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The Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual units, each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be...
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| Subject keyword(s) | Construction, Geometry, Lines and planes, Mathematics, Plane geometry |
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| Grade level | Middle School, High School, Informal Education |
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| Intended audience | Learner |
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| Resource type | Instructional Material, Reference Material |
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| Resource format | text, text/html |
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| Rights | Copyright 1999 CRC Press LLC, Copyright 1999-2005 Wolfram Research, Inc. See http://mathworld.wolfram.com/terms.html for a full terms of use statement. |
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| Title | Middle School Portal: Math and Science Pathways (MSP2) |
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| Link | http://msteacher2.org/ |
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| Description | The Middle School Portal 2: Math and Science Pathways (MSP2) supports middle grades educators with high-quality, standards-based resources and promotes collaboration and knowledge-sharing among its users. Educators use MSP2 to increase content knowledge in science, mathematics, and appropriate pedagogy for youth ages 10 to 15. MSP2 employs social networking and digital tools to foster dynamic experiences that promote creation, modification, and sharing of resources, facilitate professional development, and support the integration of technology into practice. MSP2 is a project of the Ohio State University College of Education and Human Ecology, National Middle School Association, and Education Development Center, Inc., and is funded by the National Science Foundation. The partners integrate resources, tools, and services across projects, and support multiple methods of resource discovery to meet the needs of this audience. |
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Content contained within the resource
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book 13,106 entries Last updated: Wed Jan 25 2012 Recreational Mathematics > Folding > Origami > Geometry > GeometricConstruction > Geometry > Solid Geometry > Polyhedra > Miscellaneous Polyhedra > More... Recreational Mathematics > Mathematical Art > Mathematical Images > Recreational Mathematics > Mathematics in the Arts > Mathematics in Television > NUMB3RS > Interactive Entries > Animated GIFs > MathWorld Contributors > Herrstrom, Emily > MathWorld Contributors > Lang > Interactive Entries > Interactive Demonstrations > Less... Origami Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in Mathematica by L. Zamiatina. To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming a trough. The Season 2 episode "Judgment Call" (2006) of the television crime drama NUMB3RS features Charlie discussing the types of folds in origami. Cube duplication and angle trisection can be solved using origami, although they cannot be solved using the traditional rules for geometric constructions. There are a number of recent very powerful results in origami mathematics. A very general result states that any planar straight-line drawing may be cut out of one sheet of paper by a single straight cut, after appropriate folding (Demaine et al. 1998, 1999; O'Rourke 1999). Another result is that any polyhedron may be wrapped with a sufficiently large square sheet of paper. This implies that any connected, planar, polygonal region may be covered by a flat origami folded from a single square of paper. Moreover, any 2-coloring of the faces may be realized with paper whose two sides are those colors (Demaine et al. 1999; O'Rourke 1999). Huzita (1992) has formulated what is currently the most powerful known set of origami axioms (Hull). 1. Given two points and , we can fold a line connecting them. 2. Given two points and , we can fold onto . 3. Given two lines and , we can fold line onto . 4. Given a point and a line , we can make a fold perpendicular to passing through the point . 5. Given two points and and a line , we can make a fold that places onto and passes through the point . 6. Given two points and and two lines and , we can make a fold that places onto line and places onto line . A seventh axiom overlooked by Huzita was subsequently discovered by Hatori in 2002 (Lang). 