1425:. 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
1367:, 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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1408:, 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.
301:, 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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463:. Methods for folding most regular polygons up to and including the regular 19-gon have been developed. A regular
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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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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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2080:"Modelling the folding of paper into three dimensions using affine transformations"
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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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354:: 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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1508:, folding of manufacturing instruments, and surgery by tiny origami robots.
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A diagram showing the first and last step of how origami can double the cube
3109:, PhD thesis, Department of Computer Science, University of Waterloo, 2001.
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1953:
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1777:"a comparison between straight edge and compass constructions and origami"
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1226:, treating the folds as hinges joining two flat, rigid surfaces, such as
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2565:"From Flapping Birds to Space Telescopes: The Modern Science of Origami"
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392:. Further references and technical results are discussed in Part II of
42:
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2764:
Demaine, Erik (2001). "Recent Results in Computational Origami" (PDF).
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3227:
2987:
Felton, S.; Tolley, M.; Demaine, E.; Rus, D.; Wood, R. (2014-08-08).
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2151:. Lecture Notes in Computer Science. Vol. 3763. pp. 19–33.
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is a solution to the problem, and several others have been proposed.
1158:
Therefore, BQ:CQ=k:1 implies AP:BP=k:2 for a positive real number k.
3155:
2388:
Thomas C. Hull (2002). "The Combinatorics of Flat Folds: a Survey".
1823:, Tech. Report 618, The Institute of Space and Astronautical Science
2123:
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3119:. Science Networks. Historical Studies. Vol. 59. Birkhäuser.
2398:
2180:
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.
161:
94:
29:
1999:
K. Haga, Origamics, Part 1, Nippon Hyoron Sha, 1999 (in Japanese)
4862:
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
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3171:
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
4974:
3745:
3335:
3195:
Origami Design Secrets: Mathematical Methods for an Ancient Art
2829:. Julia code animating kabuto is in example 3.4. 31 March 2024.
2116:
Mathematical Origami: Another View of Alhazen's Optical Problem
620:
then a number of other lengths are also rational functions of
3304:
3116:
A History of Folding in Mathematics: Mathematizing the Margins
1821:
Method of packaging and deployment of large membranes in space
339:. Related problems when the creases are orthogonal are called
3175:. University of Tsukuba, Japan: World Scientific Publishing.
2875:
1662:
T. Sundara Row (1917). Beman, Wooster; Smith, David (eds.).
1809:, 1981, and online at the British Origami Society web site.
170:
In 1986, Messer reported a construction by which one could
3173:
Origamics: Mathematical Explorations Through Paper Folding
2691:
Hull, Thomas (2002). "In search of a practical map fold".
2114:
Alperin, Roger C. (2002). "Ch.12". In Hull, Thomas (ed.).
1574:"Solving cubics with creases: the work of Beloch and Lill"
1901:(10): 284–285 – via Canadian Mathematical Society.
1347:
is the minimum length of the paper (or other material),
1336:{\displaystyle L={\tfrac {\pi t}{6}}(2^{n}+4)(2^{n}-1)}
1186:. Using a marked straightedge in this way is called a
45:
or paper folding has received a considerable amount of
1857:"Miura folding: Applying origami to space exploration"
1273:
3309:
3291:
Britney Gallivan has solved the Paper Folding Problem
3237:"An Overview of Mechanisms and Patterns with Origami"
1864:
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
1837:. Japan Aerospace Exploration Agency. Archived from
1648:"Lecture: Recent Results in Computational Origami".
174:, which is impossible with Euclidean constructions.
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2592:"Numberphile: How to Trisect an Angle with Origami"
1230:, has great practical importance. For example, the
1148:{\displaystyle {\frac {BQ}{CQ}}={\frac {2AP}{BP}}.}
626:
238:brought to the theoretical origami the language of
4759:The Drawing of Geometric Patterns in Saracenic Art
2901:"How Origami Is Revolutionizing Industrial Design"
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129:of instruction by diagram was introduced in 1961.
4894:Goudreau Museum of Mathematics in Art and Science
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2806:MIT News | Massachusetts Institute of Technology
2245:"Countdown: A case study in origami programming"
2243:Bird, Richard; Mu, Shin-Cheng (September 2005).
1355:is the number of folds possible. The distances
282:in only the case of single-vertex construction.
2776:"A Computational Algorithm for Origami Design"
2500:
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4889:European Society for Mathematics and the Arts
4063:Mathematica: A World of Numbers... and Beyond
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2989:"A method for building self-folding machines"
1086:Haga's theorems are generalized as follows:
8:
4043:List of works designed with the golden ratio
2534:"A Note on Haga's theorems in paper folding"
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2107:
2007:
2005:
27:Overview of the mathematics of paper folding
2558:
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2294:"One-, Two-, and Multi-Fold Origami Axioms"
2292:Lang, Robert J.; Alperin, Roger C. (2009).
1938:. University of Cambridge. + plus magazine.
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1623:"origami - History of origami | Britannica"
414:classical construction problems of geometry
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4103:Cathedral of Saint Mary of the Assumption
3241:International Journal of Space Structures
2450:. Cambridge: Cambridge University Press.
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1264:
1170:Doubling the cube: PB/PA = cube root of 2
1119:
1096:
1094:
818:
805:
803:
775:
762:
760:
718:
716:
677:
675:
654:
555:
544:
467:-gon can be constructed by paper folding
267:
247:
2964:"Webb and Origami - Webb Telescope/NASA"
2827:"Julia and Projective Geometric Algebra"
1984:"How to Divide the Side of Square Paper"
1977:
1975:
1564:
1562:
1197:
1165:
493:
325:
317:
199:
131:
4649:Vier BĂĽcher von Menschlicher Proportion
3316:
3301:Introduction to Statistics with Origami
3216:"Folding optimal polygons from squares"
2590:Dancso, Zsuzsanna (December 12, 2014).
