21. Folding And Unfolding (Erik Demaine) origami (paper folding) has lead to an intriguing collection of problems to study David Eppstein s geometry Junkyard page on origami links to many http://theory.lcs.mit.edu/~edemaine/folding/

Erik Demaine 's Folding and Unfolding Page Introduction Folding and unfolding is an exciting area of geometry. It is attractive in the way that problems and even results can be easily understood, with little knowledge of mathematics or computer science, yet the solutions are difficult and involve many sophisticated techniques. The general sort of problem considered is how a particular object (e.g., linkage, piece of paper, polyhedron, or protein) can be reconfigured or folded according to a few constraints, which depend on the object being folded and the problem of interest. In particular, we are interested in efficient algorithms for characterizing foldability, and finding efficient folding processes, or in proving that such algorithms are impossible. There is a wide range of folding and unfolding problems, some going back several centuries and still unsolved, like unfolding convex polyhedra, while others are more recent like protein folding. In the last few years, there has been tremendous progress on many of the fundamental problems in folding and unfolding, yet some of the most important questions still remain open. This leaves the area in an exciting state. Many results in folding and unfolding can be characterized in the following way. My favorite type of results are

22. Polyhedra And Paper Folding Polyhedra and the geometry of paper folding. Dec 95; Hanson, Robert M.; Molecular origami Precision Scale Models from paper, University Science Books, http://www.stolaf.edu/people/cederj/Courses.dir/Geo.dir/bib-356/node4.html

Next: Transformation Geometry Up: Index of Topics Previous: Impossible Geometric Constructions Polyhedra and Paper Folding Ball, Walter, Mathematical Recreations and Essays Baer, Steve, Zome Primer: elements of zonohedra geometry: two and three dimensional growths of stars with five fold symmetry QA491.B28 (oversize) Faulkner, J. Earl, ``Paper Folding as a Technique in Visualizing a Certain Class of Transformations,'' Mathematics Teacher 68, May `75, 376-77. Geretschlager, Robert, ``Euclidean Constructions and the Geometry of Origami,'' Mathematics Magazine , Dec '95 Hanson, Robert M.; Molecular Origami: Precision Scale Models from Paper, University Science Books, Sausalito, CA 1995. Hilton and Pederson, Approximating Any Regular Polygon by Folding Paper , Mathematics Magazine, May '83 Johnson, Donovan, Paper Folding for the Mathematics Class Miyazaki, Klji, An Adventure in Multidimensional Space: the art and geometry of polygons, polyhedra, and polytopes Olson, Alton, Mathematics Through Paper Folding , NCTM 1975.

23. Jpib's Directory Origami paper folding Not really origami, manual to learn geometry and maths through paper models of geometric shapes puzzles from FreepaperToys.com http://www.linkagogo.com/go/Members/jpib/Origami

Jpib's directory Directory Home LinkaGoGo Origami Directory ... Origami Powered by link a GoGo null Search just this folder the whole directory Comments Jean-Pierre Prieto's Origami resources on the web. Contact: jp.prieto at free.fr. Links Most Popular Complex Origami Diagrams - Not really complex, but very funny tablecards and other folds - includes "Pteranodon goes human". Ben Ball's Origami! - Very useful crease patterns teaching - don't forget to look at the models, too ! Make Yo'self a dodecahedron! - It's all in the title. Anniefolds diagrams - spintop diagrams - Annie has also a very nice collection of "Joisel" masks. Oriland - Oriland is the biggest, newest, and most imaginative paper world ever - so it says. Refreshing ! Alphabetical Benjamin Yytan's Learning Origami - A good one too. -= origami-art.org =- - -= origami-art.org =- Alex Barber's origami.com - A great resource for diagrams. Alex Bateman's Origami Page. - All about tessellations. Andy's Tessellation Page - Regular and Semi-Regular Tessellations in Paper. Anita's Origami - Simple but effective folds.

