
	Where results make sense

Topic: Mathematics of paper folding

Related Topics

Paper folding

Origami

In the News (Tue 29 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Chapter 9: Mathematics
		The first step after cutting out the colored pattern, is to fold it carefully along the lines that separate the colors, and along the lines that connect the points of the diamonds.
		Some of the folds will eventually go inwards, and some outwards, but in this step we are folding the paper back and forth both ways, and creasing the paper well.
		Now that the paper is nicely creased along the fold lines, it is time to spread some glue on the first little triangle that says "Glue here".
	sci-toys.com /scitoys/scitoys/mathematics/paper_ring.html (500 words)

	A survey of paper cutting, folding and tearing in mathematics textbooks for prospective elementary school teachers. - ...
		Through paper folding, children use their hands to follow a prescribed set of steps, and Froebel believed strongly that successful completion of the steps resulted in objects that children found both pleasing and mathematical in nature.
		During the course of reviewing the ten texts for instances of origami, the members of the panel that enumerated and classified the opportunities to fold paper, made several comments concerning the diverse presentations and lack of *~unified approach to origami that was generally evident in the texts.
		Wet folding, a procedure wherein heavy paper is folded while wet and thus allows the folder to sculpt a model with soft curves and 3-D forms, appears to have considerable potential in educational applications (Wu, 2001).
	www.encyclopedia.com /doc/1G1-105477685.html (3597 words)

	Paper-Folding-Fractals (construct)
		Fold the far edge of the strip to the left (figure 2b).
		Fold a second strip of paper in-half, and to the left as you did in level 1 (figure 3a, 3b).
		Again, folding to the left, fold this second strip in half a second time so that it is now only 1/4th its original length (figure 3c).
	www.cs.unm.edu /~joel/PaperFoldingFractal/construct.html (709 words)

	PAPER FOLDING AND PROPORTIONS IN POLYGONS
		One of the purposes of this paper is defining a common message to improve the image of mathematics in contemporary society.
		Therefore the strip of paper is placed in parallel with the diagonal of the square and tangent to its sides.
		What implies that the paper folding process is correct because the side algebraically obtained of the octagon has exactly the same value as the side of the octagon obtained from the smallest side of the strip.
	members.tripod.com /vismath7/proceedings/reyes.htm (788 words)

	Paper folding - Wikipedia, the free encyclopedia
		Paper folding is the art of folding paper; it is known in many societies that use paper.
		In much of the West, the term origami is used synonymously with paper folding, though the term properly only refers to the art of paper folding in Japan.
		Curiosity: It was formerly thought that it was impossible to fold a sheet of paper in half more than 7 times; usually it is difficult to reach even 6 times.
	en.wikipedia.org /wiki/Paper_folding (130 words)

	Mathematics of paper folding - Wikipedia, the free encyclopedia
		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 mathematical equations.
		Paper folds can be constructed to solve equations up to degree 4.
		Folding a flat model from a crease pattern has been proven by Marshall Bern and Barry Hayes to be NP complete.
	en.wikipedia.org /wiki/Mathematics_of_paper_folding (397 words)

	Paper Folding Geometry
		Folding paper is analogous to mirroring one half of a plane in a crease.
		As in the usual Geometry, the distinction is being made between experimentation with the physical paper and the abstract theory of "paper folding".
		The model for a paper sheet is a piece of the plane; folding is an isometry of the part of the plane on one side of the fold to another, the fold being the curve of fixed points of this isometry.
	www.cut-the-knot.org /pythagoras/PaperFolding/index.shtml (1161 words)

	Folding Paper in Half Twelve Times
		Limiting equations were derived for the case of folding in alternate directions and for the case of folding in a single direction using a long strip of paper.
		The exact limit for single direction folding case was derived, based on the accumulative limiting effects induced by every fold in the folding process.
		Britney derived folding limits in December of 2001 and folded paper in half 12 times in January of 2002, while a junior in High School.
	pomonahistorical.org /12times.htm (871 words)

	Mathematics Through Paper Folding. (Site not responding. Last check: )
		This booklet is a revised edition of Donovan Johnson's "Paper Folding for the Mathematics Class" (ED 077 711).
		It begins with directions for folding basic constructions such as as a straight line, the line perpendicular to a given line passing through a given point, and the bisector of an angle.
		Subsequent chapters cover concepts related to reflections, circle relationships, star and polygon constructions, symmetry, conic sections, algebra by paper folding, polygons constructed by typing paper knots, and recreations such as the Mobius strip and pop-up dodecahedra.
	www.eric.ed.gov /sitemap/html_0900000b800f9a63.html (104 words)