2334:
2332:
1558:
590:{\displaystyle BQ={\frac {2AP}{1+AP}}.}
2899:Magazine, Smithsonian; Morrison, Jim.
2832:
1487:Lawrence Livermore National Laboratory
1184:compass and straightedge constructions
841:{\displaystyle {\frac {1+x^{2}}{1+x}}}
2760:
2758:
2732:
2730:
2489:"Origami and Geometric Constructions"
1952:Bern, Marshall; Hayes, Barry (1996).
1448:Massachusetts Institute of Technology
455:, and special rectangles such as the
7:
3589:Geometric Exercises in Paper Folding
2668:Korpal, Gaurish (25 November 2015).
2043:How to Free Your Inner Mathematician
1915:, H. Huzita ed. (1989), pp. 251–261.
1665:Geometric Exercises in Paper Folding
1644:
1642:
1593:10.4169/amer.math.monthly.118.04.307
1512:figures from cheeseburger wrappers.
1423:computational intractability results
790:{\displaystyle {\frac {1-x^{2}}{2}}}
424:— are proven to be unsolvable using
78:Geometric Exercises in Paper Folding
4834:Journal of Mathematics and the Arts
3610:A History of Folding in Mathematics
2424:"Robert Lang folds way-new origami"
2084:Linear Algebra and Its Applications
1960:. ACM, New York. pp. 175–183.
1082:A generalization of Haga's theorems
368:A sheet can never penetrate a fold.
4848:Making Mathematics with Needlework
1406:California Institute of Technology
747:{\displaystyle {\frac {1-x}{1+x}}}
111:', later used in the sixth of the
25:
4674:I quattro libri dell'architettura
2249:Journal of Functional Programming
2182:The American Mathematical Monthly
1986:. Japan Origami Academic Society.
1670:The Open Court Publishing Company
1351:is the material's thickness, and
703:{\displaystyle {\frac {2x}{1+x}}}
600:The function changing the length
4953:
4952:
4198:Self-portrait in a Convex Mirror
3882:
3319:
2570:. Usenix Conference, Boston, MA.
1954:"The complexity of flat origami"
3510:Alexandrov's uniqueness theorem
2149:Automated Deduction in Geometry
1934:Newton, Liz (1 December 2009).
1749:. Chrysalis Books. p. 18.
1456:University of California Irvine
498:BQ is always rational if AP is.
4909:National Museum of Mathematics
4661:Regole generali d'architettura
2705:10.1080/10724117.2002.11975147
2680:(3). Teachers of India: 20–23.
2505:Geretschläger, Robert (2008).
1489:to develop a solution for the
1330:
1311:
1308:
1289:
1:
3448:Regular paperfolding sequence
2097:10.1016/S0024-3795(01)00608-5
1734:. Louisiana State University.
1703:George Edward Martin (1997).
1581:American Mathematical Monthly
1542:Regular paperfolding sequence
1419:efficient decision algorithms
418:trisecting an arbitrary angle
4434:Garden of Cosmic Speculation
3596:Geometric Folding Algorithms
3363:Mathematics of paper folding
2644:"Siggraph: "Curved Origami""
2447:Geometric folding algorithms
1400:, a research-scientist from
443:square, such as equilateral
395:Geometric Folding Algorithms
119:to be solved using origami.
1730:Robert Carl Yeates (1949).
1500:Other applications include
5098:
5051:three-phase electric power
5016:artificial neural networks
3951:Islamic geometric patterns
3646:Margherita Piazzola Beloch
3296:Overview of Origami Axioms
3266:"Origami Mathematics Page"
3113:Friedman, Michael (2018).
3050:Brewin, Bob (2004-05-10).
2622:. CRC Press. p. 225.
2342:The Mathematics of Origami
2040:D'Agostino, Susan (2020).
1491:James Webb Space Telescope
405:
157:solar panels on spacecraft
64:
5006:
5001:"Mathematics of" articles
4948:
4772:A Mathematician's Apology
3880:
3417:Yoshizawa–Randlett system
3125:10.1007/978-3-319-72487-4
2262:10.1017/S0956796805005642
1709:. Springer. p. 145.
1174:The classical problem of
474:is a product of distinct
242:, with an extension from
127:Yoshizawa–Randlett system
75:T. Sundara Row published
4884:The Bridges Organization
3617:Origami Polyhedra Design
3249:10.1260/0266-3511.27.1.1
3076:"The Origami Resolution"
2924:Gibbons, Jeremy (2003).
2839:: CS1 maint: location (
2532:Hiroshi Okumura (2014).
2456:10.1017/CBO9780511735172
2359:Lang, Robert J. (2004).
2225:Gibbons, Jeremy (2003).
1202:Trisecting the angle CAB
1081:
489:
426:compass and straightedge
294:the context of origami.
285:In 2002, Alperin solved
204:Mountain-valley counting
107:showed that use of the '
5026:cyclic redundacy checks
4746:The Grammar of Ornament
4698:Nature's Harmonic Unity
4608:De prospectiva pingendi
3107:"Folding and Unfolding"
3052:"Computational Origami"
3013:10.1126/science.1252610
2856:"Computational Origami"
2670:"Folding Paper in Half"
2563:Lang, Robert J (2008).
2048:Oxford University Press
1706:Geometric constructions
1652:. Retrieved 2022-05-08.
1627:Encyclopedia Britannica
4899:Institute For Figuring
4811:The 'Life' of a Carpet
4636:A Treatise on Painting
3407:Napkin folding problem
3273:Paper Folding Geometry
3222:79(4): 272–280, 2006.
