1414:. A universality result defines the bounds of possibility given a particular model of folding. For example, a large enough piece of paper can be folded into any tree-shaped origami base, polygonal silhouette, and polyhedral surface. When universality results are not attainable, efficient decision algorithms can be used to test whether an object is foldable in polynomial time. Certain paper-folding problems do not have efficient algorithms. Computational intractability results show that there are no such polynomial-time algorithms that currently exist to solve certain folding problems. For example, it is NP-hard to evaluate whether a given crease pattern folds into any flat origami.
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folding of bases. Computational origami results either address origami design or origami foldability. 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.
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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
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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
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In 2014, researchers at the
Massachusetts Institute of Technology, Harvard University, and the Wyss Institute for Biologically Inspired Engineering published a method for building self-folding machines and credited advances in computational origami for the project's success. Their origami-inspired
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In 2017, Erik
Demaine of the Massachusetts Institute of Technology and Tomohiro Tachi of the University of Tokyo published a new universal algorithm that generates practical paper-folding patterns to produce any 3-D structure. The new algorithm built upon work that they presented in their paper in
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can be solved using origami. This construction is due to Peter Messer: 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
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and
Masamori Sakamaki demonstrated a novel map-folding technique whereby the folds are made in a prescribed parallelogram pattern, which allows the map to be expandable without any right-angle folds in the conventional manner. Their pattern allows the fold lines to be interdependent, and hence the
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Applications of computational origami have been featured by various production companies and commercials. Lang famously worked with Toyota Avalon to feature an animated origami sequence, Mitsubishi
Endeavor to create a world entirely out of origami figures, and McDonald's to form numerous origami
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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
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Computational origami has contributed to applications in robotics, engineering, biotechnology & medicine, industrial design. Applications for origami have also been developed in the study of programming languages and programming paradigms, particular in the setting of functional programming.
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is another of the classical problems that cannot be solved using a compass and unmarked ruler but can be solved using origami. This construction, which was reported in 1980, is due to
Hisashi Abe. The angle CAB is trisected by making folds PP' and QQ' parallel to the base with QQ' halfway in
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The side of a square can be divided at an arbitrary rational fraction in a variety of ways. Haga's theorems say that a particular set of constructions can be used for such divisions. Surprisingly few folds are necessary to generate large odd fractions. For instance
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1999 that first introduced a universal algorithm for folding origami shapes that guarantees a minimum number of seams. The algorithm will be included in
Origamizer, a free software for generating origami crease patterns that was first released by Tachi in 2008.
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in 1989. 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
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of spherical optics. In the same paper, Alperin showed a construction for a regular heptagon. In 2004, was proven algorithmically the fold pattern for a regular heptagon. Bisections and trisections were used by
Alperin in 2005 for the same construction.
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In 2009, Alperin and Lang extended the theoretical origami to rational equations of arbitrary degree, with the concept of manifold creases. This work was a formal extension of Lang's unpublished 2004 demonstration of angle quintisection.
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at all points on its surface, and only folds naturally along lines of zero curvature. Curved surfaces that can't be flattened can be produced using a non-folded crease in the paper, as is easily done with wet paper or a fingernail.
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There are several software design tools that are used for origami design. Users specify the desired shape or functionality and the software tool constructs the fold pattern and/or 2D or 3D model of the result. Researchers at the
1356:, in December 2001. In January 2002, she folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times in the same direction, debunking a long-standing myth that paper cannot be folded in half more than eight times.
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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
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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
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asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete cut. The solution, known as the fold-and-cut theorem, states that any shape with straight sides can be obtained.
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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
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In late 2001 and early 2002, Britney
Gallivan proved the minimum length of paper necessary to fold it in half a certain number of times and folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times.
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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.
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Paper-folding problems are classified as either origami design or origami foldability problems. There are predominantly three current categories of computational origami research: universality results,
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have developed and posted publicly available tools in computational origami. TreeMaker, ReferenceFinder, OrigamiDraw, and Origamizer are among the tools that have been used in origami design.
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can be packed into a very compact shape. In 1985 Miura reported a method of packaging and deployment of large membranes in outer space, and as early as 2012 this technique had been applied to
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Robu, Judit; Ida, Tetsuo; Ţepeneu, Dorin; Takahashi, Hidekazu; Buchberger, Bruno (2006). "Computational Origami Construction of a Regular Heptagon with Automated Proof of Its Correctness".
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map can be unpacked in one motion by pulling on its opposite ends, and likewise folded by pushing the two ends together. No unduly complicated series of movements are required, and folded
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Robert Lang participated in a project with researchers at EASi Engineering in Germany to develop automotive airbag folding designs. In the mid-2000s, Lang worked with researchers at the
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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
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robot was reported to fold itself in 4 minutes and walk away without human intervention, which demonstrated the potential for autonomous self-controlled assembly in robotics.
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290:, 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.
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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
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The maximum number of times an incompressible material can be folded has been derived. With each fold a certain amount of paper is lost to potential folding. The
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In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms.
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Animation of folds to make a Samurai helmet, also called a kabuto. (On a laptop computer, Julia and GLMakie generated the 66 second .mp4 video in 10 seconds.)
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is the problem of whether a square or rectangle of paper can be folded so the perimeter of the flat figure is greater than that of the original square.
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It follows from this that every vertex has an even number of creases, and therefore also the regions between the creases can be colored with two colors.
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published "Houdini's Paper Magic," which described origami techniques that drew informally from mathematical approaches that were later formalized.
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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
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origami is a technique evolved by Yoshizawa that allows curved folds to create an even greater range of shapes of higher order complexity.
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There are other software solutions associated with building computational origami models using non-paper materials such as Cadnano in
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Benedetto Scimemi, Regular Heptagon by Folding, Proceedings of Origami, Science and Technology, ed. H. Huzita., Ferrara, Italy, 1990
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which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the
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In 1980 a construction was reported which enabled an angle to be trisected. Trisections are impossible under Euclidean rules.
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triangles. Aligning the two points on the two lines is another neusis construction as in the solution to doubling the cube.
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or Kawasaki-Justin theorem: at any vertex, the sum of all the odd angles (see image) adds up to 180 degrees, as do the even.
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Justin, Jacques, "Resolution par le pliage de l'equation du troisieme degre et applications geometriques", reprinted in
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must be expressed in the same units, such as inches. This result was derived by Britney Gallivan, a high schooler from
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system. Row demonstrated an approximate trisection of angles and implied construction of a cube root was impossible.
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In 1999, a theorem due to Haga provided constructions used to divide the side of a square into rational fractions.
