61. Tom's Combinatorial Geometry Class and edge coloring, spherical geometry, solid angles and Analytic Modelling of paperFolding The classic edge and compass axioms, origami axioms, analytic paper http://web.merrimack.edu/hullt/combgeom/combgeom.html

(Jun Maekawa's Devil, folded from one square of paper, no cutting) Here are some assignments: ASSIGNMENT 1 Due Jan. 31 ASSIGNMENT 2 Due Feb. 14 ... Due April 25 Here are some handouts: (all are in PDF format) Handout 1 Handout 2 Handout 3 Handout 4 MA 323A Combinatorial Geometry! Class Notes: Notes on Planar Graph Theory Notes on how to do Proofs! Notes on spherical Buckyballs Part 1 and Part 2 Notes on Graph Colorings Notes on Spherical and Convex Polyhedral Geometry Notes on Axiomatic Origami Constructions ... Notes on Reflections are Foldings, Part 2 Hi! This page was developed in the Spring of 2000 for the course Combinatorial Geometry at Merrimack College. It is currently being revised and updated for the Spring 2002 offering of this course. Below is a draft of the syllabus. Combinatorics (the mathematics of counting) and geometry have many strong connections. We will explore these connections by looking at 3D polyhedral geometry and the mathematics of paperfolding (origami). In particular, we'll cover... Convex Polyhedral Geometry: construction of paper models, Euler's Formula, planar duality, coloring theorems, Hamilton cycles, Buckyball classification and edge coloring, spherical geometry, solid angles, Descartes' Theorem, surfaces of higher genus, convex polytopes, simplices, the Generalized Euler's Formula.

62. Folding And Unfolding (Erik Demaine) as soon as we have written the paper. references on the web about folding and unfolding axiomatizationof geometric constructions with origami, and instructions 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

63. Wrapping Polyhedra (Erik Demaine) you can read our paper ``folding Flat Silhouettes and New Results in ComputationalOrigami.'' The latest version appears in Computational geometry Theory and http://theory.lcs.mit.edu/~edemaine/wrapping/

Erik Demaine 's Folding and Unfolding Wrapping Polyhedra Given a shape, what is the most efficient way to wrap it with a piece of paper? The shape might be a flat polygon, and the goal is to fold that silhouette, or the shape might be the surface of a three-dimensional polyhedron. A wrapping is any folding of the piece of paper (without ripping or stretching) into precisely the desired shape. The folding is allowed to cover the shape with multiple layers of paper, but no paper can be strictly inside or outside the shape. Another twist on the problem is that the paper may be colored differently on its two sides. For example, wrapping paper typically has a patterned side and a white side, and kami (common origami paper) typically has a colored side and a white side. If you are wrapping a gift, you probably want only the patterened side of the paper showing. If you are folding origami, say out of paper that is black on one side and white on the other, you might want to use the two colors to produce a particular effect of colors showing at different places in the shape. For example, it is possible to fold a single square of paper into a black-and-white checkerboard or a zebra with its stripes. The above zebra is an origami model designed by John Montroll. It appears in his book

64. Citations: Folding And Cutting Paper - Demaine, Demaine, Lubiw (ResearchIndex) 1. 1 paper folding Many problems related to the folding and unfolding Several interestingalgorithmic questions relating to origami have attracted the http://citeseer.nj.nec.com/context/919782/70943

5 citations found. Retrieving documents... E. D. Demaine, M. L. Demaine, and A. Lubiw. Folding and cutting paper . In Proc. Japan Conf. Discrete Comput. Geom., Lecture Notes Comput. Sci., Tokyo, Japan, December 1998. Springer-Verlag. Home/Search Document Details and Download Summary Related Articles ... Check This paper is cited in the following contexts: A Motion Planning Approach to Folding: From Paper Craft to.. - Song, Amato (2000) (1 citation) (Correct) ....Before describing our approach, we first review related work in computational geometry and computational biology. 1. 1 Paper folding Many problems related to the folding and unfolding of polyhedral objects have recently attracted the attention of the computational geometry community (see, e.g. ) One class of problems concerns itself with the constructability of certain polygonal or polyhedral structures. Several interesting algorithmic questions relating to origami have attracted the attention of computational geometers, who have obtained some remarkable results (see, e.g. 6, 16, .... E. D. Demaine, M. L. Demaine, and A. Lubiw.