2072:Belcastro, Sarah-Marie
1936:"The power of origami"
1745:Nick Robinson (2004).
1532:Napkin folding problem
1442:
1337:
1239:napkin folding problem
1203:
1171:
1149:
842:
791:
748:
704:
663:
591:
499:
331:
330:Angles around a vertex
323:
276:
256:
240:affine transformations
205:
167:
140:
115:, allowed the general
100:
51:mathematical equations
38:
4780:George David Birkhoff
4754:Ernest Hanbury Hankin
4622:De divina proportione
4602:Piero della Francesca
4581:Leon Battista Alberti
4168:Piero della Francesca
3807:Hyperboloid structure
2926:"Origami Programming"
2361:"Angle Quintisection"
2227:"Origami Programming"
1884:Peter Messer (1986).
1689:Houdini's Paper Magic
1460:University of Tsukuba
1440:
1386:Computational origami
1338:
1201:
1169:
1150:
843:
792:
749:
705:
664:
628:Haga's first theorem
592:
497:
329:
321:
277:
257:
232:Sarah-Marie Belcastro
203:
165:
135:
98:
33:
5082:NP-complete problems
4705:Frederik Macody Lund
4576:Filippo Brunelleschi
4457:Hamid Naderi Yeganeh
4319:La condition humaine
3567:Fold-and-cut theorem
3523:Steffen's polyhedron
3387:Huzita–Hatori axioms
3377:Big-little-big lemma
3220:Mathematics Magazine
3144:Mathematics Magazine
2905:Smithsonian Magazine
2307:. pp. 383–406.
1841:on 25 November 2005.
1553:Notes and references
1433:Software & tools
1372:fold-and-cut problem
1263:
1246:developable surfaces
1093:
802:
759:
715:
674:
653:
543:
408:Huzita–Hatori axioms
402:Huzita–Justin axioms
372:Paper exhibits zero
266:
246:
113:Huzita–Hatori axioms
105:Margharita P. Beloch
5077:Mathematics and art
4930:Mathematical beauty
4855:Rhythm of Structure
4798:Gödel, Escher, Bach
4594:De re aedificatoria
4225:The Ancient of Days
3844:Projective geometry
3773:Mathematics and art
3515:Flexible polyhedron
3243:27(1): 1–14, 2012.
3005:2014Sci...345..644F
2542:Forum Geometricorum
2348:. Carleton College.
1894:Crux Mathematicorum
1796:"The Miura-Ori map"
1464:University of Tokyo
1402:Stanford University
1248:that are not flat.
1194:Trisecting an angle
1188:neusis construction
629:
337:NP-complete problem
5031:general relativity
4935:Patterns in nature
4792:Douglas Hofstadter
4418:Desmond Paul Henry
4408:Bathsheba Grossman
4340:The Swallow's Tail
4261:Giorgio de Chirico
4133:Sydney Opera House
3988:Croatian interlace
3696:Toshikazu Kawasaki
3519:Bricard octahedron
3494:Yoshimura buckling
3392:Kawasaki's theorem
2802:"Origami anything"
2157:10.1007/11615798_2
2118:. pp. 83–93.
2014:Weisstein, Eric W.
1819:Miura, K. (1985),
1794:Bain, Ian (1980),
1775:Hull, Tom (1997).
1544:(for example, the
1443:
1333:
1287:
1204:
1172:
1145:
838:
787:
744:
700:
659:
627:
587:
500:
374:Gaussian curvature
363:Kawasaki's theorem
332:
324:
272:
252:
206:
168:
141:
101:
67:History of origami
41:The discipline of
39:
5059:
5058:
4968:
4967:
4821:
4820:
4785:Aesthetic Measure
4656:Sebastiano Serlio
4630:Leonardo da Vinci
4520:
4519:
4512:Margaret Wertheim
4173:Leonardo da Vinci
3739:
3738:
3603:Geometric Origami
3474:Paper bag problem
3397:Maekawa's theorem
3233:Dureisseix, David
3212:Dureisseix, David
3204:978-1-56881-194-9
3182:978-981-283-490-4
3134:978-3-319-72486-7
2999:(6197): 644–646.
2738:"The Origami Lab"
2629:978-1-56881-714-9
2518:978-0-9555477-1-3
2508:Geometric Origami
2465:978-0-521-85757-4
2409:978-1-56881-181-9
2313:10.1201/b10653-38
2166:978-3-540-31332-8
1982:Hatori, Koshiro.
1853:Nishiyama, Yutaka
1756:978-1-84340-105-6
1747:The Origami Bible
1716:978-0-387-98276-2
1286:
1176:doubling the cube
1162:Doubling the cube
1140:
1114:
1079:
1078:
836:
785:
742:
698:
662:{\displaystyle x}
582:
431:Geometric Origami
422:doubling the cube
352:Maekawa's theorem
287:Alhazen's problem
275:{\displaystyle R}
255:{\displaystyle R}
16:(Redirected from
5089:
4995:
4988:
4981:
4972:
4956:
4955:
4806:Nikos Salingaros
4529:
4497:Hiroshi Sugimoto
4447:Robert Longhurst
4393:Helaman Ferguson
4348:Crockett Johnson
4277:Circle Limit III
4246:Danseuse au café
4153:
4123:Pyramid of Khufu
3886:
3766:
3759:
3752:
3743:
3676:David A. Huffman
3641:Roger C. Alperin
3544:Source unfolding
3412:Pureland origami
3356:
3349:
3342:
3333:
3324:
3323:
3315:
3269:
3228:10.2307/27642951
3208:
3186:
3167:
3138:
3103:Demaine, Erik D.
