[38][9] The angle CAB is trisected by making folds PP' and QQ' parallel to the base with QQ' halfway in between. Then point P is folded over to lie on line AC and at the same time point A is made to lie on line QQ' at A'. The angle A'AB is one third of the original angle CAB. This is because PAQ, A'AQ and A'AR are three congruent triangles. Aligning the two points on the two lines is another neusis construction as in the solution to doubling the cube.[40][9]</div><div></div><div></div><div>The problem of rigid origami, treating the folds as hinges joining two flat, rigid surfaces, such as sheet metal, has great practical importance. For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites.</div><div></div><div></div><div>The placement of a point on a curved fold in the pattern may require the solution of elliptic integrals. Curved origami allows the paper to form developable surfaces that are not flat.[41] Wet-folding origami is a technique evolved by Yoshizawa that allows curved folds to create an even greater range of shapes of higher order complexity.</div><div></div><div></div><div>Computational origami is a branch of computer science that is concerned with studying algorithms for solving paper-folding problems. In the early 1990s, origamists participated in a series of origami contests called the Bug Wars in which artists attempted to out-compete their peers by adding complexity to their origami bugs. Most competitors in the contest belonged to the Origami Detectives, a group of acclaimed Japanese artists.[44] Robert Lang, a research-scientist from Stanford University and the California Institute of Technology, also participated in the contest. The contest helped initialize a collective interest in developing universal models and tools to aid in origami design and foldability.[44]</div><div></div><div> 9738318194</div>
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