65. American International School In Zagreb, Croatia: General Information Information on the mathematics of paperfolding, including the geometry and origamiaxioms formulated by Italian-Japanese mathematician Humiaki Huzita. http://www.asz.tel.hr/asz/libword.htm

American International School of Zagreb Library Word of the Month O RIGAMI = paper folding Japanese word, meaning ‘to fold’ (oru) ’paper’ (kami); the art of folding objects out of paper without cutting, pasting, or decorating. Its early history is unknown, but it seems to have developed from the older art of folding cloth. There are two types of Japanese folds: figures used in ceremonial etiquette, and objects such as animals, flowers, furniture, and human figures. (Merriam-Webster’s Collegiate Encyclopedia) Some interesting origami web sites: Joseph Wu’s Origami Page Picture galleries of origami models and diagrams showing how to make some of them ( www.origami.vancouver.bc.ca Paper Folding A fun site to learn about origami, the art of paper folding! Learn to fold, see the origami insects and dinosaurs, learn about origami history and a lot more ( www.paperfolding.com Origami Mathematics Information on the mathematics of paper-folding, including the geometry and origami axioms formulated by Italian-Japanese mathematician Humiaki Huzita. There is also a section

66. Washington Parent Article In the Western world, origami is mostly practiced by In the United States, paper foldinghas also been explored who appreciate the art for its simple geometry. http://www.washingtonparent.com/articles/0203/origami.htm

March 2002 Folding is Fun Creating Origami Architecture at the National Building Museum Jill Wiley White, M.A.T. and Ayumu Ota, M.A.T. Make your next family experience memorable with a trip to the National Building Museum’s annual Festival of Origami Architecture. On Saturday, March 30, families can explore the Japanese craft of paper folding, known as origami. Why origami at the National Building Museum? Because origami allows people to explore the basic design of all forms, including those found in architecture and the landscape. History of Origami The art of paper folding originated in China but was elevated into an artform by the Japanese after it was brought to that country in the sixth century. The Japanese coined the term origami ( ori means "to fold" and kami means "paper"). In the eighth to twelfth centuries, because paper was such an expensive commodity, origami was practiced and used only by the Japanese nobility. They exchanged origami creations as good luck tokens and used them at wedding ceremonies to wrap sake (rice wine) bottles with paper butterflies representing the bride and groom. By the 1500s when paper became more affordable, origami was practiced by all Japanese and was used as a means of identifying one’s social status. Aristocrats folded differently from farmers and peasants. People knew their position and folded accordingly. In the 1600s-1800s, one of the high periods of Japanese art and culture, origami emerged into a true artform. Japanese publications of this period document the creation of basic origami forms that we know today.

67. Origami Web Resources For Students literally meaning to fold (oru) paper (kami the ancient art of papar folding origami paperfolding fun Jopephy Wu's Origmai Page origami Sociteit Nederland http://www.stemnet.nf.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.

68. Paper Airplane Web Sites Information and photos on origami. General Information on Aerodynamic Designs.An Introduction to Aerodynamics. Lesson Plans for paper Airplanes. http://teams.lacoe.edu/documentation/classrooms/amy/geometry/projects/airplane/a

Hang Time Paper Airplane Contest Find out more about Paper Airplanes from these sites: Build the Best Paper Airplane in the World Illustrated assembly and flying instructions. Paper Airplane Hanger Page Information on paper airplane safety, tips on making and flying paper airplanes, and a step by step guide to building planes. Paper Airplane Launcher Two students figure out a way to launch their paper airplanes. Paper Air Machines Step-by-step instruction for the expert to the novice on creating wide varieties of complex aircraft from a simple sheets of white standard paper. Paper Airplane of the Month Lots of different designs for you to try. Paper Scale Models This is a commercial site, but it show the level of sophistication you can reach with paper models. Sabertooth Step-by-step design of this favorite paper airplane. World Record Paper Airplane Download the instructions for creating Ken Blackburn's 1994 Guiness world record paper airplane. Paper Airplane Software KittyHawk Software Shareware Software for 4 different models of paper airplanes you can download and try.