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A practical problem is how to fold a map so that it may be manipulated with minimal effort or movements. The
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Assigning a crease pattern mountain and valley folds in order to produce a flat model has been proven by
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can be generated with three folds; first halve a side, then use Haga's theorem twice to produce first
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The edge with the crease mark is considered a marked straightedge, something which is not allowed in
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In 2005, principles and concepts from mathematical and computational origami were applied to solve
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for a 2×2 grid of squares: there are eight different ways to fold such a map along its creases
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Bird, Richard; Mu, Shin-Cheng (September 2005). "Countdown: A case study in origami programming".
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The Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education
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Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 1996)
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The first complete statement of the seven axioms of origami by French folder and mathematician
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Michael J Winckler; Kathrin D Wold; Hans Georg Bock (2011). "Hands-on Geometry with Origami".
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Schneider, Jonathan (December 10, 2004). "Flat-Foldability of Origami Crease Patterns" (PDF).
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Origami USA: We are the American national society devoted to origami, the art of paperfolding
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is a rigid fold that has been used to deploy large solar panel arrays for space satellites.
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Geretschlager, Robert (1995). "Euclidean Constructions and the Geometry of Origami".
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and others first attempted to write computer code that would solve origami problems.
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Origami: Fourth International Meeting of Origami Science, Mathematics, and Education
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Bertschinger, Thomas H.; Slote, Joseph; Spencer, Olivia Claire; Vinitsky, Samuel.
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was written in 1986, but were overlooked until the first six were rediscovered by
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problems. There are three mathematical rules for producing flat-foldable origami
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Proceedings of the First International Meeting of Origami Science and Technology
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343:: at any vertex the number of valley and mountain folds always differ by two.
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for a Miura fold. The parallelograms of this example have 84° and 96° angles.
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A diagram showing the first and last step of how origami can double the cube
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1766:"a comparison between straight edge and compass constructions and origami"
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2564:"From Flapping Birds to Space Telescopes: The Modern Science of Origami"
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381:. Further references and technical results are discussed in Part II of
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Demaine, Erik (2001). "Recent Results in Computational Origami" (PDF).
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Felton, S.; Tolley, M.; Demaine, E.; Rus, D.; Wood, R. (2014-08-08).
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2140:. Lecture Notes in Computer Science. Vol. 3763. pp. 19–33.
2010:
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is a solution to the problem, and several others have been proposed.
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Therefore, BQ:CQ=k:1 implies AP:BP=k:2 for a positive real number k.
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Thomas C. Hull (2002). "The Combinatorics of Flat Folds: a Survey".
1812:, Tech. Report 618, The Institute of Space and Astronautical Science
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3118:. Science Networks. Historical Studies. Vol. 59. Birkhäuser.
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Alperin, Roger C. (2005). "Trisections and Totally Real Origami".
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length PB will then be the cube root of 2 times the length of AP.
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1988:
K. Haga, Origamics, Part 1, Nippon Hyoron Sha, 1999 (in Japanese)
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Viewpoints: Mathematical Perspective and Fractal Geometry in Art
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Haga, Kazuo (2008). Fonacier, Josefina C; Isoda, Masami (eds.).
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for folding paper in half in a single direction was given to be
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3194:
Origami Design Secrets: Mathematical Methods for an Ancient Art
2828:. Julia code animating kabuto is in example 3.4. 31 March 2024.
2105:
Mathematical Origami: Another View of Alhazen's Optical Problem
609:
then a number of other lengths are also rational functions of
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3115:
A History of Folding in Mathematics: Mathematizing the Margins
1810:
Method of packaging and deployment of large membranes in space
328:. Related problems when the creases are orthogonal are called
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3174:. University of Tsukuba, Japan: World Scientific Publishing.
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T. Sundara Row (1917). Beman, Wooster; Smith, David (eds.).
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1798:, 1981, and online at the British Origami Society web site.
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In 1986, Messer reported a construction by which one could
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Origamics: Mathematical Explorations Through Paper Folding
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Hull, Thomas (2002). "In search of a practical map fold".
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Alperin, Roger C. (2002). "Ch.12". In Hull, Thomas (ed.).
1563:"Solving cubics with creases: the work of Beloch and Lill"
1890:(10): 284–285 – via Canadian Mathematical Society.
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is the minimum length of the paper (or other material),
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1325:{\displaystyle L={\tfrac {\pi t}{6}}(2^{n}+4)(2^{n}-1)}
1175:. Using a marked straightedge in this way is called a
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or paper folding has received a considerable amount of
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1846:"Miura folding: Applying origami to space exploration"
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Britney Gallivan has solved the Paper Folding Problem
3236:"An Overview of Mechanisms and Patterns with Origami"
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International Journal of Pure and Applied Mathematics
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The accompanying diagram shows Haga's first theorem:
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Dividing a Segment into Equal Parts by Paper Folding
1826:. Japan Aerospace Exploration Agency. Archived from
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1637:"Lecture: Recent Results in Computational Origami".
163:, which is impossible with Euclidean constructions.
2591:"Numberphile: How to Trisect an Angle with Origami"
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1219:, has great practical importance. For example, the
1137:{\displaystyle {\frac {BQ}{CQ}}={\frac {2AP}{BP}}.}
227:brought to the theoretical origami the language of
4758:The Drawing of Geometric Patterns in Saracenic Art
2900:"How Origami Is Revolutionizing Industrial Design"
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2276:: CS1 maint: DOI inactive as of September 2024 (
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2805:MIT News | Massachusetts Institute of Technology
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1344:is the number of folds possible. The distances
271:in only the case of single-vertex construction.
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2775:"A Computational Algorithm for Origami Design"
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4888:European Society for Mathematics and the Arts
4062:Mathematica: A World of Numbers... and Beyond
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2988:"A method for building self-folding machines"
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1075:Haga's theorems are generalized as follows:
4042:List of works designed with the golden ratio
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2533:"A Note on Haga's theorems in paper folding"
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16:Overview of the mathematics of paper folding
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2293:"One-, Two-, and Multi-Fold Origami Axioms"
2291:Lang, Robert J.; Alperin, Roger C. (2009).
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1927:. University of Cambridge. + plus magazine.
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1612:"origami - History of origami | Britannica"
403:classical construction problems of geometry
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4102:Cathedral of Saint Mary of the Assumption
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2449:. Cambridge: Cambridge University Press.
