81. Origami: Complexity In Crease (Again) origami is the art of folding uncut sheets of paper, usually squares, into decorative origami geometric constructions are part of a family of pure http://pr.caltech.edu/periodicals/EandS/articles/LXVII1/origami.html

by Robert J. Lang Over a decade ago, I wrote an article for magazine about origami, the Japanese art of paper folding, and its appeal to scientists and mathematicians. Toward the end of the article, in a fit of wild speculation, I asked: Little did this would-be futurist know what the following decade would bring. The past 10 years have seen an astonishing cross-fertilization of ideas between origami, math, and computer science. We have origami solutions to ancient problems, such as how to double a cube or trisect an angle, and origami solutions to new ones, including how to fold airbags to fit into steering columns, or telescope mirrors to fit into spacecraft. And certain origami crease patterns have been found to encode some of the hardest problems known to computer science. But most remarkably, yes, there is indeed a computer program that can, in 30 seconds or so, design origami models more complex than anything conceived over the previous thousand years. When I wrote that article in 1989, the field of origami mathematics was almost nonexistent, but over the past 10 years, researchers from many fields have developed the principles that led to that program and to the application of origami to real-world engineering problems.

82. Opera Directory Gift boxes hand made by folding paper or fabric using origami techniques. masks, geometric shapes, oversized folds, and tessellations. origami Models http://portal.opera.com/directory/?cat=8670

83. ARTTalk - FREE Copy - Vol. 11, No. 11 - Paper Craft the Japanese with the origins of origami, paper folding actually originated Originally, only geometric folding was allowed because of the religious http://www.arttalk.com/archives/vol-11/artv1111-3.htm

.com ...the link between you, the visual artist, and the manufacturer of art materials. Established 1990 SIGN UP - FREE ARTtalk e-Newsletter Search all of ARTtalk!! Paper Craft OrigamiBeauty in Simplicity Although most people credit the Japanese with the origins of origami, paper folding actually originated in China around the first century. Sometime during the sixth century it arrived in Japan and was immediately embraced by the aristocracy. The Japanese named it "ori," meaning "to fold," and "gami," meaning "paper." Because paper had great value and was rare, its use was limited to the rich. But when the Japanese perfected inexpensive ways to produce paper, origami assumed a new role in the social structure of the country. Everyone could enjoy the simplicity, economy and "Zen" of folding. From the 1600's to the late 1800's there was a flourishing of Japanese arts and culture, and it was during this time that origami attained its most glorious stature. For generations, all patterns and directions for folding were handed down from one generation to the next. Historical documentation occurred when two texts, one written in 1797 ( How To Fold 1000 Cranes ) and one in 1845 ( Window on Midwinter ) documented the early Japanese "bases" or designs.

84. The Geometry Junkyard: Unfolded Polyhedra This geometry Forum problem of the week asks for the number of different hexominoes, How to form regular polyhedra from folded strips of paper? http://www.ics.uci.edu/~eppstein/junkyard/unfold.html

Unfolded Polyhedra A common way of making models of polyhedra is to unfold the faces into a planar pattern, cut the pattern out of paper, and fold it back up. Is this always possible? Arranging six squares . This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube. Build your own polyhedra , Paul Bourke. Dodecahedron calendar , generated by a postscript program. The 85 foldings of the Latin cross , E. Demaine et al. Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes , Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke, cs.CG/0007019. Find all polytopes . Koichi Hirata's web software for finding all ways of gluing the edges of a polygon so that it can fold into a convex polytope. Flexagons . Folded paper polyiamonds which can be "flexed" to show different sets of faces. See also Harold McIntosh's flexagon papers , including copies of the original 1962 Conrad-Hartline papers, also mirrored on Erik Demaine's website HyperGami program for unfolding polyhedra, also described in

85. The Fold-and-Cut Problem (Erik Demaine) Take a piece of paper, fold it into any flat origami, and make one complete straight cut (ie, a cut along a line). Now unfold the pieces, and see what you http://theory.lcs.mit.edu/~edemaine/foldcut/

Erik Demaine 's Folding and Unfolding The Fold-and-Cut Problem Take a piece of paper, fold it into any flat origami, and make one complete straight cut (i.e., a cut along a line). Now unfold the pieces, and see what you get. Are all shapes possible? Refering back to the original sheet of paper, what patterns of cuts can be achieved by this process? Several examples are now available for viewing and printing. Background The first published reference to folding and cutting of which we are aware is a Japanese book, Wakoku Chiyekurabe (Mathematical Contests), by Kan Chu Sen, published in 1721. This book contains a variety of problems for testing mathematical intelligence. One of the problems asks to fold a rectangular piece of paper flat and make one complete straight cut, so as to make a typical Japanese crest called sangaibisi, which translates to ``three folded rhombics.'' The author gives a solution consisting of a sequence of simple folds, each of which folds along a line. Scanned images of the relevant pages in the book are available. Another early reference to folding and cutting is a July 1873 article ``National Standards and Emblems'' in Harper's New Monthly Magazine , volume 47, number 278. This article tells the story about Betsy Ross and her relation to the American flag. It claims that in 1777, George Washington and a committee of the Congress showed Betsy Ross plans for a flag with thirteen six-pointed stars, and asked whether she could make such a flag. She said that she would be willing to try, but suggested that the stars should have five points. To support her argument, she showed how easily such a star could be made, by folding a sheet of paper and making one cut with scissors. The committee decided to accept her changes, and George Washington made a new drawing, which Betsy Ross followed to make the first American flag.

