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Origami


Origami great rhombicosidodecahedron
Origami icosahedron
Origami icosidodecahedron
Origami crane animation

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 the Wolfram Language by L. Zamiatina.

OrigamiFolds

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 p_1 and p_2, we can fold a line connecting them.

2. Given two points p_1 and p_2, we can fold p_1 onto p_2.

3. Given two lines l_1 and l_2, we can fold line l_1 onto l_2.

4. Given a point p_1 and a line l_1, we can make a fold perpendicular to l_1 passing through the point p_1.

5. Given two points p_1 and p_2 and a line l_1, we can make a fold that places p_1 onto l_1 and passes through the point p_2.

6. Given two points p_1 and p_2 and two lines l_1 and l_2, we can make a fold that places p_1 onto line l_1 and places p_2 onto line l_2.

A seventh axiom overlooked by Huzita was subsequently discovered by Hatori in 2002 (Lang).

7. Given a point p_1 and two lines l_1 and l_2, we can make a fold perpendicular to l_2 that places p_1 onto line l_1.


See also

Flat Origami, Folding, Geometric Construction, Map Folding, Stamp Folding, Stomachion, Tangram

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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 Your Times, 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/.

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Origami

Cite this as:

Weisstein, Eric W. "Origami." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Origami.html

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