69. From The Origami-l Archives - David Lister On Folding From Non-square Paper The accepted constraint in origami has always been that the paper should be a To summarise,the square has become the classical shape of paper for folding. http://www.worthhall.demon.co.uk/theory/lister/nosquare.htm

Folding from non-square paper I thought I would set down a few random thoughts on folding from non-square as opposed to square paper, but I don't intend it to be a reasoned thesis! 1.A few months ago, I wrote in Origami-L about John Smith's ideas on "Origami Profiles" which analyses how each individual's preferences in folding fit into the general scheme of things. The theory accepts that everyone is entitles to adopt whatever rules for folding he or she chooses. 2. Following from this a folder can choose to fold from a square or from a triangle or from A4 or a pentagon or a rectangle or a rhombus or a long ribbon of paper half and inch wide and ten yards long. He may even prefer silk ribbon or even string. Whether other people would include in their own concepts of Origami is another thing. 3. If you accept that cutting is legitimate (and cutting, too, fits into John Smith's Profiles of Origami), your can convert a square of paper into any shape you like. Or you can trim your dollar bill into a square. Or you can chop off any of those surplus bits of paper that get in the way. (No, I accept that most people who like to use scissors don't look at it in this extreme way, but I assert the possibility.) 4. Without even using scissors, you can convert most simple shapes of paper into most other shapes by folding alone. You can fold a square to make a triangle, or a hexagon or a 3 X 7 rectangle, even A4. So having done that, in theory, you can go on to fold anything that can be folded from a square equally from a Dollar bill or anything that can be folded from a Dollar bill from a square. I write "in theory" advisedly, because all the preliminary folding to get the shape makes the model bulky and difficult or even impossible to fold in practice.

70. Ask Jeeves: Search Results For "Make Origami Cube" Related Topics In http//db.uwaterloo.ca/~eddemain/folding/ 10. Japan Links origami can be anything, from the simple paper crane figured above to http://webster.directhit.com/webster/search.aspx?qry=Make Origami Cube

71. Origami Paper Background origami is the art of paper folding. In fact, origami can be considered a subsetof the overall In this paper, we will describe a set of general strategies http://rachel.ns.purchase.edu/~Jeanine/origami/oriback.htm

Background Origami is the art of paper folding. By a sequence of folds, a flat piece of paper is turned into a stylized animal, flower, box, or other recognizable object, generally 3-dimensional and often with moving parts or serving a utilitarian purpose. The final object is called a 'model'. Origami is associated with Japan, but it is practiced all over the world. The classical models include the water bomb, crane, and flapping bird. In recent times unit or modular origami, in which geometric constructions are built up from so-called modules, has become popular. Origami is both a craft and an art. Origami as practiced in the United States and elsewhere, has developed a certain culture, largely influenced by Lillian Oppenheimer, Alice Gray, and Michael Shall who over time established Origami USA . In this culture, everyone is potentially a teacher as well as a student; a high value is placed on sharing. Similarly, care is taken to giving credit to creators, people who add variations to models, teachers, collectors, and people who write down directions and diagrams. It is to be noted that the recommended practice in origami circles goes beyond the letter of the law concerning intellectual property. Certain attributes of the nature of origami and this culture provide the potential for its use in teaching (and doing) mathematics. Creating an origami model involves following a procedure.

