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Folding


There are many mathematical and recreational problems related to folding. Origami, the Japanese art of paper folding, is one well-known example.

It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).

FoldingPentagon

Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular pentagon (illustrated above), hexagon, heptagon, octagon, and decagon using paper folding.

The least number of folds required to create an n-gon for n>=4 is not known, but some bounds are. In particular, every set of n points is the image of a suitable regular n-gon under at most F(n) folds, where

 F(n)<={1/2(3n-2)   for n even; 1/2(3n-3)   for n odd.
(1)

The first few values are 0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, ... (OEIS A007494).

The points accessible from c by a single fold which leaves a_1, ..., a_n fixed are exactly those points interior to or on the boundary of the intersection of the circles through c with centers at a_i, for i=1, ..., n. Given any three points in the plane a, b, and c, there is an equilateral triangle with polygon vertices x, y, and z for which a, b, and c are the images of x, y, and z under a single fold.

Given any four points in the plane a, b, c, and d, there is some square with polygon vertices x, y, z, and w for which a, b, c, and d are the images of x, y, z, and w under a sequence of at most three folds. In addition, any four collinear points are the images of the polygon vertices of a suitable square under at most two folds. Every five (six) points are the images of the polygon vertices of suitable regular pentagon (hexagon) under at most five (six) folds.

Assuming it were possible to fold paper without restriction, the height of a piece of folded paper would double in thickness each time it was folded. Since one sheet of typical 20-pound paper has a thickness of about 0.1 millimeter, folding 50 times (if this were physically possible, which of course it is not) would produce a wad of height 1.13×10^(11) meters, and folding one more time would make the stack higher than the distance between the Earth and Sun.

Paper folder Britney Gallivan

The function

 L=1/6pid(2^n+4)(2^n-1)
(2)

gives the loss function for folding paper in half, where L is the minimum possible length of the material, d is the thickness, and n is the possible number of folds in a given direction. This formula indicates how much "normalized" paper has been lost for n folds, and thus sets a limit for the number of times things of finite thickness can be folded in one direction (Pomona Valley Historical Society). For n=0, 1, 2, ... the sequence L/(pid) gives 0, 1, 4, 14, 50, 186, 714, ... (OEIS A076024). The formula was derived by high school student Britney Gallivan in December of 2001. Britney then proceeded to set a new world record by folding first gold foil and then paper in half a whopping 12 times in January of 2002, thus debunking the assertions of Math@Home and PBS Kids that paper cannot be folded in half more than eight times. Britney and her feat were mentioned in the Season 1 episode "Identity Crisis" (2005) of the television crime drama NUMB3RS.


See also

Alexandrov's Theorem, Flexagon, Folding Function, Map Folding, Origami, Rudin-Shapiro Sequence, Stamp Folding

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References

Bern, M.; Demaine, E.; Eppstein, D.; and Hayes, B. "A Disk-Packing Algorithm for an Origami Magic Trick." In Proc. Third Internat. Meeting of Origami Science, Math, and Education, Monterey, California, March 2001. pp. 17-28.Bern, M.; Demaine, E.; Eppstein, D.; and Hayes, B. "A Disk-Packing Algorithm for an Origami Magic Trick." Proc. Internat. Conference of Fun with Algorithms, Isola d'Elba, Italy, June 1998. pp. 32-42.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.Demaine, E. K. and Demaine, M. L. "Fold-and-Cut Magic." In Tribute to a Mathemagician (Ed. B. Cipra, E. D. Demaine, M. L. Demaine, and T. Rodgers). Wellesley, MA: A K Peters, pp. 23-30, 2004.Demaine, E. K.; Demaine, M. L.; and Lubiw, A. "Folding and Cutting Paper." It Revised Papers from the Japan Conference on Discrete and Computation Geometry (Ed. J. Akiyama, M. Kano, and M. Urabe). Tokyo, Japan, pp. 104-117, 1998.Demaine, E. K.; Demaine, M. L.; and Lubiw, A. "Folding and Straight Cut Suffice." In Proc. 10th Annual ACM-SIAM Symposium Disc. Alg. Baltimore, MD: ACM Press, pp. 891-892, 1999.Gallivan, B. C. "How to Fold Paper in Half Twelve Times: An 'Impossible Challenge' Solved and Explained." Pomona, CA: Historical Society of Pomona Valley, 2002.Gardner, M. "Mathematical Games: Recreations Involving Folding and Cutting Sheets of Paper." Sci. Amer. 202, 161-170, Jun. 1960.Gardner, M. "Paper Cutting." Ch. 5 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 58-69, 1966.Gardner, M. "Single-Cut Stunts." In Encyclopedia of Impromptu Magic. Chicago, IL: Magic, Inc., pp. 424-428, 1978.Gurkewitz, R. and Arnstein, B. Multimodular Origami Polyhedra: Archimedeans, Buckyballs, and Duality. New York: Dover, 2003.Hilton, P.; Holton, D.; and Pedersen, J. "Paper-Folding and Number Theory." Ch. 4 in Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 87-142, 1997.Houdini, H. Houdini's Paper Magic: The Whole Art of Performing with Paper, Including Paper Tearing, Paper Folding and Paper Puzzles. New York: Dutton, 1922.Klein, F. "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, p. 42, 1980.Legman, G. Bibliography of Paper-Folding. Malvern, England: Priory Press, 1952.Loe, G. M. Paper Capers. Chicago, IL: Magic, Inc., 1955.Math@Home. "Amazing Paperfolding Fact." http://educ.queensu.ca/~fmc/june2002/PaperFact.htm. June 2002."National Standards and Emblems." Harper's New Monthly Magazine 47, No. 278, 171-181, July 1873.PBS Kids. "Paper Fold." http://pbskids.org/zoom/activities/phenom/paperfold.html.Peterson, I. "MathTrek: Folding Paper in Half--Twelve Times." Jan. 24, 2004. http://www.sciencenews.org/20040124/mathtrek.asp.Pomona Valley Historical Society. "How to Fold Paper in Half Twelve Times." http://pomonahistorical.org/12times.htm.Sabinin, P. and Stone, M. G. "Transforming n-gons by Folding the Plane." Amer. Math. Monthly 102, 620-627, 1995.Sloane, N. J. A. Sequences A007494 and A076024 in "The On-Line Encyclopedia of Integer Sequences."Sundra Row, T. Geometric Exercises in Paper Folding. New York: Dover, 1966.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 191-192, 1991.Wertheim, M. "Origami as the Shape of Things to Come." The New Your Times, Section F, Column 1, p. 1. Feb. 15, 2005.

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Folding

Cite this as:

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

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