3090:
3089:
3087:
3086:
3080:Damn Interesting
3072:
3066:
3065:
3063:
3062:
3047:
3041:
3040:
2984:
2978:
2977:
2975:
2974:
2960:
2954:
2953:
2951:
2950:
2944:"Airbag Folding"
2939:
2933:
2932:
2930:
2921:
2915:
2914:
2912:
2911:
2896:
2890:
2889:
2887:
2886:
2872:
2866:
2865:
2863:
2862:
2851:
2845:
2844:
2838:
2830:
2823:
2817:
2816:
2814:
2813:
2798:
2792:
2789:
2783:
2782:
2780:
2771:
2765:
2762:
2753:
2752:
2750:
2749:
2734:
2725:
2724:
2688:
2682:
2681:
2665:
2659:
2658:
2656:
2655:
2646:. Archived from
2640:
2634:
2633:
2615:
2609:
2608:
2606:
2604:
2578:
2572:
2571:
2569:
2560:
2551:
2550:
2538:
2529:
2523:
2522:
2502:
2493:
2492:
2484:
2478:
2477:
2442:O'Rourke, Joseph
2438:Demaine, Erik D.
2434:
2428:
2427:
2420:
2414:
2413:
2401:
2385:
2379:
2378:
2376:
2374:
2365:
2356:
2350:
2349:
2347:
2336:
2327:
2326:
2298:
2289:
2283:
2282:
2264:
2240:
2234:
2233:
2231:
2222:
2216:
2215:
2204:10.2307/30037438
2197:
2177:
2171:
2170:
2144:
2138:
2137:
2111:
2102:
2101:
2099:
2090:(1–3): 273–282.
2068:
2062:
2061:
2037:
2028:
2027:
2026:
2009:
2000:
1997:
1988:
1987:
1979:
1970:
1969:
1949:
1940:
1939:
1931:
1925:
1922:
1916:
1909:
1903:
1902:
1890:
1881:
1872:
1871:
1861:
1849:
1843:
1842:
1831:
1825:
1824:
1816:
1810:
1805:. Reproduced in
1804:
1791:
1785:
1784:
1772:
1761:
1760:
1742:
1736:
1735:
1727:
1721:
1720:
1700:
1694:
1693:
1680:
1674:
1673:
1659:
1653:
1646:
1637:
1636:
1634:
1633:
1619:
1613:
1612:
1578:
1566:
1342:
1340:
1339:
1334:
1323:
1322:
1301:
1300:
1288:
1282:
1274:
1218:Related problems
1207:Angle trisection
1154:
1152:
1151:
1146:
1141:
1139:
1131:
1120:
1115:
1113:
1105:
1097:
1075:
1074:
1070:
1064:
1063:
1059:
1053:
1052:
1048:
1042:
1041:
1037:
1031:
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1026:
1018:
1017:
1013:
1007:
1006:
1002:
996:
995:
991:
985:
984:
980:
974:
973:
969:
961:
960:
956:
950:
949:
945:
939:
938:
934:
928:
927:
923:
917:
916:
912:
904:
903:
899:
893:
892:
888:
882:
881:
877:
871:
870:
866:
860:
859:
855:
847:
845:
844:
839:
837:
835:
824:
823:
822:
806:
796:
794:
793:
788:
786:
781:
780:
779:
763:
753:
751:
750:
745:
743:
741:
730:
719:
709:
707:
706:
701:
699:
697:
686:
678:
668:
666:
665:
660:
630:
596:
594:
593:
588:
583:
581:
567:
556:
532:
531:
527:
522:
521:
517:
512:
511:
507:
461:silver rectangle
457:golden rectangle
322:Two-colorability
281:
279:
278:
273:
261:
259:
258:
253:
188:regular heptagon
71:In 1893, Indian
21:
18:Flat-foldability
5097:
5096:
5092:
5091:
5090:
5088:
5087:
5086:
5062:
5061:
5060:
5055:
5002:
4999:
4969:
4964:
4944:
4940:Sacred geometry
4913:
4879:Ars Mathematica
4867:
4817:
4728:
4681:
4668:Andrea Palladio
4564:
4557:De architectura
4516:
4472:Antoine Pevsner
4452:Jeanette McLeod
4403:Susan Goldstine
4352:
4211:
4205:
4142:
4128:Sagrada FamĂlia
4089:
4031:
3899:Algorithmic art
3887:
3878:
3874:Wallpaper group
3812:Minimal surface
3775:
3770:
3740:
3735:
3721:Joseph O'Rourke
3656:Robert Connelly
3629:
3576:
3555:
3498:
3484:Schwarz lantern
3469:Modular origami
3452:
3421:
3365:
3360:
3330:
3318:
3310:
3260:
3257:
3205:
3191:Lang, Robert J.
3189:
3183:
3170:
3156:10.2307/2690924
3141:
3135:
3112:
3099:
3097:Further reading
3094:
3093:
3084:
3082:
3074:
3073:
3069:
3060:
3058:
3049:
3048:
3044:
2986:
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2800:
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2790:
2786:
2778:
2773:
2772:
2768:
2763:
2756:
2747:
2745:
2736:
2735:
2728:
2690:
2689:
2685:
2674:At Right Angles
2667:
2666:
2662:
2653:
2651:
2642:
2641:
2637:
2630:
2617:
2616:
2612:
2602:
2600:
2589:
2586:Wayback Machine
2579:
2575:
2567:
2562:
2561:
2554:
2536:
2531:
2530:
2526:
2519:
2511:. UK: Arbelos.