86. Activity Subject: Teacher uses a 6 or 8 inch square of paper (recycled, waxed, or origami) to demonstrate regarding geometric terms revealed with each fold of the paper. http://www.scopeonarope.lsu.edu/classroom/lessonplans/ORIGAMI/georigami.html

A Tactile Geometric Experience SOARING at LSU SOAR Summer Workshop Grade Level Author's Name: Linda S. Bonnette Activity Subject: Geometry LA State Standards: G-4-E, G-6-E, G-2-M, G-4-M and Art: CE-1VA-E6, CE-1VA-M6 Summary of Exercise: Students will construct a three dimensional geometrical model known as the Sonobe Unit. Students will follow along while the teacher demonstrates using SOAR Set-Up 3 (1X Stand and View). This construction will promote geometric concepts and communication in verbal, visual, tactile, and written formats. The students will use their math journals to write a description of the Sonobe Unit and illustrate each fold. Students may then construct more elaborate units and/or teach another student or class how to fold the Sonobe Unit. Materials: SOAR , TV, VCR, videotape, recycled telephone book paper, gift wrap paper, or old magazines cut into 6 or 8 inch squares (6 sheets per student) to practice before using good origami paper, popsicle sticks (to "crisply crease" each fold), Origami paper (various colors), and waxed paper Procedure: Setup the SOAR with the 1X lens using the stand on a flat surface. Teacher uses a 6 or 8 inch square of paper (recycled, waxed, or origami) to demonstrate each fold of the Sonobe Unit. The "Alphabet" of origami should be demonstrated before students start work on the Sonobe Unit. The teacher should be asking questions regarding geometric terms revealed with each fold of the paper. These may include terms such as: parallel, perpendicular, congruent, right angle, triangle, square, rectangle, etc.

87. NTTI Lesson: Patty Paper And Origami A Tool For All Students If origami paper is used each student may make one unit and the class When folded along a line of symmetry and cut there will be two right trapezoids. http://www.thirteen.org/edonline/nttidb/lessons/kc/paperkc.html

PATTY PAPER AND ORIGAMI: A TOOL FOR ALL STUDENTS Grades 8-10 Tangrams and symmetry will be explored using patty paper. Using a basic building block in origami, polygons and three dimensional figures will be engineered. Landscape of Geometry Program 7 - "Cracking Up" The Riddle of the Wizard's Oak Program 4- "Shapes make shapes" Mathematical Eye Series II Program 4- "Paper Engineering" Tacile experience will be used to demonstrate geometry. These activities are designed to include all students and learning styles. Students will each create a tangram puzzle and use it to discuss topics of form and area. Students will create a building block that can be used to make origami. A piece of patty paper for each student, and a pair of scissors for each pair will be needed for the tangram. For the extention a piece of centimeter graph paper for each student may be used. For the origami, each student will need thirty pieces of square paper. If origami paper is used each student may make one unit and the class complete the stellated icosahedron. Tangram The patty paper and scissors are passed out. Students review the vocabulary of square, isosceles right triangle, rhombus, trapezoid, and parallelogram.

88. Origami Tag Search Results - Shadows.com paperfolding.com. 2 users Last action 7/7/2005 by aprice The geometry Junkyard origami. 1 users - Last action 6/17/2005 by tagcrawler http://www.shadows.com/tags/origami

Welcome to Shadows create an account help log in Tag: origami Search Google Narrow Results: amazing andersen art asian ... domestic Top Users of this Tag: tagcrawler riddle aprice Calufraxis ... dpanos My results Please Log in to see items you have tagged. What is Shadows? Shadows is a community based service that gives you the power to create your own search engine, find sites that interest you, and share your entire Internet experience with anyone on the web. Learn More -> Click here to quickly begin using tags, comments, and ratings. Community Results Sort by: most tags most recent Displaying 1 to 20 of 69 results paperfolding.com 2 users - Last action 7/7/2005 by aprice Tagged as: origami paper folding ericandersen ... Paper Forest 2 users - Last action 7/29/2005 by aprice Tagged as: paper origami models folding ... Kim's Crane 1 users - Last action 7/22/2005 by elvesoncrack Tagged as: origami crafts asian kimscrane.com/catalog ... Money Origami 1 users - Last action 7/19/2005 by riddle Tagged as: origami money dollar members.cox.net/crandall11/money/ ... Origami: Diagrams 1 users - Last action 7/18/2005 by Calufraxis Tagged as: origami crafts cool dev.origami.com/diagram.cfm

89. Origami Math As students use their fine motor skills to fold and crease paper into fun their origami projects to look at the interesting patterns and geometric http://teacher.scholastic.com/products/instructor/April04_origami.htm