24. Joseph Wu Origami origami is the Japanese name for the art of paper folding. Scientists, architects, and mathematicians explore the geometry of origami for its own beauty http://www.origami.as/Info/intro.php

home articles What is origami? Origami , the Japanese name for the art of paper folding, comes from the Japanese verb oru (to fold) and the noun kami (paper). The word "origami" is now commonly used around the world. A finished origami figure is called a model , the method for folding a model is called a design , and drawn instructions for a model is called a set of diagrams . An origami artist is usually called a paperfolder The only requirement for origami is a piece of paper, making it one of the most accessible arts. Almost any paper may be used, but standard "origami paper" is thin, strong, and holds a crease very well. It is also usually white on one side and colored on the other side, and is cut into 15 cm squares (about 6 inches). Some origami artists also experiment with other materials, and have folded models out of cardboard, various types of cloth, wire mesh, sheet metal, and even sheets of pasta. The basic technique of origami is folding, and many complex folds have been developed. The simplest fold is the valley fold , where a flat piece of paper is folded towards the paperfolder. When this fold is unfolded, the crease line forms a valley shape. Closely related is the

25. Origami Links Featuring their book paper folding which explains how to use paper folding to teach geometry in Mathematics and geometry. Date added October, 25, 1995 http://www.origami.as/link.php?category_id=002001&action=show_list&order_by=owne

26. Origami (reflection) To learn more about Buckminster Fuller s geometry, check out the Design Science Since origami is the art of paper folding, I created a design that http://www.scottkim.com/inversions/gallery/origami.html

ORIGAMI. S YMMETRY. Reflection about a vertical axis. I NSPIRATION. Commissioned by high school mathematics teacher David Masunaga for a talk about origami, and published in Peter Engel's book Origami from Angelfish to Zen. S TORY. In April 1988 I attended the 25th anniversary reunion of the Design Science program at Harvard University's Carpenter Center for the Visual Arts . Although I hadn't actually attended the program, let alone Harvard, I felt such kinship with this merry band of artist-mathematicians that I showed up anyway. The program, founded by Harvard professor Arthur Loeb, is a uniquely interdisciplinary of 3d polyhedral geometry and sculpture. Equal parts art, mathematics and engineering, the program has produced one of the most interesting collection of alumnae I've ever seen. Talks at the event included: Peggy Weil talking about her vision of the Weatherium: a multi-story inverted polyhedral globe that you view from a platform floating on the inside, with a live projected image of earth's surface as seen from weather satellites. Amy Edmundson, who wrote a most illuminating book about Buckminster Fuller's geometry called A Fuller Explanation, giving a hands-on workshop on building structures out of toothpicks and marshmallows. You quickly learn that in order to build anything stable you have to use triangulated structures like tetrahedra.

27. K's Origami : History Of Origami We did not call paper folding origami in Japan until Showa era. Elementary geometry is taught in the origami of forms of knowledge. http://origami.ousaan.com/library/historye.html

K's Origami Fractional Library History of Origami Before Origami We know very little about the origin of origami. Some say origami originated in China around 2000 years ago. But it is probably wrong. This opinion is based on the conjecture that origami started right after the invention of paper, for which we have no evidence. The paper of Former Han dynasty shows no trace of origami. The Chinese character for paper, zhi, originally stood for writing material made of silk. The origin of the Japanese word for paper, kami, is said to be birch tree, kaba, or strips of wood or bamboo, kan. Both of them were also writing material. These suggest that paper was primarily writing material, not folding. Others say origami originated from Japan in Heian era. Again, it is probably wrong. They refer to a story of Abe-no Seimei who made a paper bird and turned it to a real one, or another story about Fujiwara-no Kiyosuke who sent his ex-girlfriend a fake frog. There is no reason, however, for believing that they folded paper to make them. In Japan, we use wrapping paper called tatogami or tato. Today we mainly wrap kimono with it. It actually dates back to Heian era. But it is by no means an example of origami, since it is folded just squarely.

28. Articles: Folding, By Bryan Clair origami. People have been folding paper for centuries. Constructions like these are reminiscent of high school geometry, where you used a straightedge http://www.strangehorizons.com/2002/20020311/folding.shtml

CONTENTS Art Gallery Articles Columns ... Workshops Folding By Bryan Clair 11 March 2002 S o, you like web magazines? You believe paper will become obsolete, resulting in acres of beautiful forest? Well, forget it. Go get a ream, and let's waste it with idle mathematical diversions. It's time to learn the ins and outs of paper folding. Origami People have been folding paper for centuries. Not long after paper was invented in China, the Chinese were folding it. In Japan it became the art of origami, first practiced only by the elite, and then by the masses as paper became cheap and abundant. The patterns used to fold square sheets into representational figures propagated as oral traditions for centuries, though only a few simple designs survived to the earliest printed works. The best known origami design is the crane, and in fact the oldest surviving book on origami is the Hiden Senbazuru Orikata (How to Fold One Thousand Cranes). The book begins with basic instructions for folding the origami crane. With beautiful calligraphy and illustrations, it then shows dozens of ways to fold many cranes from one sheet of paper. In these multi-crane folds, the sheet is cut into small squares which remain joined by thin strips at the corners. Then the dexterous artist folds the individual cranes without tearing the strips. Hiden Senbazuru Orikata, page 15-2. The dark cuts divide the square into one large square for the mother crane and six smaller squares for the children. Notice the cuts do not quite separate the squares.