2504:
2503:
2496:
2486:
2485:
2481:
2466:
2436:
2435:
2431:
2422:
2421:
2417:
2410:
2387:
2386:
2382:
2372:
2370:
2368:langorigami.com
2363:
2358:
2357:
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2345:
2338:
2337:
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2323:
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2290:
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2237:
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2224:
2223:
2219:
2179:
2178:
2174:
2167:
2146:
2145:
2141:
2134:
2113:
2112:
2105:
2076:Hull, Thomas C.
2070:
2069:
2065:
2058:
2039:
2038:
2031:
2012:
2011:
2010:
2003:
1998:
1991:
1981:
1980:
1973:
1951:
1950:
1943:
1933:
1932:
1928:
1923:
1919:
1910:
1906:
1888:
1883:
1882:
1875:
1859:
1851:
1850:
1846:
1833:
1832:
1828:
1818:
1817:
1813:
1807:British Origami
1793:
1792:
1788:
1774:
1773:
1764:
1757:
1744:
1743:
1739:
1732:Geometric Tools
1729:
1728:
1724:
1717:
1702:
1701:
1697:
1682:
1681:
1677:
1661:
1660:
1656:
1647:
1640:
1631:
1629:
1621:
1620:
1616:
1576:
1570:Hull, Thomas C.
1568:
1567:
1560:
1555:
1518:
1479:
1435:
1414:
1388:
1314:
1292:
1275:
1261:
1260:
1222:The problem of
1220:
1196:
1164:
1132:
1121:
1106:
1098:
1091:
1090:
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1072:
1068:
1067:
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1035:
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989:
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967:
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953:
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943:
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936:
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921:
920:
914:
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909:
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886:
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874:
868:
864:
863:
857:
853:
852:
825:
814:
807:
800:
799:
771:
764:
757:
756:
731:
720:
713:
712:
687:
679:
672:
671:
651:
650:
624:. For example:
568:
557:
541:
540:
529:
525:
524:
519:
515:
514:
509:
505:
504:
492:
490:Haga's theorems
484:powers of three
476:Pierpont primes
440:
410:
404:
345:crease patterns
316:
311:
264:
263:
244:
243:
172:double the cube
99:The Beloch fold
69:
63:
28:
23:
22:
15:
12:
11:
5:
5095:
5093:
5085:
5084:
5079:
5074:
5064:
5063:
5057:
5056:
5054:
5053:
5048:
5043:
5038:
5028:
5023:
5018:
5013:
5007:
5004:
5003:
5000:
4998:
4997:
4990:
4983:
4975:
4966:
4965:
4963:
4962:
4949:
4946:
4945:
4943:
4942:
4937:
4932:
4927:
4921:
4919:
4915:
4914:
4912:
4911:
4906:
4901:
4896:
4891:
4886:
4881:
4875:
4873:
4869:
4868:
4866:
4865:
4858:
4851:
4844:
4837:
4829:
4827:
4823:
4822:
4819:
4818:
4816:
4815:
4814:
4813:
4803:
4802:
4801:
4789:
4788:
4787:
4777:
4776:
4775:
4763:
4762:
4761:
4751:
4750:
4749:
4736:
4734:
4730:
4729:
4727:
4726:
4725:
4724:
4722:The Greek Vase
4714:
4713:
4712:
4702:
4701:
4700:
4689:
4687:
4683:
4682:
4680:
4679:
4678:
4677:
4665:
4664:
4663:
4653:
4652:
4651:
4644:Albrecht DĂĽrer
4641:
4640:
4639:
4627:
4626:
4625:
4613:
4612:
4611:
4599:
4598:
4597:
4590:
4578:
4572:
4570:
4566:
4565:
4563:
4562:
4561:
4560:
4548:
4547:
4546:
4535:
4533:
4526:
4522:
4521:
4518:
4517:
4515:
4514:
4509:
4507:Roman Verostko
4504:
4499:
4494:
4489:
4484:
4482:Alba Rojo Cama
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4444:
4439:
4438:
4437:
4428:Charles Jencks
4425:
4420:
4415:
4413:George W. Hart
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4366:
4360:
4358:
4354:
4353:
4351:
4350:
4345:
4344:
4343:
4336:
4324:
4323:
4322:
4310:
4309:
4308:
4301:
4294:
4287:
4280:
4268:
4263:
4258:
4257:
4256:
4249:
4240:Jean Metzinger
4237:
4236:
4235:
4228:
4215:
4213:
4207:
4206:
4204:
4203:
4202:
4201:
4189:
4187:Albrecht DĂĽrer
4184:
4183:
4182:
4170:
4165:
4159:
4157:
4150:
4144:
4143:
4141:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4099:
4097:
4091:
4090:
4088:
4087:
4080:
4073:
4066:
4059:
4052:
4045:
4039:
4037:
4033:
4032:
4030:
4029:
4024:
4019:
4014:
4013:
4012:
4002:
3997:
3996:
3995:
3990:
3985:
3975:
3974:
3973:
3968:
3963:
3958:
3948:
3943:
3938:
3933:
3928:
3927:
3926:
3921:
3916:
3906:
3904:Anamorphic art
3901:
3895:
3893:
3889:
3888:
3881:
3879:
3877:
3876:
3871:
3866:
3861:
3860:
3859:
3854:
3846:
3841:
3836:
3835:
3834:
3832:Camera obscura
3829:
3819:
3814:
3809:
3804:
3799:
3794:
3789:
3783:
3781:
3777:
3776:
3771:
3769:
3768:
3761:
3754:
3746:
3737:
3736:
3734:
3733:
3728:
3726:Tomohiro Tachi
3723:
3718:
3713:
3708:
3703:
3701:Robert J. Lang
3698:
3693:
3691:Humiaki Huzita
3688:
3683:
3678:
3673:
3671:Rona Gurkewitz
3668:
3666:Martin Demaine
3663:
3658:
3653:
3648:
3643:
3637:
3635:
3631:
3630:
3628:
3627:
3620:
3613:
3606:
3599:
3592:
3584:
3582:
3578:
3577:
3575:
3574:
3569:
3563:
3561:
3557:
3556:
3554:
3553:
3552:
3551:
3549:Star unfolding
3546:
3541:
3536:
3526:
3512:
3506:
3504:
3500:
3499:
3497:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3460:
3458:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3429:
3427:
3423:
3422:
3420:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3382:Crease pattern
3379:
3373:
3371:
3367:
3366:
3361:
3359:
3358:
3351:
3344:
3336:
3329:
3328:
3308:
3307:
3298:
3293:
3288:
3279:
3270:
3256:
3255:External links
3253:
3252:
3251:
3230:
3209:
3203:
3197:. A K Peters.