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1840:
1159:Doubling the cube: PB/PA = cube root of 2
456:-gon can be constructed by paper folding
3259:
3111:
2963:"Webb and Origami - Webb Telescope/NASA"
2826:"Julia and Projective Geometric Algebra"
2660:
2552:
1973:"How to Divide the Side of Square Paper"
1966:
1964:
1863:
1834:
1553:
1551:
1425:
1374:
1186:
1154:
482:
314:
306:
188:
150:
120:
83:
18:
4648:Vier BĂĽcher von Menschlicher Proportion
3770:
3300:Introduction to Statistics with Origami
3215:"Folding optimal polygons from squares"
2922:
2589:Dancso, Zsuzsanna (December 12, 2014).
2333:
2331:
2213:
2168:
2102:
1801:
1752:
1541:
1421:
579:{\displaystyle BQ={\frac {2AP}{1+AP}}.}
390:
5063:
3048:
2898:Magazine, Smithsonian; Morrison, Jim.
2683:
2666:
2588:
2429:
2351:
2231:
2129:
1979:
1931:
1922:
1894:
1776:
1476:Lawrence Livermore National Laboratory
1182:
1173:compass and straightedge constructions
830:{\displaystyle {\frac {1+x^{2}}{1+x}}}
4973:
3744:
3334:
2759:
2757:
2731:
2729:
2488:"Origami and Geometric Constructions"
2001:
1961:
1941:Bern, Marshall; Hayes, Barry (1996).
1916:
1807:
1727:
1548:
1437:Massachusetts Institute of Technology
444:, and special rectangles such as the
3588:Geometric Exercises in Paper Folding
3169:
2689:
2667:Korpal, Gaurish (25 November 2015).
2561:
2485:
2357:
2328:
2032:How to Free Your Inner Mathematician
1904:, H. Huzita ed. (1989), pp. 251–261.
1816:
1782:
1763:
1654:Geometric Exercises in Paper Folding
1644:
1633:
1631:
1582:10.4169/amer.math.monthly.118.04.307
1557:
1501:figures from cheeseburger wrappers.
1412:computational intractability results
1150:
779:{\displaystyle {\frac {1-x^{2}}{2}}}
413:— are proven to be unsolvable using
67:Geometric Exercises in Paper Folding
4833:Journal of Mathematics and the Arts
3609:A History of Folding in Mathematics
2423:"Robert Lang folds way-new origami"
2073:Linear Algebra and Its Applications
1949:. ACM, New York. pp. 175–183.
1671:
1206:
1071:A generalization of Haga's theorems
357:A sheet can never penetrate a fold.
13:
4847:Making Mathematics with Needlework
3095:
2754:
2726:
1970:
1395:California Institute of Technology
736:{\displaystyle {\frac {1-x}{1+x}}}
478:
100:', later used in the sixth of the
14:
5092:
4673:I quattro libri dell'architettura
3253:
2940:
2852:
2234:Journal of Functional Programming
2171:The American Mathematical Monthly
1975:. Japan Origami Academic Society.
1659:The Open Court Publishing Company
1628:
1340:is the material's thickness, and
692:{\displaystyle {\frac {2x}{1+x}}}
589:The function changing the length
4952:
4951:
4197:Self-portrait in a Convex Mirror
3881:
3318:
2772:
2569:. Usenix Conference, Boston, MA.
1943:"The complexity of flat origami"
426:
3509:Alexandrov's uniqueness theorem
3067:
3042:
2979:
2955:
2934:
2916:
2891:
2867:
2846:
2818:
2793:
2784:
2766:
2635:
2610:
2573:
2479:
2415:
2380:
2225:
2207:
2138:Automated Deduction in Geometry
1923:Newton, Liz (1 December 2009).
1738:. Chrysalis Books. p. 18.
1465:
1445:University of California Irvine
487:BQ is always rational if AP is.
302:
297:
4908:National Museum of Mathematics
4660:Regole generali d'architettura
2704:10.1080/10724117.2002.11975147
2679:(3). Teachers of India: 20–23.
2504:Geretschläger, Robert (2008).
1712:
1685:
1665:
1604:
1478:to develop a solution for the
1319:
1300:
1297:
1278:
1:
3447:Regular paperfolding sequence
2086:10.1016/S0024-3795(01)00608-5
1723:. Louisiana State University.
1692:George Edward Martin (1997).
1570:American Mathematical Monthly
1531:Regular paperfolding sequence
1408:efficient decision algorithms
407:trisecting an arbitrary angle
4433:Garden of Cosmic Speculation
3595:Geometric Folding Algorithms
3362:Mathematics of paper folding
2643:"Siggraph: "Curved Origami""
2446:Geometric folding algorithms
1389:, a research-scientist from
432:square, such as equilateral
384:Geometric Folding Algorithms
108:to be solved using origami.
7:
1719:Robert Carl Yeates (1949).
1504:
1489:Other applications include
1400:
615:
10:
5097:
5050:three-phase electric power
5015:artificial neural networks
3950:Islamic geometric patterns
3645:Margherita Piazzola Beloch
3295:Overview of Origami Axioms
3265:"Origami Mathematics Page"
3112:Friedman, Michael (2018).
3049:Brewin, Bob (2004-05-10).
2621:. CRC Press. p. 225.
2341:The Mathematics of Origami
2029:D'Agostino, Susan (2020).
1480:James Webb Space Telescope
394:
146:solar panels on spacecraft
53:
49:
5005:
5000:"Mathematics of" articles
4947:
4916:
4870:
4824:
4771:A Mathematician's Apology
4731:
4684:
4567:
4530:
4523:
4355:
4208:
4154:
4145:
4092:
4034:
3890:
3879:
3778:
3632:
3579:
3558:
3501:
3455:
3424:
3416:Yoshizawa–Randlett system
3368:
3124:10.1007/978-3-319-72487-4
2247:10.1017/S0956796805005642
1698:. Springer. p. 145.
1163:The classical problem of
463:is a product of distinct
231:, with an extension from
116:Yoshizawa–Randlett system
64:T. Sundara Row published
4883:The Bridges Organization
3616:Origami Polyhedra Design
3248:10.1260/0266-3511.27.1.1
3075:"The Origami Resolution"
2923:Gibbons, Jeremy (2003).
2838:: CS1 maint: location (
2531:Hiroshi Okumura (2014).
2455:10.1017/CBO9780511735172
2358:Lang, Robert J. (2004).
2214:Gibbons, Jeremy (2003).
1191:Trisecting the angle CAB
415:compass and straightedge
283:the context of origami.
