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Topic: Paper folding

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Mathematics of paper folding

Origami

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	Crane Folding Paper : Origami
		Origami : 1000 Crane Folding Paper : EF21K1-F051
		Origami : 1000 Crane Folding Paper : EF31K1-F052
		Origami : Crane Glow-in-dark Folding Paper : EG11K1-F042
	www.opane.com /crfopa.html (249 words)

	Paper Folding Geometry (Site not responding. Last check: 2007-11-06)
		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)

	Origami - Wikipedia, the free encyclopedia
		"folding paper") is the art of paper folding.
		A paper art star on a window, an example of a form of origami known as modular origami, in which the model is constructed from multiple pieces of paper.
		As these papers are floppy to start with, they are often backcoated or resized with methylcellulose or wheat paste before folding.
	en.wikipedia.org /wiki/Origami (1592 words)

	Folding Paper Cranes
		Fold the paper in half to the "east" to form a rectangle.
		Fold the paper in half to the "north" to form a new rectangle.
		Taper the diamond at its legs by folding the top layer of each side in the direction of the arrows along lines a-f and a-e so that they meet at the center line.
	www.sadako.com /howtofold.html (703 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)

	Problem D - Paper Folding
		In this problem, the size of the paper is 10485761048576 units (by pure coincidence, this means that the length of the sides of the paper* after up to 20 foldings will always be an integer).
		For instance, if your first fold is a 't', the top half of the paper is folded so the top half of the backside of the paper appears on the front.
		A folding sequence is described as a string containing between 1 and 20 characters from the set of valid folding characters ('l', 'r', 't', 'b', 'L', 'R', 'T', 'B').
	acm.uva.es /p/v106/10612.html (542 words)

	Paper Napkin Folding Instructions
		These premium disposable paper napkins are soft to the touch and sturdy to hold their shape for display and your individual artistry.
		To fold the napkin, lay it on a flat surface, and fold in half widthwise so the folded edge is near you.
		Fold the bottom third of the napkin up to the middle, roll it into a cone shape, and insert it into a glass.
	freenapkinfolding.com (794 words)

	Paper Folding Machine / Paper Folders / Paper Folder - Paperfolder Co.
		Folds 1 to 3 sheets of letter size (8.5" x 11") paper into a letter fold.
		Folds 1 to 3 sheets of A4 or letter size (8.5" x 11") paper into a letter fold or half fold.
		Folds letter size paper in all folds, legal size half only.
	www.paperfolder.com /Folders.html (514 words)

	Muse: Champion Paper-Folder
		If you start with a standard sheet of paper, after seven folds, the wad is thicker that it is wide, and it takes too much strength to fold it again.
		Paper didn't appear to work, so she decided to use gold foil—only 11 millionths of an inch thick.
		If she used paper that was the same thickness as regular paper, she calculated, she would need a roll that was nearly 4000 feet long (about three-quarters of a mile) to be able to fold it 12 times.
	www.sciencenewsforkids.org /pages/puzzlezone/muse/muse0704.asp (456 words)

	The Institute For Figuring // Online Exhibit: Mathematical Paper Folding
		RL: There’s both relevance and differences here, because paper folding is two-dimensional and a protein is roughly a one-dimensional shape, a linear chain with a bunch of joints in the chain.
		Protein folding is actually much more complicated than paper in that folds can happen only at certain angles and there are bits that stick together if you get them close.
		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.
	www.theiff.org /oexhibits/paper02.html (2565 words)

	Paper Folding (Site not responding. Last check: 2007-11-06)
		If a large sheet of paper is folded in half, then in half again, etc, with all the folds parallel, then opened up flat, there are a series of parallel creases, some pointing up and some down, dividing the paper into fractions of the original length.
		The right half is folded over so it lies on top of the left half, then the right half of the new double sheet is folded on top of the left, to form a 4-thick sheet, and so on, for N folds.
		From this view, the only unchanged part of the original paper is the piece containing the ``start edge'', and this piece will be horizontal, with the ``start edge'' on the left.
	acm.uva.es /p/v1/177.html (364 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)

	Articles: Folding, by Bryan Clair
		Folding paper is a very simple operation, but leads to surprisingly precise constructions.
		Just fold it once anywhere, then fold again at a right angle, lining up the two halves of the first fold to make sure the angle is perfectly 90°.
		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.
	www.strangehorizons.com /2002/20020311/folding.shtml (2609 words)

	Folding and Unfolding (Erik Demaine)
		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.
		In fact some of my early work in "folding" was on rock folding or the study of tectonic plates, in particular plate subduction, at the Geodynamics Group at Dalhousie University.
	theory.lcs.mit.edu /~edemaine/folding (1403 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The Techko Paper Folder is one of the best bargains we have seen in 15 years of selling folding machines.
		This is the machine to get for volume paper folding: church bulletins, brochures, self mailers, etc., It folds up to legal size paper (in half only) and has a duty cycle (recommended maximum number of pieces folded) of 25,000 folds per month.
		Fold types are C, V and double parallel (fold in 1/2 and then in 1/2 again).
	www.letterfolder.com /indexold.htm (1237 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)