3187:
3181:
3168:
3150:(5): 357–371.
3139:
3133:
3110:
3098:
3095:
3092:
3091:
3067:
3042:
2979:
2955:
2934:
2916:
2891:
2867:
2846:
2818:
2808:. 22 June 2017
2793:
2784:
2774:Lang, Robert.
2766:
2754:
2742:The New Yorker
2726:
2683:
2660:
2635:
2628:
2610:
2573:
2552:
2524:
2517:
2494:
2479:
2464:
2429:
2415:
2408:
2380:
2351:
2328:
2321:
2284:
2255:(5): 679–702.
2235:
2217:
2188:(3): 200–211.
2172:
2165:
2139:
2132:
2124:10.1201/b15735
2103:
2063:
2056:
2050:. p. 22.
2029:
2001:
1989:
1971:
1941:
1926:
1917:
1904:
1886:"Problem 1054"
1873:
1844:
1826:
1811:
1786:
1781:origametry.net
1762:
1755:
1737:
1722:
1715:
1695:
1684:Houdini, Harry
1675:
1654:
1638:
1614:
1587:(4): 307–315.
1557:
1556:
1554:
1551:
1550:
1549:
1539:
1534:
1529:
1524:
1517:
1514:
1478:
1475:
1434:
1431:
1413:
1410:
1387:
1384:
1332:
1329:
1326:
1321:
1317:
1313:
1310:
1307:
1304:
1299:
1295:
1291:
1285:
1281:
1278:
1271:
1268:
1232:Miura map fold
1219:
1216:
1195:
1192:
1163:
1160:
1156:
1155:
1144:
1138:
1135:
1130:
1127:
1124:
1118:
1112:
1109:
1104:
1101:
1083:
1080:
1077:
1076:
1065:
1054:
1043:
1032:
1020:
1019:
1008:
997:
986:
975:
963:
962:
951:
940:
929:
918:
906:
905:
894:
883:
872:
861:
849:
848:
834:
831:
828:
821:
817:
813:
810:
797:
784:
778:
774:
770:
767:
754:
740:
737:
734:
729:
726:
723:
710:
696:
693:
690:
685:
682:
669:
658:
647:
646:
643:
640:
637:
634:
598:
597:
586:
580:
577:
574:
571:
566:
563:
560:
554:
551:
548:
491:
488:
469:if and only if
439:
436:
406:Main article:
403:
400:
370:
369:
366:
360:
359:
358:
315:
312:
310:
307:
271:
251:
195:Robert J. Lang
183:Humiaki Huzita
179:Jacques Justin
146:Also in 1980,
137:Crease pattern
117:cubic equation
62:
59:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5094:
5083:
5080:
5078:
5075:
5073:
5070:
5069:
5067:
5052:
5049:
5047:
5044:
5042:
5041:paper folding
5039:
5036:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5011:apportionment
5009:
5008:
5005:
4996:
4991:
4989:
4984:
4982:
4977:
4976:
4973:
4961:
4960:
4951:
4950:
4947:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4925:Droste effect
4923:
4922:
4920:
4916:
4910:
4907:
4905:
4904:Mathemalchemy
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4876:
4874:
4872:Organizations
4870:
4864:
4863:
4859:
4857:
4856:
4852:
4850:
4849:
4845:
4843:
4842:
4841:Lumen Naturae
4838:
4836:
4835:
4831:
4830:
4828:
4824:
4812:
4809:
4808:
4807:
4804:
4800:
4799:
4795:
4794:
4793:
4790:
4786:
4783:
4782:
4781:
4778:
4774:
4773:
4769:
4768:
4767:
4764:
4760:
4757:
4756:
4755:
4752:
4748:
4747:
4743:
4742:
4741:
4738:
4737:
4735:
4731:
4723:
4720:
4719:
4718:
4715:
4711:
4708:
4707:
4706:
4703:
4699:
4696:
4695:
4694:
4693:Samuel Colman
4691:
4690:
4688:
4684:
4676:
4675:
4671:
4670:
4669:
4666:
4662:
4659:
4658:
4657:
4654:
4650:
4647:
4646:
4645:
4642:
4638:
4637:
4633:
4632:
4631:
4628:
4624:
4623:
4619:
4618:
4617:
4614:
4610:
4609:
4605:
4604:
4603:
4600:
4596:
4595:
4591:
4589:
4588:
4584:
4583:
4582:
4579:
4577:
4574:
4573:
4571:
4567:
4559:
4558:
4554:
4553:
4552:
4549:
4545:
4542:
4541:
4540:
4537:
4536:
4534:
4530:
4527:
4523:
4513:
4510:
4508:
4505:
4503:
4502:Daina Taimiņa
4500:
4498:
4495:
4493:
4490:
4488:
4487:Reza Sarhangi
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4436:
4435:
4431:
4430:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4398:Peter Forakis
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4370:
4367:
4365:
4362:
4361:
4359:
4355:
4349:
4346:
4342:
4341:
4337:
4335:
4334:
4330:
4329:
4328:
4327:Salvador DalĂ
4325:
4321:
4320:
4316:
4315:
4314:
4313:René Magritte
4311:
4307:
4306:
4302:
4300:
4299:
4295:
4293:
4292:
4288:
4286:
4285:
4284:Print Gallery
4281:
4279:
4278:
4274:
4273:
4272:
4269:
4267:
4264:
4262:
4259:
4255:
4254:
4253:L'Oiseau bleu
4250:
4248:
4247:
4243:
4242:
4241:
4238:
4234:
4233:
4229:
4227:
4226:
4222:
4221:
4220:
4219:William Blake
4217:
4216:
4214:
4208:
4200:
4199:
4195:
4194:
4193:
4190:
4188:
4185:
4181:
4180:
4179:Vitruvian Man
4176:
4175:
4174:
4171:
4169:
4166:
4164:
4163:Paolo Uccello
4161:
4160:
4158:
4154:
4151:
4149:
4145:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4100:
4098:
4096:
4092:
4086:
4085:
4084:Pi in the Sky
4081:
4079:
4078:
4074:
4072:
4071:
4067:
4065:
4064:
4060:
4058:
4057:
4056:Mathemalchemy
4053:
4051:
4050:
4046:
4044:
4041:
4040:
4038:
4034:
4028:
4025:
4023:
4020:
4018:
4015:
4011:
4008:
4007:
4006:
4003:
4001:
3998:
3994:
3991:
3989:
3986:
3984:
3981:
3980:
3979:
3976:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3953:
3952:
3949:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3925:
3924:Vastu shastra
3922:
3920:
3917:
3915:
3914:Geodesic dome
3912:
3911:
3910:
3907:
3905:
3902:
3900:
3897:
3896:
3894:
3890:
3885:
3875:
3872:
3870:
3867:
3865:
3862:
3858:
3855:
3853:
3850:
3849:
3847:
3845:
3842:
3840:
3839:Plastic ratio
3837:
3833:
3830:
3828:
3827:Camera lucida
3825:
3824:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3784:
3782:
3778:
3774:
3767:
3762:
3760:
3755:
3753:
3748:
3747:
3744:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3638:
3636:
3632:
3626:
3625:
3621:
3619:
3618:
3614:
3612:
3611:
3607:
3605:
3604:
3600:
3598:
3597:
3593:
3591:
3590:
3586:
3585:
3583:
3579:
3573:
3572:Lill's method
3570:
3568:
3565:
3564:
3562:
3560:Miscellaneous
3558:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3531:
3530:
3527:
3524:
3520:
3516:
3513:
3511:
3508:
3507:
3505:
3501:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3479:Rigid origami
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3461:
3459:
3457:3d structures
3455:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3428:
3426:Strip folding
3424:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3374:
3372:
3368:
3364:
3357:
3352:
3350:
3345:
3343:
3338:
3337:
3334:
3327:
3322:
3317:
3313:
3306:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3283:
3280:
3278:
3274:
3271:
3267:
3263:
3259:
3258:
3254:
3250:
3246:
3242:
3238:
3234:
3231:
3229:
3225:
3221:
3217:
3213:
3210:
3206:
3200:
3196:
3192:
3188:
3184:
3178:
3174:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3140:
3136:
3130:
3126:
3122:
3118:
3117:
3111:
3108:
3104:
3101:
3100:
3096:
3081:
3077:
3071:
3068:
3057:
3056:Computerworld
3053:
3046:
3043:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2983:
2980:
2969:
2968:webb.nasa.gov
2965:
2959:
2956:
2945:
2938:
2935:
2927:
2920:
2917:
2906:
2902:
2895:
2892:
2881:
2877:
2871:
2868:
2857:
2850:
2847:
2842:
2836:
2828:
2822:
2819:
2807:
2803:
2797:
2794:
2788:
2785:
2777:
2770:
2767:
2761:
2759:
2755:
2743:
2739:
2733:
2731:
2727:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2693:Math Horizons
2687:
2684:
2679:
2675:
2671:
2664:
2661:
2650:on 2017-05-08
2649:
2645:
2639:
2636:
2631:
2625:
2621:
2614:
2611:
2599:
2598:
2593:
2587:
2583:
2577:
2574:
2566:
2559:
2557:
2553:
2548:
2544:
2543:
2535:
2528:
2525:
2520:
2514:
2510:
2509:
2501:
2499:
2495:
2490:
2483:
2480:
2475:
2471:
2467:
2461:
2457:
2453:
2449:
2448:
2443:
2439:
2433:
2430:
2425:
2419:
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1224:rigid origami
1217:
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1190:in geometry.
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438:Constructions
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4826:Publications
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4717:Jay Hambidge
4710:Ad Quadratum
4709:
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4616:Luca Pacioli
4606:
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4467:Hinke Osinga
4462:István Orosz
4432:
4423:Anthony Hill
4378:Scott Draves
4373:Erik Demaine
4357:Contemporary
4338:
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4108:Hagia Sophia
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4009:
3931:Computer art
3909:Architecture
3869:Tessellation
3852:Architecture
3802:Golden ratio
3731:Eve Torrence
3661:Erik Demaine
3622:
3615:
3608:
3601:
3594:
3587:
3581:Publications
3443:Möbius strip
3433:Dragon curve
3370:Flat folding
3362:
3305:Mario Cigada
3286:cut-the-knot
3277:cut-the-knot
3262:Dr. Tom Hull
3240:
3219:
3194:
3172:
3147:
3143:
3115:
3083:. Retrieved
3079:
3070:
3059:. Retrieved
3055:
3045:
2996:
2992:
2982:
2971:. Retrieved
2967:
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2947:. Retrieved
2937:
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2904:
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2883:. Retrieved
2879:
2870:
2859:. Retrieved
2849:
2821:
2810:. Retrieved
2805:
2796:
2787:
2769:
2746:. Retrieved
2744:. 2007-02-12
2741:
2699:(3): 22–24.