29. Origami Glossary folding geometry, The angles which arise naturally (or not! Modular origami, Where many sheets of paper are folded into (often) identical units or http://www.britishorigami.org.uk/academic/glossary.php

Origami Glossary Here are some useful origami terms and phrases. Please do not reproduce this list, even in part, but you may link to it. If you have any additions, please send them to Nick Robinson . There are currently 117 glossary entries A rectangle with has sides in the proportion of 1: root 2 - the diagonal of a square forms the long side of an A proportion rectangle Action model A design which "does" something when finished, such as a banger or paper plane Adult origami Origami designs of a sexual nature AEP Associacion Espanol Pairoflexia - Spanish folding society Back-coating Gluing two different sheets (such as foil and tissue paper) together to form a single sheet Base A combination of folds that can be used as a starting point for creating Bird base A classic base formed by petal folding both sides of a preliminary base Book fold a fold that involves folding one side of a square to the opposite side BOGS British Origami Gourmet Society - a group of paperfolders who practise folding during meals (see poppadom) BOS British Origami Society Blintz to fold all four corners of a square into the centre Box- pleating A technique developed by American folder Neal Elias which pleats the paper and collapses into narrow points Bronco sinkin A legendary origami figure CDO Centro Diffusione Origami - Italian origami Society Chapeaugraphy The folding of a felt ring into shapes representing hats Circular origami As the name suggests, the use of circular paper as the starting point for origami

30. Math On The Street Where S The Math? Mathematics in origami. People who spend time folding paper often ask One popular approach is to develop origami geometry as an axiomatic system and http://math.serenevy.net/?page=Origami-WhereMath

31. Math On The Street Using Origami To Teach Math Links Bill Wagner has a lesson plan that uses simple paper folding to develop concepts He gives examples of mathematical origami questions in geometry, http://math.serenevy.net/?page=Origami-TeachingLinks

32. Origami The geometry Junkyard origami The Japanese art of paper folding is obviously geometrical in nature. Some origami masters have looked at constructing http://www.reference.com/Dir/Recreation/Crafts/Origami/

Dictionary Thesaurus Encyclopedia Web Home Premium: Sign up Login YOUR AD HERE Dictionary ... Encyclopedia - Web Directory Web Directory Top Recreation Crafts / Origami Paper Airplanes The Underground Origami Page A place for erotic and objectionable origami. Joseph Wu's Origami Page Origami is the Japanese name for the art of paper folding. However, it has now become a beloved artform for people around the world. This site is the most comprehensive collection of origami materials available on the World Wide Web. Paperfolders Around the Lower Mainland (PALM) PALM is Vancouver's own origami club, dedicated to the promotion of the art and craft of origami. Origami and Mental Health Therapy Page This page looks at using origami as a tool in therapy for treating mental health patients. Included - people involved in origami in history, and some mental health topics. Pictures and origami links also included Origami Stop in and tour the section on origami, the art of paper-folding. British Origami SocietyThe BOS is a society devoted to the art of paper-folding. We have over 600 members worldwide and publish a bi-monthly magazine. World's Largest Paper Crane World Peace Project The crane symbolizes Good Luck and Well Being In addition to the crane being a symbol of peace.

33. MSN Encarta - Origami origami, art of folding paper to create threedimensional figures of animals and may have first practiced paper folding as a means of studying geometry. http://encarta.msn.com/encyclopedia_1741500677/Origami.html

MSN Home My MSN Hotmail Shopping ... Sign In Web Search: Encarta Subscriber Sign In Help Home ... Upgrade your Encarta Experience Search Encarta Upgrade your Encarta Experience Spend less time searching and more time learning. Learn more Tasks Find in this article Print Preview Related Items Japan Paper more... Further Reading Search for books and more related to Origami Encarta Search Search Encarta about Origami Advertisement document.write(' Origami Encyclopedia Article Multimedia 1 item Origami , art of folding paper to create three-dimensional figures of animals, people, objects, and abstract shapes. Origami is a Japanese word that combines the verb oru (to fold) and the noun kami (paper). The only material required for origami is a piece of paper. Almost any paper may be used, but folding is easiest if the paper is thin, strong, and able to hold a crease. Standard origami paper is usually cut into 15-cm (6-in) squares; it is generally plain white on one side, with a color or decorative pattern on the other. Some origami artists experiment with other materials, including cardboard, cloth, wire mesh, sheet metal, and even pasta. To fold origami, a person only needs to learn a few standard types of folds. These folds are used to form