274:In 2002, Alperin solved
193:Mountain-valley counting
96:showed that use of the '
5025:cyclic redundacy checks
4745:The Grammar of Ornament
4697:Nature's Harmonic Unity
4607:De prospectiva pingendi
3106:"Folding and Unfolding"
3051:"Computational Origami"
3012:10.1126/science.1252610
2855:"Computational Origami"
2669:"Folding Paper in Half"
2562:Lang, Robert J (2008).
2250:(inactive 2024-09-25).
2037:Oxford University Press
1695:Geometric constructions
1641:. Retrieved 2022-05-08.
1616:Encyclopedia Britannica
4898:Institute For Figuring
4810:The 'Life' of a Carpet
4635:A Treatise on Painting
3406:Napkin folding problem
3272:Paper Folding Geometry
3221:79(4): 272–280, 2006.
2061:Belcastro, Sarah-Marie
1925:"The power of origami"
1734:Nick Robinson (2004).
1521:Napkin folding problem
1431:
1326:
1228:napkin folding problem
1192:
1160:
1138:
831:
780:
737:
693:
652:
580:
488:
320:
319:Angles around a vertex
312:
265:
245:
229:affine transformations
194:
156:
129:
104:, allowed the general
89:
40:mathematical equations
27:
4779:George David Birkhoff
4753:Ernest Hanbury Hankin
4621:De divina proportione
4601:Piero della Francesca
4580:Leon Battista Alberti
4167:Piero della Francesca
3806:Hyperboloid structure
2925:"Origami Programming"
2360:"Angle Quintisection"
2216:"Origami Programming"
1873:Peter Messer (1986).
1678:Houdini's Paper Magic
1449:University of Tsukuba
1429:
1375:Computational origami
1327:
1190:
1158:
1139:
832:
781:
738:
694:
653:
617:Haga's first theorem
581:
486:
318:
310:
266:
246:
221:Sarah-Marie Belcastro
192:
154:
124:
87:
22:
5081:NP-complete problems
4704:Frederik Macody Lund
4575:Filippo Brunelleschi
4456:Hamid Naderi Yeganeh
4318:La condition humaine
3566:Fold-and-cut theorem
3522:Steffen's polyhedron
3386:Huzita–Hatori axioms
3376:Big-little-big lemma
3219:Mathematics Magazine
3143:Mathematics Magazine
2904:Smithsonian Magazine
2306:. pp. 383–406.
1830:on 25 November 2005.
1542:Notes and references
1422:Software & tools
1361:fold-and-cut problem
1252:
1235:developable surfaces
1082:
791:
748:
704:
663:
642:
532:
397:Huzita–Hatori axioms
391:Huzita–Justin axioms
361:Paper exhibits zero
255:
235:
102:Huzita–Hatori axioms
94:Margharita P. Beloch
5076:Mathematics and art
4929:Mathematical beauty
4854:Rhythm of Structure
4797:Gödel, Escher, Bach
4593:De re aedificatoria
4224:The Ancient of Days
3843:Projective geometry
3772:Mathematics and art
3514:Flexible polyhedron
3242:27(1): 1–14, 2012.
3004:2014Sci...345..644F
2541:Forum Geometricorum
2347:. Carleton College.
1883:Crux Mathematicorum
1785:"The Miura-Ori map"
1453:University of Tokyo
1391:Stanford University
1237:that are not flat.
1183:Trisecting an angle
1177:neusis construction
618:
326:NP-complete problem
5030:general relativity
4934:Patterns in nature
4791:Douglas Hofstadter
4417:Desmond Paul Henry
4407:Bathsheba Grossman
4339:The Swallow's Tail
4260:Giorgio de Chirico
4132:Sydney Opera House
3987:Croatian interlace
3695:Toshikazu Kawasaki
3518:Bricard octahedron
3493:Yoshimura buckling
3391:Kawasaki's theorem
2801:"Origami anything"
2146:10.1007/11615798_2
2107:. pp. 83–93.
2003:Weisstein, Eric W.
1808:Miura, K. (1985),
1783:Bain, Ian (1980),
1764:Hull, Tom (1997).
1533:(for example, the
1432:
1322:
1276:
1193:
1161:
1134:
827:
776:
733:
689:
648:
616:
576:
489:
363:Gaussian curvature
352:Kawasaki's theorem
321:
313:
261:
241:
195:
157:
130:
90:
56:History of origami
30:The discipline of
28:
5058:
5057:
4967:
4966:
4820:
4819:
4784:Aesthetic Measure
4655:Sebastiano Serlio
4629:Leonardo da Vinci
4519:
4518:
4511:Margaret Wertheim
4172:Leonardo da Vinci
3738:
3737:
3602:Geometric Origami
3473:Paper bag problem
3396:Maekawa's theorem
3232:Dureisseix, David
3211:Dureisseix, David
3203:978-1-56881-194-9
3181:978-981-283-490-4
3133:978-3-319-72486-7
2998:(6197): 644–646.
2737:"The Origami Lab"
2628:978-1-56881-714-9
2517:978-0-9555477-1-3
2507:Geometric Origami
2464:978-0-521-85757-4
2408:978-1-56881-181-9
2312:10.1201/b10653-38
2155:978-3-540-31332-8
1971:Hatori, Koshiro.
1842:Nishiyama, Yutaka
1745:978-1-84340-105-6
1736:The Origami Bible
1705:978-0-387-98276-2
1275:
1165:doubling the cube
1151:Doubling the cube
1129:
1103:
1068:
1067:
825:
774:
731:
687:
651:{\displaystyle x}
571:
420:Geometric Origami
411:doubling the cube
341:Maekawa's theorem
276:Alhazen's problem
264:{\displaystyle R}
244:{\displaystyle R}
5088:
4994:
4987:
4980:
4971:
4970:
4955:
4954:
4805:Nikos Salingaros
4528:
4527:
4496:Hiroshi Sugimoto
4446:Robert Longhurst
4392:Helaman Ferguson
4347:Crockett Johnson
4276:Circle Limit III
4245:Danseuse au café
4152:
4151:
4122:Pyramid of Khufu
3885:
3765:
3758:
3751:
3742:
3741:
3675:David A. Huffman
3640:Roger C. Alperin
3543:Source unfolding
3411:Pureland origami
3355:
3348:
3341:
3332:
3331:
3323:
3322:
3314:
3268:
3227:10.2307/27642951
3207:
3185:
3166:
3137:
3102:Demaine, Erik D.