2696:
2692:
2686:
2677:
2673:
2663:
2652:. Retrieved
2648:the original
2638:
2619:
2613:
2601:. Retrieved
2595:
2582:Ghostarchive
2580:Archived at
2576:
2546:
2540:
2527:
2507:
2482:
2446:
2432:
2418:
2389:
2383:
2371:. Retrieved
2367:
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2195:math/0408159
2185:
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1957:
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1870:(2): 269–279
1867:
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1839:the original
1829:
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1630:. Retrieved
1626:
1617:
1584:
1580:
1546:dragon curve
1510:
1499:
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1480:
1477:Applications
1468:
1452:Georgia Tech
1444:
1427:
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605:
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333:
314:Flat folding
309:Pure origami
303:
298:
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229:
225:
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207:
192:
176:
169:
152:
145:
142:
124:
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87:
83:kindergarten
76:
70:
55:
47:mathematical
40:
4766:G. H. Hardy
4569:Renaissance
4539:Polykleitos
4477:Tony Robbin
4388:John Ernest
4383:Jan Dibbets
4333:Crucifixion
4156:Renaissance
4010:Mathematics
3983:Celtic knot
3946:Fractal art
3848:Proportion
3822:Perspective
3716:KĹŤryĹŤ Miura
3711:Jun Maekawa
3686:KĂ´di Husimi
3402:Map folding
3326:Mathematics
1537:Map folding
1506:RNA origami
1502:DNA origami
1471:DNA origami
1398:Robert Lang
1250:Wet-folding
1228:sheet metal
390:NP-complete
386:Barry Hayes
341:map folding
218:NP-complete
214:Barry Hayes
148:KĹŤryĹŤ Miura
109:Beloch fold
35:Map folding
5066:Categories
5021:bookmaking
4740:Owen Jones
4587:De pictura
4492:Oliver Sin
4442:Andy Lomas
4291:Relativity
4022:String art
3936:Fiber arts
3817:Paraboloid
3706:Anna Lubiw
3539:Common net
3464:Miura fold
3085:2022-05-08
3061:2022-05-08
2973:2022-05-08
2949:2022-05-08
2910:2022-05-08
2885:2022-05-08
2861:2022-05-08
2812:2022-05-08
2748:2022-05-09
2654:2008-10-08
2603:October 2,
2549:: 241–242.
2487:Tom Hull.
2373:16 January
1835:"2D Array"
1632:2022-05-08
1380:Miura fold
1365:California
65:See also:
4551:Vitruvius
4525:Theorists
4305:Waterfall
4210:19th–20th
4138:Taj Mahal
4118:Parthenon
4095:Buildings
4049:Continuum
4017:Sculpture
3993:Interlace
3787:Algorithm
3624:Origamics
3503:Polyhedra
3021:0036-8075
2876:"Cadnano"
2721:126397750
2620:Origami 5
2399:1307.1065
2301:Origami 4
2271:1469-7653
2022:MathWorld
2017:"Folding"
1325:−
1277:π
1212:congruent
769:−
725:−
523:and then
449:pentagons
445:triangles
416:— namely
299:Countdown
230:In 2002,
208:In 1996,
153:Miura-ori
88:In 1922,
4959:Category
4686:Romantic
4364:Max Bill
4298:Reptiles
4113:Pantheon
4070:Octacube
4036:Artworks
3978:Knotting
3966:Muqarnas
3864:Symmetry
3792:Catenary
3780:Concepts
3681:Tom Hull
3651:Yan Chen
3534:Blooming
3438:Flexagon
3193:(2003).
3037:18415193
3029:25104380
2835:cite web
2713:25678354
2584:and the
2444:(2007).
2279:46359986
2212:30037438
2078:(2002).
1855:(2012),
1572:(2011).
1522:Flexagon
1516:See also
1412:Research
1404:and the
1393:Bug Wars
1343:, where
459:and the
453:hexagons
236:Tom Hull
103:In 1936
5072:Origami
4918:Related
4532:Ancient
4266:Man Ray
4212:Century
4148:Artists
4005:Origami
3919:Pyramid
3797:Fractal
3164:2690924
3001:Bibcode
2993:Science
2942:TASON.
2880:cadnano
2854:TASON.
2597:YouTube
2474:2354878
1966:1381938
1609:2540978
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61:History
43:origami
5046:Sudoku
4733:Modern
4369:Martin
4232:Newton
4027:Tiling
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3941:4D art
3634:People
3489:Sonobe
3312:Portal
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1421:, and
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482:, and
388:to be
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4000:Music
3956:Girih
3892:Forms
3857:Human
3160:JSTOR
3033:S2CID
2929:(PDF)
2779:(PDF)
2717:S2CID
2709:JSTOR
2568:(PDF)
2537:(PDF)
2394:arXiv
2364:(PDF)
2346:(PDF)
2297:(PDF)
2275:S2CID
2230:(PDF)
2208:JSTOR
2190:arXiv
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1860:(PDF)
1605:S2CID
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412:Some
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3177:ISBN
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3017:ISSN
2841:link
2624:ISBN
2605:2021
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2404:ISBN
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2267:ISSN
2161:ISBN
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608:is
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