Scholastic Home About Us Site Map Search ... Product Information Origami Math "Unfold" math learning with hands-on fun and activities By Karen Baicker Origami Resources The art of origami is truly hands-on learning. As students use their fine motor skills to fold and crease paper into fun shapes and structures, they build skills involving spatial reasoning, following precise directions in sequence, fractions, geometry, and more. Best of all, the results-fantastic frogs, dragonflies, birds-are works of art! Your students can try out their own origami creations with the Whale Reproducible and Frog Reproducible , below. Here are some tips on teaching with origami: Prepare for the Lesson Teach The Lesson Key to Using Origami Symbols Mini-Reproducible Folding Fundamentals Karen Baicker is the author of Instructor . To order a copy of this book, call 1-800-SCHOLASTIC. Download the Key to Using Origami Symbols Reproducible. To open the Reproducible, you will need Adobe Acrobat Reader software. If you do not have this software already installed, click here to download it FREE Download the Wonderful Whale! Origami Reproducible.

90. Geometric Paper Folding: Dr. David Huffman Geometric paper folding Dr. David Huffman. Nov 1996. Horiz Bar. David Huffman has been creating some very complex and original folded structures. http://www.sgi.com/misc/grafica/huffman/

Geometric Paper Folding: Dr. David Huffman Nov 1996 David Huffman has been creating some very complex and original folded structures. He works with both straight and curved folds, using mathematical techniques that he has developed over many years. Mr. Huffman teaches at the University of California at Santa Cruz. Here are three examples of his work. The piece above was created by 4 parabolic curved folds that meet in a central square. This "earthquake" pattern consists of many straight folds. The position of each fold was calculated and plotted on the surface by hand. Here 4 free hand curves were used to make this unusual surface. A very simple surface can be made by making two folds in the center, one straight and one curved. These two folds conspire to remove material from the center of the sheet. A symmetric saddle structure results.

91. From Fred Galvin Galvin@math.ukans.edu Subject Re Straight Dmitry Fuchs and Serge Tabachnikov, More on Paperfolding , American Mathematical Monthly, Robert Euclidean constructions and the geometry of origami. http://www.math.niu.edu/~rusin/known-math/99/paperfold

From: Fred Galvin

92. Origamido Studio: Teach With a wide range of experience teaching origami, papermaking, and paper sculpting to Students will fold multiples of origami unit puzzle pieces and http://www.origamido.com/what-we-do/teach.html

Origami at the Studio We offer instruction in all areas of origami and at all levels of skill. Instruction in other areas of paper art, paper craft, and papermaking can also be arranged. At the Studio, examine our collections of fantastic Origami creations, enjoy a class, and explore the neighborhood Arts District as well! More... Origami at your location Author and artist, Michael LaFosse is one of the most seasoned and respected presenters of the paper arts. With a wide range of experience teaching origami, papermaking, and paper sculpting to individuals and groups, he can tailor instructional and/or entertaining programs to people of all ages and abilities, from beginner to master level. More... Papermaking We agree that part of origami's greatness lies in the fact that you can take any piece of paper and with a few folds, form a wonderful creation. However, there are, for special work, many advantages to making your own paper... More... Origami at The Studio We offer instruction in all areas of origami and at all levels of skill. Instruction in other areas of paper art , paper craft, and papermaking can also be arranged. Although we are a working studio, visitors are welcome whenever our "Open" flag is up. We recommend phoning ahead to check our schedule if you will be traveling a distance to visit us, or would like to arrange for a class.

93. Froebel USA History / Biography (14) origami / Paperfolding (9) Parenting (4) Mathematical origami Mathematical origami Buy 1 Mathematical origami now, $13.50 http://rs184.securehostserver.com/~froebelu/catalog/default.php/cPath/22_34

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94. History Of Origami origami (pronounced ori-GA-me) is the Japanese art of paperfolding. The Moors used paperfolding to create geometric figures because their religion http://library.thinkquest.org/5402/history.html

History of Origami by Callie and Jeremy If you can dream it, you can make it so. ...Belva Davis Origami (pronounced or-i-GA-me) is the Japanese art of paperfolding. "Ori" is the Japanese word for folding and "kami" is the Japanese word for paper. That is how origami got its name. However, origami did not start in Japan. It began in China in the first or second century and then spread to Japan sometime during the sixth century. At first, there was very little paper available so only the rich could afford to do paperfolding. The Japanese found useful purposes for their origami. For example, the Samurai (sa-MURE-ay) would exchange gifts with a form known as a noshi (NO-shee). This was a paper folded with a strip of dried fish or meat. It was considered a good luck token. Also, the Shinto Noblemen would celebrate weddings by wrapping glasses of sake or rice wine in butterfly forms that had been folded to represent the bride and groom. As easier papermaking methods were developed, paper became less expensive. Origami became a popular art for everyone, no matter if they were rich or poor. However, the Japanese people have always been very careful not to waste anything. They have always saved even the tiniest scraps of paper and used them for folding origami models. For centuries there were no written directions for folding origami models. The directions were taught to each generation and then handed down to the next. This form of art became part of the cultural heritage of the Japanese people. In 1797

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