72. Edward Crankshaw's 'The East And West Of Origami' Paper appeals to the puzzlesolving branch of mathematics as wells as in the teaching ofgeometry. There is more to origami than just the art of folding paper. http://fly.hiwaay.net/~ejcranks/arth193b.html

You're at Home About Us Origami / East and West of Origami Paper 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 . When combined, kami becomes gami and the word is origami. It shares the same origin but has taken two distinct avenues of development since its creation. Those two avenues of origami have merged together in the twentieth century. There is more to origami today than just paper-folding. Origami has a long history in the east and west and a sophistication which is more artistic than a mere craft. The following is the story of the east and west in origami. The Basic Materials The Japanese learned about paper-making in the early seventh century. They gained their knowledge of paper from a Buddhist monks who came to Japan from China through Korea and the books that they brought. This started about 538 A.D. The Japanese began making paper of their own by about 610 A.D. It is believed that Dokyo, a Buddhist monk, brought knowledge of paper-making as well as painting and ink-making. He was even a doctor and became the chief physician to Empress Shotoku. The Eastern Avenue Figure 1: Flower wrapping resembling noshi wrapping The Muromachi period in Japan lasted from 1338 to 1573. The folding style of this period represented a persons place in society. Paper was now inexpensive enough to be used by everyone and origami became a means of social stratification. It was a period of military rule and two schools of origami existed. The samurai followed the Ise school while peasants and farmers followed the Ogasawara school. There has been 400 years of mothers teaching their children how to fold paper.

73. David Mitchell's Origami Heaven - Modular Origami Design Encyclopaedia - Standar A4 paper is a good approximation of a silver of 1sqrt3 and its natural folding geometryyields angles at present, rather underused in modular origami design. http://www.mizushobai.freeserve.co.uk/rectangles.htm

David Mitchell's Origami Heaven Useful Rectangles and Folding Geometries This page introduces various rectangles that are of proven (or potential) use as starting shapes in modular origami. Whilst all folding geometries can be obtained from all rectangles, the optimum folding solution for a particular module will often be found by starting from the paper shape that most naturally yields the angles required. The Square The square is undoubtedly the most versatile starting shape in modular origami. This is not only because of its symmetry but because it easily yields angles that are useful in modelling many types of polyhedral forms. A method of making a square from any other rectangle is given in the FAQ The Silver Rectangle The Silver Rectangle (also known as the pure or DIN rectangle) has sides in the proportion of 1:sqrt2. Its natural folding geometry yields angles of 110 / 70 / 55 degrees. These angles are found in the structure of the tetrahedron, the cube and the cuboctahedron as well as in the many interesting forms known as rhombic polyhedra.

74. Geometry Resources Cubes, Cones, Cylinders, Spheres by Tana Hoban. Exploring Triangles paperfoldinggeometry, Jo Phillips. Right Angles paper-folding geometry, Jo Phillips. http://www.eho.org/geometry_resources.htm

Add Your Support Group Update Your Support Group Search WWW Search eho.org Receive the EHO Newsletter Volunteers Needed Join the Work of EHO April Featured Book Homeschooling and the Voyage of Self-Discovery document.write(code); Message Boards Feature Articles Support Resource Center ... Search Geometry Resources (Many of these books are out of print. You should find them in your local library or through inter library loan.) Books for Younger Children Shapes , John J. Reiss. Shapes: Slides, Curves and Corners , Illa Podendorf. Three Sides and the Round One , Margaret Friskey. Beginning to Learn About Shapes , Richard L. Allington. The Secret Birthday Message , Eric Carle. Books: The Silly Story of Goldie Locks , Grace Maccarone. The Shapes Game , Paul Rogers. A Cloak for the Dreamer , Aileen Friedman.

75. Exploratorium Magazine: Paper: Page 2 Cutting is the least aesthetic folding operation of destroys the wholeness of thepaper, writes Jeremy unicyclist who leads San Francisco's origami club, the http://www.exploratorium.edu/exploring/paper/paper2.html

Volume 23, Number 2 Exploring Origami - Page 2 of 5 In Japan, legend says that a person who makes a thousand origami cranes will live a long life. The legend took on new meaning in 1955, when Sadako Sasaki (shown above and below), a twelve-year-old girl who was exposed to radiation during the bombing of Hiroshima, contracted leukemia. While in the hospital, she tried to fold 1,000 cranes but died before she could finish. Sadako's statue now stands in Hiroshima's Peace Memorial Park, where it is decorated every year with thousands of origami cranes, folded by children from around the world, as a symbol of peace. Click here for a larger image of Sadako Sasaki. Click here to visit www.sadako.com he art of paper folding, like the art of papermaking, began in China, but it spread to Japan by the sixth century A.D. Over the centuries, it became an integral part of Japanese culture. Paper butterflies symbolized the bride and groom at weddings; folding a thousand paper cranes became a traditional way to ensure a long and healthy life. Animals were an especially popular theme, as "folders" developed more and more ingenious ways to make a little diamond of folded paper sprout legs or wings - and even to make it float, hop, or fly. Even so, Japanese origami evolved very slowly. According to American origami expert Peter Engel, Japanese folders invented about 150 traditional origami figures in a millennium of folding.