3089:
3088:
3086:
3085:
3079:Damn Interesting
3071:
3065:
3064:
3062:
3061:
3046:
3040:
3039:
2983:
2977:
2976:
2974:
2973:
2959:
2953:
2952:
2950:
2949:
2943:"Airbag Folding"
2938:
2932:
2931:
2929:
2920:
2914:
2913:
2911:
2910:
2895:
2889:
2888:
2886:
2885:
2871:
2865:
2864:
2862:
2861:
2850:
2844:
2843:
2837:
2829:
2822:
2816:
2815:
2813:
2812:
2797:
2791:
2788:
2782:
2781:
2779:
2770:
2764:
2761:
2752:
2751:
2749:
2748:
2733:
2724:
2723:
2687:
2681:
2680:
2664:
2658:
2657:
2655:
2654:
2645:. Archived from
2639:
2633:
2632:
2614:
2608:
2607:
2605:
2603:
2577:
2571:
2570:
2568:
2559:
2550:
2549:
2537:
2528:
2522:
2521:
2501:
2492:
2491:
2483:
2477:
2476:
2441:O'Rourke, Joseph
2437:Demaine, Erik D.
2433:
2427:
2426:
2419:
2413:
2412:
2400:
2384:
2378:
2377:
2375:
2373:
2364:
2355:
2349:
2348:
2346:
2335:
2326:
2325:
2297:
2288:
2282:
2281:
2275:
2267:
2249:
2229:
2223:
2222:
2220:
2211:
2205:
2204:
2193:10.2307/30037438
2186:
2166:
2160:
2159:
2133:
2127:
2126:
2100:
2091:
2090:
2088:
2079:(1–3): 273–282.
2057:
2051:
2050:
2026:
2017:
2016:
2015:
1998:
1989:
1986:
1977:
1976:
1968:
1959:
1958:
1938:
1929:
1928:
1920:
1914:
1911:
1905:
1898:
1892:
1891:
1879:
1870:
1861:
1860:
1850:
1838:
1832:
1831:
1820:
1814:
1813:
1805:
1799:
1794:. Reproduced in
1793:
1780:
1774:
1773:
1761:
1750:
1749:
1731:
1725:
1724:
1716:
1710:
1709:
1689:
1683:
1682:
1669:
1663:
1662:
1648:
1642:
1635:
1626:
1625:
1623:
1622:
1608:
1602:
1601:
1567:
1555:
1331:
1329:
1328:
1323:
1312:
1311:
1290:
1289:
1277:
1271:
1263:
1207:Related problems
1196:Angle trisection
1143:
1141:
1140:
1135:
1130:
1128:
1120:
1109:
1104:
1102:
1094:
1086:
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777:
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769:
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752:
742:
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734:
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730:
719:
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695:
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688:
686:
675:
667:
657:
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619:
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583:
582:
577:
572:
570:
556:
545:
521:
520:
516:
511:
510:
506:
501:
500:
496:
450:silver rectangle
446:golden rectangle
311:Two-colorability
270:
268:
267:
262:
250:
248:
247:
242:
177:regular heptagon
60:In 1893, Indian
5096:
5095:
5091:
5090:
5089:
5087:
5086:
5085:
5061:
5060:
5059:
5054:
5001:
4998:
4968:
4963:
4943:
4939:Sacred geometry
4912:
4878:Ars Mathematica
4866:
4816:
4727:
4680:
4667:Andrea Palladio
4563:
4556:De architectura
4515:
4471:Antoine Pevsner
4451:Jeanette McLeod
4402:Susan Goldstine
4351:
4210:
4204:
4141:
4127:Sagrada FamĂlia
4088:
4030:
3898:Algorithmic art
3886:
3877:
3873:Wallpaper group
3811:Minimal surface
3774:
3769:
3739:
3734:
3720:Joseph O'Rourke
3655:Robert Connelly
3628:
3575:
3554:
3497:
3483:Schwarz lantern
3468:Modular origami
3451:
3420:
3364:
3359:
3329:
3317:
3309:
3256:
3204:
3190:Lang, Robert J.
3182:
3155:10.2307/2690924
3134:
3098:
3096:Further reading
3093:
3092:
3083:
3081:
3073:
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2777:
2771:
2767:
2762:
2755:
2746:
2744:
2735:
2734:
2727:
2688:
2684:
2673:At Right Angles
2665:
2661:
2652:
2650:
2641:
2640:
2636:
2629:
2615:
2611:
2601:
2599:
2585:Wayback Machine
2578:
2574:
2566:
2560:
2553:
2535:
2529:
2525:
2518:
2510:. UK: Arbelos.
2502:
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2465:
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2430:
2421:
2420:
2416:
2409:
2385:
2381:
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2369:
2367:langorigami.com
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2208:
2167:
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2156:
2134:
2130:
2123:
2101:
2094:
2065:Hull, Thomas C.
2058:
2054:
2047:
2027:
2020:
1999:
1992:
1987:
1980:
1969:
1962:
1939:
1932:
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1877:
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1839:
1835:
1822:
1821:
1817:
1806:
1802:
1796:British Origami
1781:
1777:
1762:
1753:
1746:
1732:
1728:
1721:Geometric Tools
1717:
1713:
1706:
1690:
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1670:
1666:
1649:
1645:
1636:
1629:
1620:
1618:
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1605:
1565:
1559:Hull, Thomas C.