	Articles: Folding, by Bryan Clair
		Folding paper is a very simple operation, but leads to surprisingly precise constructions.
		One fold that's been around for centuries is the bellows, used in old-style cameras and as wind power for accordions and church organs.
		Bryan Clair is a professor of mathematics at Saint Louis University.
	www.strangehorizons.com /2002/20020311/folding.shtml (2590 words)

	Articles Category: Reference And Education - ArticleClick.com - Free Articles Directory
		Folding paper may not seem to be very challenging but as the projects advance, origami can in fact be quire complicated and complex.
		Certain set shapes were fashioned from folded paper for special occasions like weddings, while serrated strips of white paper were used to adorn sacred objects in the shrines, a practice that continues to this day.
		Paper folding was virtually unheard of outside of Spain and Argentina at this point in time but by the 1030's Miguel's followers had helped to spread his art to the people of South America.
	www.articleclick.com /Category/Reference-and-Education/220 (2839 words)

	Origami Summary
		In general, these designs begin with a square sheet of paper, whose sides may be different colors, and usually proceed without cutting the paper.
		Folding a flat model from a crease pattern has been proven by Marshall Bern and Barry Hayes to be NP complete.
		For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites.
	www.bookrags.com /Origami (2048 words)

	Exponential growth
		Take a sheet of paper of the ordinary variety - letter size for the Americans, A4 for the rest of the world - and fold it into half.
		If you would have been able to fold it 10 times, it would be as thick as the width of your hand.
		Then he folded that long thick strand in half, and pulled the dough out again into its original length, so that two thinner strands now passed from one hand to the other.
	raju.varghese.org /articles/powers2.html (1231 words)

	The origami polygon cutting theorem
		The difference arizes when a fold line and a cut line coincide, the points on the fold are not removed with a scissor cut (see figure 2).
		Because each wall of the (subdivided) corridor is perpendicular to the cut edges, folding at the skeleton edge causes the two incident perpendicular edges in the wall to fold to a common line.
		Folding this collection of accordions flat onto one plane reduces to folding a tree (figure 8).
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2003/EricBiunno (2402 words)

	Paper Folding Activity
		It is important to build a bridge between the technology representing the piece of paper and the actual physical piece of paper.
		If you fold the paper to the 1/2 mark of line segment EF, what is the property of point I? If you fold the paper to the 1/3 mark of line segment EF, what is the property of point I? What is the relation between the gray shaded area and the striped shaded area?
		Once more have the students use their sheet of paper and the vocabulary they have used to identify the vertices, triangles, angles and lengths, to describe the relations that exist between the triangles formed from the fold.
	mathforum.org /alejandre/escot/folding.html (644 words)

	math
		Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions.
		The Japanese art of paper folding is obviously geometrical in nature.
		Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon.
	www.homestead.com /origamidesigns/math.html (1972 words)

	Folding and Unfolding (Erik Demaine)
		Origami mathematics is a recent branch of mathematics (whose major study started circa 1980) that studies the properties of origami, such as what patterns you might get when you unfold a flat origami.
		The basic constraints in folding a piece of paper are that the paper is folded continuously (no ripping), while preserving distances along its surface (no stretching), and not causing the paper to self-intersect (no crossing).
		Folding silhouettes and wrapping polyhedra: We show that "origami design is always possible," in the sense that if you desire a given polyhedron (or a flat silhouette) you can fold it out of a sufficiently large piece of paper.
	theory.lcs.mit.edu /~edemaine/folding (1403 words)

	The Institute For Figuring // An Interview with Robert Lang
		Robert Lang is a pioneer in the emerging field of computational origami, a branch of mathematics that explores the formal properties and potentialities of folded paper.
		Mathematically, trisecting an angle is the equivalent of solving a cubic equation—an equation involving x to the power of three.
		But the fundamental theory of folding is the same, and if you can develop general concepts that apply across dimensions—from one-dimensional to two-dimensional, and even higher-dimensional problems—then the results that you derive are going to be applicable to these very fundamental issues like protein folding and biological activity.
	theiff.org /publications/cab17-lang.html (2607 words)

	Origami Mathematics Page
		These pages are an attempt to begin collecting information on the mathematics of paper folding.
		Clearly there is an origami geometry at work when paper is folded.
		Such geometry, this mathematics of origami, has been studied extensively by origamists, mathematicians, scientists and artists.
	www.merrimack.edu /~thull/OrigamiMath.html (442 words)