	Designing Paper
		Paper folding is an age-old art, dating back to the pre-history Chinese civilizations -- more commonly known as Origami.
		Paul creates this folded paper sculpture from a piece of paper about 8.5 by 11.0 inches which goes through a metamorphosis to end with this wonderful finished sculpture.
		Their flagship holiday folding project is a sleigh and reindeer, all of which come with their own downloadable PDF files, instructions and coloring guides.
	www.graphic-design.com /DTG/Design/folding.html (1566 words)

	Creative Papercrafts : Products : Iris Folding : Techniques
		Using a cut-out image, fill in the image with folded strips of paper laid clockwise or counter-wise and taped in place around the image opening, working inwards until completed with a solid center, the iris of the eye.
		Layers of strips of folded paper are taped in place behind the cut-out opening in a card.
		We have also layered rice paper between our paper folding, be sure to tear the rice paper, not fold it, and tear by running a bead of water with a brush in a straight line and gently pull the rice paper apart.
	www.creativepapercrafts.com /products/irisfolding_techniques.aspx (796 words)

	Playful Invention and Exploration - Paper Folding
		Make the next fold about 1/2 to 1 inches away from the first and parallel to it.
		Continue the accordian folds until the whole leaf is folded up.
		You'll now re-fold the leaf along those lines so that the first "v" is an "up" fold across the leaf, and the next is a "down" fold across the leaf.
	www.pienetwork.org /a2z/p/paperfolding (268 words)

	Formax paper folders and document folding machines
		Folding that took hours to process by hand can be accomplished in minutes with three simple steps: load your forms, set your fold and press start.
		Formax tabletop paper folding machines process forms at speeds of up to 11,000 forms per hour and can accommodate paper sizes up to 11"x17".
		Our high-volume folders can be joined together for tandem folding, allowing you to cross fold or create three or four folds in one pass.
	www.relyco.com /formax_paper_folders.htm (444 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)

	Card Inspirations an introduction to iris folding
		A simple paper folding technique is combined with aperture cards to give stunning results.
		The Iris folding pattern used in this demonstration is from the book Iris Folding for Christmas by Maruscha Gaasenbeek and Tine Beauveser and used by kind permission of the publishers Forte Uitgevers.
		The folded edge should be against the line of the pattern.
	www.cardinspirations.co.uk /tech/iris_folding.htm (414 words)

	Cutfold - paper folding simulation
		You can cut the paper by (clicking cut and) drawing a series of connected lines by repeatedly clicking and moving the mouse; the cutting is finalized with a double-click (or by crossing your own cutting line).
		You can fold the paper by (clicking fold and) dragging a line segment across it; the line must begin and end outside the bounds of the paper.
		a piece of paper that becomes detached from another piece to which it should be attached (by a fold or crease) when unfolded.
	www.ifactory.com /cutfold (526 words)

	Paper Folders - Paper Folding Machines - CFS Bindery Products
		The PF-P320 folder sets fold plates at the touch of a button and folds paper at speeds of up to 250 sheet per minute, making it more desirable to quick printers that have greater paper folding demands.
		These paper folders are a great fit for churches, schools, small in-plants and quick printers, the 310 paper folder offers six standard fold formats to choose from, and can fold up to 190 sheets per minute!
		The 206M paper folder is an economical, simple-to-operate, compact, and durable paper folding machine.
	www.cfsprod.com /folders1.htm (610 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)

	Mathematics of paper folding - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		The art of paper folding or origami has received a considerable amount of mathematical 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 mathematical equations.
		Folding a flat model from a crease pattern has been proven by Marshall Bern and Barry Hayes to be NP complete.
	en.wikipedia.org.cob-web.org:8888 /wiki/Mathematics_of_paper_folding (406 words)

	Iris Folding @ CircleOfCrafters.com: Learn a New Card Making Technique
		Iris folding is a fun and easy paper folding technique that can be used to make greeting cards, on scrapbook pages, or designs can be framed and hung on a wall.
		Iris folding reminds me a lot of paint-by-number, except that you are using folded strips of paper instead of paint.
		Traditionally, origami paper is used for the main design and holographic paper is used to fill in the center of the design.
	www.irisfolding.circleofcrafters.com (402 words)

	Brooklyn Origami - Paper Folding Diagrams
		Once you have folded one of the traditional origami bases you can then manipulate the base and shape it into a model using a variety of folding techniques.
		In origami diagrams the various folding techniques are indicated by symbols or groups of symbols, illustrated here with their meanings.
		The first illustration in each pair shows a piece of paper, either flat or partially folded, with the symbols indicating the folding technique to be performed.
	www.brooklynorigami.com /folding.htm (459 words)

	Paper Folders & Paper Folding Machines - paperfolderpros.com
		Most of our paper folding machines are suitably sized to serve as tabletop paper folders or desktop paper folders consuming very little space.
		Paper folding equipment of this type is perfect for any job, long or short run, in a variety of installations like churches, schools, small offices, home offices, in-plant print or copy facilities, copy shops, for personal use, for desktop publishers and more.
		an air compressor is used to lift the paper sheet to be fed with a burst of air.
	www.paperfolderpros.com (302 words)

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