1556:
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1507:
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1424:
1403:
1377:
1307:
1303:
1285:
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1249:
1211:The problem of
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947:
943:
942:
936:
932:
931:
925:
921:
920:
914:
910:
909:
903:
899:
898:
890:
886:
885:
879:
875:
874:
868:
864:
863:
857:
853:
852:
846:
842:
841:
814:
807:
803:
796:
794:
792:
789:
788:
764:
760:
753:
751:
749:
746:
745:
720:
709:
707:
705:
702:
701:
676:
668:
666:
664:
661:
660:
643:
640:
639:
613:. For example:
557:
546:
544:
533:
530:
529:
518:
514:
513:
508:
504:
503:
498:
494:
493:
481:
479:Haga's theorems
473:powers of three
465:Pierpont primes
429:
399:
393:
334:crease patterns
305:
300:
256:
253:
252:
236:
233:
232:
161:double the cube
88:The Beloch fold
58:
52:
17:
12:
11:
5:
5094:
5084:
5083:
5078:
5073:
5056:
5055:
5053:
5052:
5047:
5042:
5037:
5027:
5022:
5017:
5012:
5006:
5003:
5002:
4997:
4996:
4989:
4982:
4974:
4965:
4964:
4962:
4961:
4948:
4945:
4944:
4942:
4941:
4936:
4931:
4926:
4920:
4918:
4914:
4913:
4911:
4910:
4905:
4900:
4895:
4890:
4885:
4880:
4874:
4872:
4868:
4867:
4865:
4864:
4857:
4850:
4843:
4836:
4828:
4826:
4822:
4821:
4818:
4817:
4815:
4814:
4813:
4812:
4802:
4801:
4800:
4788:
4787:
4786:
4776:
4775:
4774:
4762:
4761:
4760:
4750:
4749:
4748:
4735:
4733:
4729:
4728:
4726:
4725:
4724:
4723:
4721:The Greek Vase
4713:
4712:
4711:
4701:
4700:
4699:
4688:
4686:
4682:
4681:
4679:
4678:
4677:
4676:
4664:
4663:
4662:
4652:
4651:
4650:
4643:Albrecht DĂĽrer
4640:
4639:
4638:
4626:
4625:
4624:
4612:
4611:
4610:
4598:
4597:
4596:
4589:
4577:
4571:
4569:
4565:
4564:
4562:
4561:
4560:
4559:
4547:
4546:
4545:
4534:
4532:
4525:
4521:
4520:
4517:
4516:
4514:
4513:
4508:
4506:Roman Verostko
4503:
4498:
4493:
4488:
4483:
4481:Alba Rojo Cama
4478:
4473:
4468:
4463:
4458:
4453:
4448:
4443:
4438:
4437:
4436:
4427:Charles Jencks
4424:
4419:
4414:
4412:George W. Hart
4409:
4404:
4399:
4394:
4389:
4384:
4379:
4374:
4365:
4359:
4357:
4353:
4352:
4350:
4349:
4344:
4343:
4342:
4335:
4323:
4322:
4321:
4309:
4308:
4307:
4300:
4293:
4286:
4279:
4267:
4262:
4257:
4256:
4255:
4248:
4239:Jean Metzinger
4236:
4235:
4234:
4227:
4214:
4212:
4206:
4205:
4203:
4202:
4201:
4200:
4188:
4186:Albrecht DĂĽrer
4183:
4182:
4181:
4169:
4164:
4158:
4156:
4149:
4143:
4142:
4140:
4139:
4134:
4129:
4124:
4119:
4114:
4109:
4104:
4098:
4096:
4090:
4089:
4087:
4086:
4079:
4072:
4065:
4058:
4051:
4044:
4038:
4036:
4032:
4031:
4029:
4028:
4023:
4018:
4013:
4012:
4011:
4001:
3996:
3995:
3994:
3989:
3984:
3974:
3973:
3972:
3967:
3962:
3957:
3947:
3942:
3937:
3932:
3927:
3926:
3925:
3920:
3915:
3905:
3903:Anamorphic art
3900:
3894:
3892:
3888:
3887:
3880:
3878:
3876:
3875:
3870:
3865:
3860:
3859:
3858:
3853:
3845:
3840:
3835:
3834:
3833:
3831:Camera obscura
3828:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3782:
3780:
3776:
3775:
3768:
3767:
3760:
3753:
3745:
3736:
3735:
3733:
3732:
3727:
3725:Tomohiro Tachi
3722:
3717:
3712:
3707:
3702:
3700:Robert J. Lang
3697:
3692:
3690:Humiaki Huzita
3687:
3682:
3677:
3672:
3670:Rona Gurkewitz
3667:
3665:Martin Demaine
3662:
3657:
3652:
3647:
3642:
3636:
3634:
3630:
3629:
3627:
3626:
3619:
3612:
3605:
3598:
3591:
3583:
3581:
3577:
3576:
3574:
3573:
3568:
3562:
3560:
3556:
3555:
3553:
3552:
3551:
3550:
3548:Star unfolding
3545:
3540:
3535:
3525:
3511:
3505:
3503:
3499:
3498:
3496:
3495:
3490:
3485:
3480:
3475:
3470:
3465:
3459:
3457:
3453:
3452:
3450:
3449:
3444:
3439:
3434:
3428:
3426:
3422:
3421:
3419:
3418:
3413:
3408:
3403:
3398:
3393:
3388:
3383:
3381:Crease pattern
3378:
3372:
3370:
3366:
3365:
3358:
3357:
3350:
3343:
3335:
3328:
3327:
3307:
3306:
3297:
3292:
3287:
3278:
3269:
3255:
3254:External links
3252:
3251:
3250:
3229:
3208:
3202:
3196:. A K Peters.
3186:
3180:
3167:
3149:(5): 357–371.
3138:
3132:
3109:
3097:
3094:
3091:
3090:
3066:
3041:
2978:
2954:
2933:
2915:
2890:
2866:
2845:
2817:
2807:. 22 June 2017
2792:
2783:
2773:Lang, Robert.
2765:
2753:
2741:The New Yorker
2725:
2682:
2659:
2634:
2627:
2609:
2572:
2551:
2523:
2516:
2493:
2478:
2463:
2428:
2414:
2407:
2379:
2350:
2327:
2320:
2283:
2240:(5): 679–702.
2224:
2206:
2177:(3): 200–211.
2161:
2154:
2128:
2121:
2113:10.1201/b15735
2092:
2052:
2045:
2039:. p. 22.
2018:
1990:
1978:
1960:
1930:
1915:
1906:
1893:
1875:"Problem 1054"
1862:
1833:
1815:
1800:
1775:
1770:origametry.net
1751:
1744:
1726:
1711:
1704:
1684:
1673:Houdini, Harry
1664:
1643:
1627:
1603:
1576:(4): 307–315.