	Recreations: Math at Canadian Content
		The annual journal of the Archimedeans, the mathematical society of the University of Cambridge.
		At Mathematics Museum (Japan) you would be surprised how interesting mathematics is. You will find exhibition rooms produced by Japanese researchers and educators.
		Mathematical Spectrum is a magazine for students and teachers of mathematics in universities, colleges and schools worldwide.
	www.canadiancontent.net /dir/Top/Science/Math/Recreations (1381 words)

	Tech Tidbit -- July 2004
		The mathematics is highly complex and the computer coding that accompanies it is as well.
		Folding, however, applies to far more than paper, and the products can be even more beautiful than the colorful paper cranes in the photo above and incredibly useful in fields ranging from the study of proteins to the production of automobile airbags.
		An airbag needs to be folded in the most efficient manner so that it will fit into a compartment, for example in an automobile's steering wheel, and inflate when needed instantly and without snagging.
	www.alteich.com /tidbits/t070104.htm (589 words)

	The Institute For Figuring // Lecture: The Mathematics of Paper Folding
		Robert Lang is one of the pioneers of the field of computational origami, the cross disciplinary marriage of mathematics and paper folding, sometimes known as origami sekkei or technical folding.
		The toolkit of the computational origamist vastly expands this repertoire through the techniques of mathematics, enabling the construction of elaborate geometrical models and startlingly realistic animals with detailed anatomical features such as wings and claws and antennae.
		It turns out that mathematically this is equivalent to the long-standing problem of how can one efficiently pack a bunch of circles into a square.
	www.theiff.org /lectures/04.html (496 words)

	Creased.com - Links (Site not responding. Last check: )
		Paper Aircraft Association - The site of Andy Chipling, who works with the Guinness Book of Records to produce the rules for official paper aircraft records.
		Paper Airplane Flier's Club - Details of how to build and throw a world record paper airplane, as well as the skyhawk and gemini designs.
		Origami Mathematics - Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions.
	www.creased.com /links/links.htm (1767 words)

	SuccessLink (Site not responding. Last check: )
		Take a piece of paper, a waxed sheet, or a section of wax paper and hold it length-wise.
		Take a piece of paper, a waxed sheet, or a section of a waxed paper and place a circle (diameter 2 inches) onto the paper middle left.
		Fold the paper up towards the circle covering the focal point hole over the circumference.
	www.successlink.org /gti/gti_lesson.asp?lid=3297 (1282 words)

	MATH 4160 -- Senior Seminar Topics
		A tile is a mathematical idealization of the tiles one might encounter on their kitchen floor.
		Tiles have been studied for scientific and mathematical purposes in more recent years and have revealed a theory that is deep and interesting while being simultaneously accessible to people without much mathematical training.
		In knot theory, a knot is a mathematical idealization of the knot you formed with your rope; that is, a knot is a simple closed curve in space.
	math.uttyler.edu /seminar_topics.htm (1270 words)

	Ma Baker's Origami and Paper Folding Web Quest Page
		Most origami is folded from an uncut square of paper.The most common sizes of square are 6-inches and 10-inches.
		Paper folding activities related to geometry help to motivate student interest in mathematics.
		The process of producing a paper figure allows students to learn to follow directions, to become motivated, to use a visual aide for better understanding of mathematical concepts, and to complete a project through their own perseverance.
	education.nmsu.edu /webquest/wq/origami/index.htm (822 words)

	Origami (折り紙)
		Make a fold on half of a square paper, the first step for folding a regular hexagon or regular pentagon.
		The methods are based on folding a 60 deg angle at center (for hexagon) and an approximate of 36 or 72 deg angle (for pentagon).
		Unfolding Mathematics with Unit Origami by Betsy Franco, Activity 6 and 8.
	web.hku.hk /~amslee/mathbook/pages/67.html (113 words)

	Geometry and Modulars :: Origami : Gourt
		Mathematical Origami - It highlights origami models and techniques with a mathematical flavour.
		Origami Mathematics - Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions.
		Teaching Mathematical Thinking Through Origami - Suggestions for using origami to teach concepts in mathematics, with diagrams of models.
	arts.gourt.com /Crafts/Origami/Geometry-and-Modulars.html (434 words)

	Smith College Engineering Education Partnership \| Teaching K-12 Students \| High School \| SSEP \| Folding and Unfolding
		In this course, the mathematics of origami (paper folding) and kirigami (paper cut-outs) is used to teach basic geometric concepts and engineering principles.
		Origami is often viewed as following a folding recipe to make interesting shapes (e.g., animals) out of paper.
		Similar reasoning is used to design cardboard boxes to hold particular objects (e.g., a CD), or to manufacture a 3D shape by folding aluminum.
	www.smith.edu /engin-eep/k12/hsfoldingpage.html (396 words)

Try your search on: Qwika (all wikis)