34. Math Trek: Paper Bags And Tricky Folds, Science News Online, July 23, 2005 It came up in the course of designing a robot that could do origami—fold pieces at a workshop on computational geometry, devoted to folding paper bags. http://www.sciencenews.org/articles/20050723/mathtrek.asp

Science News Books. Subscribe to Science News ... Science News for Kids Math Trek Paper Bags and Tricky Folds Food for Thought Money Matters in Obesity Science Safari Animal Skulls TimeLine 70 Years Ago in Science News Science News e-LETTER. ... Week of July 23, 2005; Vol. 168, No. 4 Paper Bags and Tricky Folds Ivars Peterson The humble brown paper bag, which you might use to carry lunch or lug groceries, is a remarkable contrivance. It can stand upright by itself. You don't need an extra hand to hold it open while you fill it. Yet it readily folds flat for easy storage. Fitted with handles, it becomes a handy shopping bag. A paper grocery bag has a flat bottom, allowing it stand erect without additional support. S. Norcross All this comes from an ingenious set of folds performed on a tube shaped from a roll of paper to produce a flat-bottomed bag. Suppose, however, that you had a bag made of a very stiff material, perhaps steel or plastic plates, and that each crease was some sort of hinge. In this case, you wouldn't be able to collapse the bag into its flat configuration. It turns out that, given the existing creases, it's impossible to fold a grocery bag flat without bending the facets. At the same time, the fact that the facets of a stiff paper bag resist bending and crinkling help keep the bag open or closed.

35. LII - Results For "origami" Features paper folding background, videos of the origami process, Information on the mathematics of paperfolding, including the geometry and origami http://www.lii.org/search?searchtype=subject;query=Origami;subsearch=Origami

36. March 25, 2004: Flattened Origami, Tessellations And Multidimensional Space. This is similar to a type of geometry known as flattened origami. Recreational origami paper folding in Japan around 1600 (other forms, http://skazat.com/justin/currents.cgi/2004/3/25-origamitinkerings.html

Free at last There's No Such Place As Far Away March 25, 2004: Flattened Origami, Tessellations and Multidimensional Space. I've been working on some paintings that look like this: (there's one that's not shown since it's mono chromatic and wouldn't really photograph well) The paintings' composition is created simply by folding square pieces of paper. This is similar to a type of geometry known as flattened origami. Flattened origami is a fairly new branch of math that uses the creases from folding paper to solve complex calculus problems. I don't know calculus, but I know how to fold paper. In fact, I'm sort of an expert of sorts. Just as one remembers long lost memories of a forgotten childhood with the slight whiff of a certain herb or spice (for me it's rosemary), the same can be said for tactile activities. Riding a bike after not being on one for years opens the floodgates to first memories of learning to ride a bike. It's muscle memory of a different type. Ever have a word you're trying to grasp? What do you do? You move your hands as if you're trying to feel it. If it's an object, you're trying to hold it. If it's an emotion, you're trying to act it out. When I was younger, I folded paper constantly. You may have as well. Don't remember yet? Ever make a paper snowflake, or a paper airplane? Perhaps a "cootie catcher"? Or how about making a birthday card? And let's not forget Oriental origami. (little flapping birds).

37. With Origami Desk, Users Learn To Fold Paper Into Beautiful Shapes. camera, constraints, image origami, projection, occluding, geometry, Since Chris and Andy both liked origami (Japanese paper folding) so much, http://www.clickerado.com/o/origami/origami_desk.htm

home about us advertise with us downloads ... We will learn different methods of cartooning animals. Documents organized by subject word: advertising affiliate alternative medicine animation ... zoo OrigamiMobile Mobile Communicator, National Semiconductor, Origami Mobile, video, connectivity, access device, Geode, Internet, sound, stereo, personal access device, camera, Bluetooth, Sight, CoCom. The Geode(TM) Origami(TM) Mobile Communicator is a multi-function device that combines some of today's most popular electronic products all into one package. The online world of the Internet such as surfing and email. The real world of special moments recorded, sent via email to anyone around the world, and then shared with laughter in a video call. Twist it one way for a personal access device and another way to compose an email. National Semiconductor, a leader in processor, mixed-signal, and analog technologies, develops total solutions that provide the power, connection, and human interface necessary to drive the information appliance market. origami fold, crease, beak, flap, upper, arrow, Lift, crane, head, Repeat, book fold, legs, layer, unfold, Flip.