1546:
1545:
1543:
1540:
1539:
1538:
1528:
1523:
1518:
1513:
1506:
1503:
1467:
1464:
1423:
1420:
1402:
1399:
1376:
1373:
1321:
1318:
1315:
1310:
1306:
1302:
1299:
1296:
1293:
1288:
1284:
1280:
1274:
1270:
1267:
1260:
1257:
1221:Miura map fold
1208:
1205:
1184:
1181:
1152:
1149:
1145:
1144:
1133:
1127:
1124:
1119:
1116:
1113:
1107:
1101:
1098:
1093:
1090:
1072:
1069:
1066:
1065:
1054:
1043:
1032:
1021:
1009:
1008:
997:
986:
975:
964:
952:
951:
940:
929:
918:
907:
895:
894:
883:
872:
861:
850:
838:
837:
823:
820:
817:
810:
806:
802:
799:
786:
773:
767:
763:
759:
756:
743:
729:
726:
723:
718:
715:
712:
699:
685:
682:
679:
674:
671:
658:
647:
636:
635:
632:
629:
626:
623:
587:
586:
575:
569:
566:
563:
560:
555:
552:
549:
543:
540:
537:
480:
477:
458:if and only if
428:
425:
395:Main article:
392:
389:
359:
358:
355:
349:
348:
347:
304:
301:
299:
296:
260:
240:
184:Robert J. Lang
172:Humiaki Huzita
168:Jacques Justin
135:Also in 1980,
126:Crease pattern
106:cubic equation
51:
48:
15:
9:
6:
4:
3:
2:
5093:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5066:
5051:
5048:
5046:
5043:
5041:
5040:paper folding
5038:
5035:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5010:apportionment
5008:
5007:
5004:
4995:
4990:
4988:
4983:
4981:
4976:
4975:
4972:
4960:
4959:
4950:
4949:
4946:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4924:Droste effect
4922:
4921:
4919:
4915:
4909:
4906:
4904:
4903:Mathemalchemy
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4875:
4873:
4871:Organizations
4869:
4863:
4862:
4858:
4856:
4855:
4851:
4849:
4848:
4844:
4842:
4841:
4840:Lumen Naturae
4837:
4835:
4834:
4830:
4829:
4827:
4823:
4811:
4808:
4807:
4806:
4803:
4799:
4798:
4794:
4793:
4792:
4789:
4785:
4782:
4781:
4780:
4777:
4773:
4772:
4768:
4767:
4766:
4763:
4759:
4756:
4755:
4754:
4751:
4747:
4746:
4742:
4741:
4740:
4737:
4736:
4734:
4730:
4722:
4719:
4718:
4717:
4714:
4710:
4707:
4706:
4705:
4702:
4698:
4695:
4694:
4693:
4692:Samuel Colman
4690:
4689:
4687:
4683:
4675:
4674:
4670:
4669:
4668:
4665:
4661:
4658:
4657:
4656:
4653:
4649:
4646:
4645:
4644:
4641:
4637:
4636:
4632:
4631:
4630:
4627:
4623:
4622:
4618:
4617:
4616:
4613:
4609:
4608:
4604:
4603:
4602:
4599:
4595:
4594:
4590:
4588:
4587:
4583:
4582:
4581:
4578:
4576:
4573:
4572:
4570:
4566:
4558:
4557:
4553:
4552:
4551:
4548:
4544:
4541:
4540:
4539:
4536:
4535:
4533:
4529:
4526:
4522:
4512:
4509:
4507:
4504:
4502:
4501:Daina Taimiņa
4499:
4497:
4494:
4492:
4489:
4487:
4486:Reza Sarhangi
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4435:
4434:
4430:
4429:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4397:Peter Forakis
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4369:
4366:
4364:
4361:
4360:
4358:
4354:
4348:
4345:
4341:
4340:
4336:
4334:
4333:
4329:
4328:
4327:
4326:Salvador DalĂ
4324:
4320:
4319:
4315:
4314:
4313:
4312:René Magritte
4310:
4306:
4305:
4301:
4299:
4298:
4294:
4292:
4291:
4287:
4285:
4284:
4283:Print Gallery
4280:
4278:
4277:
4273:
4272:
4271:
4268:
4266:
4263:
4261:
4258:
4254:
4253:
4252:L'Oiseau bleu
4249:
4247:
4246:
4242:
4241:
4240:
4237:
4233:
4232:
4228:
4226:
4225:
4221:
4220:
4219:
4218:William Blake
4216:
4215:
4213:
4207:
4199:
4198:
4194:
4193:
4192:
4189:
4187:
4184:
4180:
4179:
4178:Vitruvian Man
4175:
4174:
4173:
4170:
4168:
4165:
4163:
4162:Paolo Uccello
4160:
4159:
4157:
4153:
4150:
4148:
4144:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4099:
4097:
4095:
4091:
4085:
4084:
4083:Pi in the Sky
4080:
4078:
4077:
4073:
4071:
4070:
4066:
4064:
4063:
4059:
4057:
4056:
4055:Mathemalchemy
4052:
4050:
4049:
4045:
4043:
4040:
4039:
4037:
4033:
4027:
4024:
4022:
4019:
4017:
4014:
4010:
4007:
4006:
4005:
4002:
4000:
3997:
3993:
3990:
3988:
3985:
3983:
3980:
3979:
3978:
3975:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3952:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3924:
3923:Vastu shastra
3921:
3919:
3916:
3914:
3913:Geodesic dome
3911:
3910:
3909:
3906:
3904:
3901:
3899:
3896:
3895:
3893:
3889:
3884:
3874:
3871:
3869:
3866:
3864:
3861:
3857:
3854:
3852:
3849:
3848:
3846:
3844:
3841:
3839:
3838:Plastic ratio
3836:
3832:
3829:
3827:
3826:Camera lucida
3824:
3823:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3783:
3781:
3777:
3773:
3766:
3761:
3759:
3754:
3752:
3747:
3746:
3743:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3637:
3635:
3631:
3625:
3624:
3620:
3618:
3617:
3613:
3611:
3610:
3606:
3604:
3603:
3599:
3597:
3596:
3592:
3590:
3589:
3585:
3584:
3582:
3578:
3572:
3571:Lill's method
3569:
3567:
3564:
3563:
3561:
3559:Miscellaneous
3557:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3530:
3529:
3526:
3523:
3519:
3515:
3512:
3510:
3507:
3506:
3504:
3500:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3478:Rigid origami
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3460:
3458:
3456:3d structures
3454:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3429:
3427:
3425:Strip folding
3423:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3373:
3371:
3367:
3363:
3356:
3351:
3349:
3344:
3342:
3337:
3336:
3333:
3326:
3321:
3316:
3315:
3312:
3305:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3282:
3279:
3277:
3273:
3270:
3266:
3262:
3258:
3257:
3249:
3245:
3241:
3237:
3233:
3230:
3228:
3224:
3220:
3216:
3212:
3209:
3205:
3199:
3195:
3191:
3187:
3183:
3177:
3173:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3139:
3135:
3129:
3125:
3121:
3117:
3116:
3110:
3107:
3103:
3100:
3099:
3080:
3076:
3070:
3056:
3055:Computerworld
3052:
3045:
3037:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2997:
2993:
2989:
2982:
2968:
2967:webb.nasa.gov
2964:
2958:
2944:
2937:
2926:
2919:
2905:
2901:
2894:
2880:
2876:
2870:
2856:
2849:
2841:
2835:
2827:
2821:
2806:
2802:
2796:
2787:
2776:
2769:
2760:
2758:
2742:
2738:
2732:
2730:
2721:
2717:
2713:
2709:
2705:
2701:
2697:
2693:
2692:Math Horizons
2686:
2678:
2674:
2670:
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4716:Jay Hambidge
4709:Ad Quadratum
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4615:Luca Pacioli
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4461:István Orosz
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4422:Anthony Hill
4377:Scott Draves
4372:Erik Demaine
4356:Contemporary
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3930:Computer art
3908:Architecture
3868:Tessellation
3851:Architecture
3801:Golden ratio
3730:Eve Torrence
3660:Erik Demaine
3621:
3614:
3607:
3600:
3593:
3586:
3580:Publications
3442:Möbius strip
3432:Dragon curve
3369:Flat folding
3361:
3304:Mario Cigada
3285:cut-the-knot
3276:cut-the-knot
3261:Dr. Tom Hull
3239:
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3082:. Retrieved
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3058:. Retrieved
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2809:. Retrieved
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2743:. 2007-02-12
2740:
2698:(3): 22–24.