76. Untitled the first published collection of origami models, appeared from West Africa, broughtpaper folding with them included studies on the geometry of tessellation http://www.geocities.com/SoHo/Exhibit/9908/history.html

Whew, there! Did you really read all that stuff? I didn't either! :c) Home Origami Projects Join the Origami Club Join the Origami Club Webring ... E-mail the Club

77. Mathart: Connecting Math And Art What new geometry or art skills are being the golden rectangle, fractals, origami,tangrams, technical drawing, cartography, or wycinanki (Polish paper cutting http://u2.lvcm.com/esullivan/webquest.html

MATHART Connecting Geometry and Art A WebQuest for High School Students by Barbara Lierly and Elaine Sullivan Introduction Task Process References ... Teacher Introduction Scenario: I must go out of town for the last two weeks of the semester. You are going to be the teacher! Research, design and then teach a lesson to help review for the semester exam. Think about what you've learned about basic geometric figures, congruence, similarity, polygons, and transformations. It's time to get creative with math - integrate it with art. Make your lesson fun, relevant and/or unusual. Devise a lesson for your classmates to create works of art. You will be making new connections between math and art utilizing a hands on approach. Choose one of the following topics: cartography, technical drawing, tessellations, fractals, tangrams, the golden rectangle, wycinanki, or origami. Let's get started. back The Task Your task is to design and present a lesson that integrates art and geometry. Research material from a variety of online web resources. Utilize these as a starting point to expand on previously learned geometric concepts. Include the historical background and significance of the art form. Plan for a hands on, creative art project that demonstrates the art form. Instruct the class to complete the project. Your presentation must include: Research on the history of this topic What mathematician and/or artist has used this type of mathematics and in what way?

78. Articles: Folding, By Bryan Clair Because with paper folding, the angle trisection is a piece The paper starts witha line drawn, giving an and straightedge, can be done with origami, and the http://www.strangehorizons.com/2002/20020311/folding.shtml

Folding By Bryan Clair 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.

79. ÁʶRºP¯Èª««~ paper folding, paper folding which explains how to use paper folding to teach geometryin the classroom. origami Kaikan, ? . http://origami.uhome.net/new_page_11.htm

Origami Shops / Search Engines Gallery Origami House Origami Videos, Books, Handmade Papers Address: 1-33-8-216,Hakusan,Bunkyo-ku,Tokyo,113-0001 TEL 03-5684-6040 FAX 03-5684-6080 A wide range of private publications, including booklets, BOS convention packs and a limited range of paper Japanese Paper and Origami Supplies(Australia) Retailers of origami papers and books and washi System GET'S Co., Ltd. Sasuga Japanese Bookstore Origami Videos, Books, Handmade Papers Publishers: Sojusha- Origami Books, Magazine (Sojusha is no longer producing ORU. But still available through Sasuga and Sojusha Web Site) tel¡@048-654-1910¡@fax¡@048-654-1361 ¸Û¤å°ó·s¥úªÀ SEIBUNDO SHINKOSHA Publishers: Origami Books Fascinatings Folds Origamido §é¯È¹D Origami Videos, Books, Handmade Papers ¤p¬z©M¯È³Õª«À]OZU Washi Japanese paper shop Powell's Books A great place for used/out of print books. ¤é¥»¼Ö¤Ñºô¤W®Ñ©± Japanese Origami books Book Ends Many unusual and hard to find titles with possibly the widest range of origami books available in the UK.

80. Joseph Wu Origami the bases reveal that many paperfolders enjoy folding models of Therapists and teachersuse origami as a tool to Many people fold paper simply because it is fun 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

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