7. Given a point and two lines and , we can make a fold perpendicular to that places onto line . SEE ALSO: Flat Origami, Folding, Geometric Construction, Map Folding, Stamp Folding, Stomachion, Tangram REFERENCES: Alperin, R. C. "A Mathematical Theory of Origami Constructions and Numbers."New York J. Math. 6, 119-133, 2000. Andersen, E. M. "paperfolding.com." http://www.paperfolding.com/. Andersen, E. M. "Origami and Math." http://www.paperfolding.com/math/. Biddle, S. and Biddle, M. The New Origami. New York: St. Martin's Press, 1993. Brill, D. Brilliant Origami: A Collection of Original Designs. Tokyo: Japan Pub., 1996. Cerceda, A. and Palacios, V. Fascinating Origami: 101 Models by Adolfo Cerceda. New York: Dover, 1997. Demaine, E. D.; Demaine, M. L.; and Lubiw, A. "Folding and Cutting Paper." In Proc. Japan Conf. Discrete Comput. Geom. New York: Springer-Verlag, 1998. Demaine, E. D.; Demaine, M. L.; and Lubiw, A. "Folding and One Straight Cut Suffice." In Proc. 10th Ann. ACM-SIAM Sympos. Discrete Alg. (SODA'99). Baltimore, MD, pp. 891-892, Jan. 1999. Demaine, E. D.; Demaine, M. L.; and Mitchell, J. S. B. "Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computation Origami." In Proc. 15th Ann. ACM Sympos. Comput. Geom. Miami Beach, FL, pp. 105-114, June 1999. Eppstein, D. "Origami." http://www.ics.uci.edu/~eppstein/junkyard/origami.html. Fusè, T. Unit Origami: Multidimensional Transformations. Tokyo: Japan Pub., 1990. Geretschläger, R. "Euclidean Constructions and the Geometry of Origami."Math. Mag. 68, 357-371, 1995. Gurkewitz, R. "Rona's Modular Origami Polyhedra Page." http://www.wcsu.ctstateu.edu/~gurkewitz/homepage.html. Gurkewitz, R. and Arnstein, B. 3-D Geometric Origami: Modular Polyhedra. New York: Dover, 1995. Gurkewitz, R. and Arnstein, B. Multimodular Origami Polyhedra: Archimedeans, Buckyballs, and Duality. Mineola, New York: Dover, 2003. Harbin, R. Origami Step-By-Step. New York: Dover, 1998. Harbin, R. Secrets of Origami: The Japanese Art of Paper Folding. New York: Dover, 1997. Hull, T. "Origami and Geometric Construction: A Comparison between Straight Edge [sic] and Compas Constructions and Origami." http://www.merrimack.edu/~thull/omfiles/geoconst.html. Huzita, H. "Understanding Geometry through Origami Axioms." In COET91: Proceedings of the First International Conference on Origami in Education and Therapy (Ed. J. Smith). British Origami Society, pp. 37-70, 1992. Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, 1988. Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987. Kosmulski, M. "Modulus Origami--Fractals, IFS." http://hektor.umcs.lublin.pl/~mikosmul/origami/fractals.html. Lang, R. "Huzita-Hatori Axioms." http://www.langorigami.com/science/hha/hha.php4. Lang, R. "Robert J. Lang Origami." http://www.langorigami.com/. Montroll, J. Origami Inside-Out. New York: Dover, 1993. Montroll, J. Origami Sculptures, 2nd ed. Antroll Pub., 1991. Montroll, J. A Plethora of Polyhedra in Origami New York: Dover, 2002. O'Rourke, J. "Computational Geometry Column 36." SIGACT News 30,35-38, Sep. 1999. Palacios, V. Fascinating Origami: 101 Models by Alfredo Cerceda. New York: Dover, 1997. Pappas, T. "Mathematics & Paperfolding." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 48-50, 1989. Plank, J. "Jim Plank's Origami Page (Modular)." http://www.cs.utk.edu/~plank/plank/pics/origami/origami.html. Row, T. S. Geometric Exercises in Paper Folding. New York: Dover, 1966. Sakoda, J. M. Modern Origami. New York: Simon and Schuster, 1969. Simon, L.; Arnstein, B.; and Gurkewitz, R. Modular Origami Polyhedra. New York: Dover, 1999. Takahama, T. The Complete Origami Collection. Japan Pub., 1997. Tomoko, F. Unit Origami. Tokyo: Japan Publications, 1990. Wertheim, M. "Origami as the Shape of Things to Come." The New YourTimes, Section F, Column 1, p. 1. Feb. 15, 2005. Wu, J. "Joseph Wu's Origami Page." http://www.origami.vancouver.bc.ca/. Zamiatina, L. "Computer Simulations of Origami." http://library.wolfram.com/infocenter/Articles/1786/. CITE THIS AS: Weisstein, Eric W. "Origami." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Origami.html Contact the MathWorld Team © 1999-2012 Wolfram Research, Inc. | Terms of Use Origami in the Maximizing the Volume of a Cup Made from a Square Sheet of Paper ll Folding an Equilateral Triangle in a Square n-gon Polynomials Origami Flanged Pots Other Wolfram Web Resources » Wolfram Research Mathematica Home Page Mathematica Documentation Center Wolfram Demonstrations Project Online Integrator Functions Site Wolfram Tones Wolfram Science Wolfram Blog more…