2695:
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2662:
2651:. Retrieved
2647:the original
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2600:. Retrieved
2594:
2581:Ghostarchive
2579:Archived at
2575:
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2370:. Retrieved
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2184:math/0408159
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1946:
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1859:(2): 269–279
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1619:. Retrieved
1615:
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1535:dragon curve
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1466:Applications
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1441:Georgia Tech
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303:Flat folding
298:Pure origami
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165:
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134:
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76:
72:kindergarten
65:
59:
44:
36:mathematical
29:
4765:G. H. Hardy
4568:Renaissance
4538:Polykleitos
4476:Tony Robbin
4387:John Ernest
4382:Jan Dibbets
4332:Crucifixion
4155:Renaissance
4009:Mathematics
3982:Celtic knot
3945:Fractal art
3847:Proportion
3821:Perspective
3715:KĹŤryĹŤ Miura
3710:Jun Maekawa
3685:KĂ´di Husimi
3401:Map folding
3325:Mathematics
1526:Map folding
1495:RNA origami
1491:DNA origami
1460:DNA origami
1387:Robert Lang
1239:Wet-folding
1217:sheet metal
379:NP-complete
375:Barry Hayes
330:map folding
207:NP-complete
203:Barry Hayes
137:KĹŤryĹŤ Miura
98:Beloch fold
24:Map folding
5065:Categories
5020:bookmaking
4739:Owen Jones
4586:De pictura
4491:Oliver Sin
4441:Andy Lomas
4290:Relativity
4021:String art
3935:Fiber arts
3816:Paraboloid
3705:Anna Lubiw
3538:Common net
3463:Miura fold
3084:2022-05-08
3060:2022-05-08
2972:2022-05-08
2948:2022-05-08
2909:2022-05-08
2884:2022-05-08
2860:2022-05-08
2811:2022-05-08
2747:2022-05-09
2653:2008-10-08
2602:October 2,
2548:: 241–242.
2486:Tom Hull.
2372:16 January
1824:"2D Array"
1621:2022-05-08
1369:Miura fold
1354:California
54:See also:
4550:Vitruvius
4524:Theorists
4304:Waterfall
4209:19th–20th
4137:Taj Mahal
4117:Parthenon
4094:Buildings
4048:Continuum
4016:Sculpture
3992:Interlace
3786:Algorithm
3623:Origamics
3502:Polyhedra
3020:0036-8075
2875:"Cadnano"
2720:126397750
2619:Origami 5
2398:1307.1065
2300:Origami 4
2256:1469-7653
2011:MathWorld
2006:"Folding"
1314:−
1266:π
1201:congruent
758:−
714:−
512:and then
438:pentagons
434:triangles
405:— namely
288:Countdown
219:In 2002,
197:In 1996,
142:Miura-ori
77:In 1922,
4958:Category
4685:Romantic
4363:Max Bill
4297:Reptiles
4112:Pantheon
4069:Octacube
4035:Artworks
3977:Knotting
3965:Muqarnas
3863:Symmetry
3791:Catenary
3779:Concepts
3680:Tom Hull
3650:Yan Chen
3533:Blooming
3437:Flexagon
3192:(2003).
3036:18415193
3028:25104380
2834:cite web
2712:25678354
2583:and the
2443:(2007).
2264:46359986
2201:30037438
2067:(2002).
1844:(2012),
1561:(2011).
1511:Flexagon
1505:See also
1401:Research
1393:and the
1382:Bug Wars
1332:, where
448:and the
442:hexagons
225:Tom Hull
92:In 1936
5071:Origami
4917:Related
4531:Ancient
4265:Man Ray
4211:Century
4147:Artists
4004:Origami
3918:Pyramid
3796:Fractal
3163:2690924
3000:Bibcode
2992:Science
2941:TASON.
2879:cadnano
2853:TASON.
2596:YouTube
2473:2354878
1955:1381938
1598:2540978
1590:2800341
1060:⁄
1049:⁄
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902:⁄
889:⁄
878:⁄
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856:⁄
845:⁄
517:⁄
507:⁄
497:⁄
50:History
32:origami
5045:Sudoku
4732:Modern
4368:Martin
4231:Newton
4026:Tiling
3970:Zellij
3940:4D art
3633:People
3488:Sonobe
3311:Portal
3200:
3178:
3161:
3130:
3034:
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1588:
1451:, and
1410:, and
601:. Let
471:, and
377:to be
4543:Canon
3999:Music
3955:Girih
3891:Forms
3856:Human
3159:JSTOR
3032:S2CID
2928:(PDF)
2778:(PDF)
2716:S2CID
2708:JSTOR
2567:(PDF)
2536:(PDF)
2393:arXiv
2363:(PDF)
2345:(PDF)
2296:(PDF)
2260:S2CID
2219:(PDF)
2197:JSTOR
2179:arXiv
1878:(PDF)
1849:(PDF)
1594:S2CID
1566:(PDF)
401:Some
4370:and
3960:Jali
3198:ISBN
3176:ISBN
3128:ISBN
3024:PMID
3016:ISSN
2840:link
2623:ISBN
2604:2021
2512:ISBN
2459:ISBN
2403:ISBN
2374:2021
2316:ISBN
2278:link
2252:ISSN
2150:ISBN
2117:ISBN
2041:ISBN
1740:ISBN
1700:ISBN
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1359:The
1348:and
1226:The
373:and
223:and
201:and
114:The
3528:Net
3302:by
3283:at
3274:at
3244:doi
3223:doi
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3120:doi
3008:doi
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2308:doi
2242:doi
2189:doi
2175:112
2142:doi
2109:doi
2081:doi
2077:348
1578:doi
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634:PQ
605:be
597:is
593:to
409:or
251:to
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4076:Pi
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