38. Origami Web Resources For Students origami is the art of folding paper. The word is Japanese, literally meaning to fold origami Sociteit Nederland The geometry Junkyard origami http://www.cdli.ca/CITE/origami.htm

Gander Academy Origami What is Origami? Paperfolding Instructions Paper Airplanes Search Index ... Teaching Resources What is Origami The East and West of Origami Most of us will remember folding paper cups, salt cellars (we called them 'cootie' catchers or 'fortune tellers') and paper balloons as children in elementary school. There is more to origami than these simple models would lead us to believe. Origami comes from the Japanese words for folding, ori, and the Japanese word for paper, kami. History of Origami Since about the first century AD, the time when it is believed that paper was first invented in China, people have been folding paper into various shapes. The Chinese developed some simple forms, some of which survive down to this day. When the secret of paper was carried to Japan in the sixth century AD by Buddhist monks, it was quickly integrated into their culture. Origami: Fold art, geography and cultural studies into one lesson Origami, the ancient Japanese art of paper folding, can be an interesting way to combine art lessons with units on social studies, culture and even history and geography. The Ten Commandments of Origami 1.Choose suitable paper and cut to required form and size.

39. Computational Origami At The MIT He works on computational origami the geometry of paper folding which interests not only hobbyists or MIT professors, but also the manufacturing http://radio.weblogs.com/0105910/2003/01/18.html

How new technologies are modifying our way of life samedi 18 janvier 2003 Computational origami at the MIT It's the weekend, so today we'll talk about a pastime: origami. Erik Demaine must be a happy guy. At 21, he became the youngest professor at the Massachusetts Institute of Technology (MIT). And he's being paid to have fun. He works on computational origami the geometry of paper folding which interests not only hobbyists or MIT professors, but also the manufacturing industries. Steve Nadis interviewed him for the New Scientist, looking back at his itinerary. Here are some selected questions and answers. Q: What was your first real accomplishment in mathematics? A: Six years ago, when I began my PhD work in computational geometry at the University of Waterloo in Ontario, my dad remembered "the paper cut problem" from an article written in the 1960s on paper folding and mathematics. The idea is to take a piece of paper, fold it any way and as many times as you want, and then make one straight cut and see what shapes you get. The question is, are all shapes possible? I worked on this problem for two years at Dalhousie with my dad and adviser Anna Lubiw. After experimenting for a while, we realised you could make all kinds of shapes, such as butterflies, swans, hearts or stars

40. Classic Cranes History rigamitsuru.com The art of folding paper cranes into a masterpiece, View my photo To the mathematician, the beauty of origami is its simple geometry. http://www.origami-tsuru.com/crane7.htm

Introduction FOLDING: A COMPACT HISTORY From the book " Origami from Angelfish to Zen" by Peter Engel Paperfolding originated in China around the first or second century A.D. and reached Japan by the sixth century. The Japanese called this new art form origami (the name coined from ori, "to fold," and gami, "paper") and cultivated it as an art of understatement. Origami suggests; it implies without announcing outright, intimates with out brashness. It exists best in a kind of light the Japanese call ke, a soft, gentle light for intimate occasions. Why use a bright light when you can see in a dim one? Why shout when you can whisper? For that matter, why draw the entire bamboo tree when a few brush strokes suffice? Just as a three-line haiku evokes a setting or a season, the placement of a rock and a pond in a Japanese garden recalls the universe. It is a short imaginative leap from the rock to a mountain, from the pond to the sea. Origami is an art of economy. A few simple creases evoke an animal; modify the sequence slightly, and an entirely new beast appears. To the Japanese sensibility, the success of a completed origami figure depends on the creator's eye for form, structure, and proportion. Does it capture the creature's true form, the placement of its head and limbs, the shape of its shoulders and hips? Does it suggest the animal's motion, its stride, glide, or gallop? And finally, is the paper figure a mere likeness of the original, or does it delve deeper, into its essential character?

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