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Origami Paper Folding Geometry: more detail

Mathematical Origami: Geometrical Shapes by Paper Folding by David Mitchell, 1997-07

Fun with figures, by Mae Blacker Freeman, 1946

Origamics: Mathematical Explorations Through Paper Folding by Kazuo Haga, Josefina C. Fonacier, et all 2008-09-11

Amazing Origami by Kunihiko Kasahara, 2002-03-28

Ornamental Origami: Exploring 3D Geometric Designs by Meenakshi Mukerji, 2008-12-01

Marvelous Modular Origami by Meenakshi Mukerji, 2007-04-24

Explore Folding of the Circle: Series Book 3 (Explore Folding of the Circle, Book 3) by Bradford Hansen-Smith, 2007

Geometric Origami by Robert Geretschlager, 2008-10-14

lists with details

Formal Geometry "Math Art"
Forum Suzanne Alejandre Mandelbrot Chopping Broccoli ThinkQuest Library of EntriesPaper-Folding-Fractals (home) origami paper folding geometry Origami
http://www.biglake.k12.mn.us/HS_Page/Student bookmarks/mathart.html
Origami Mathematics Page
Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions. Photo gallery of completed modular, geometric, and tessellation models.
http://web.merrimack.edu/~thull/OrigamiMath.html

Extractions: These pages are an attempt to begin collecting information on the mathematics of paper folding. Anyone who has practiced origami has probably, at one time or another, unfolded an origami model and marveled at the intricate crease pattern which forms the "blueprint" of the fold. Clearly there are some rules at play in these collection of creases. Clearly there is an origami geometry at work when paper is folded. Unfortunately, much of the above-mentioned work is new, and at the time of this writing there are few good references for this type of information. These pages will try to help solve this problem by providing an extensive bibliography for origami-math, list upcoming lectures and events, and offer instructions and tutorials for select topics. However, this is an on-going project! These pages are still in their infancy, and any comments or suggestions (or offers to help!) would be greatly appreciated! In March of 2001 the 3rd International Meeting of Origami Science and Technology (3OSME) was held. See the above link for the program listing, pictures, and information on the upcoming proceedings. Browse our Origami Math Bibliography Tutorial: Origami Geometric Constructions
Ma Baker's Origami And Paper Folding Web Quest Page
paper folding paper folding is as paper folding, the student paper folding. Overhead transparency masters set. OVERVIEW. INTRODUCTION. INTENDED AUDIENCE. BASIC FOLDS. geometry
http://education.nmsu.edu/webquest/wq/origami

Extractions: Ma Baker's Origami and Paper Folding Web Quest Page Origami is the art of folding paper into decorative objects. The term origami is the Japanese word for folded paper . There are about 100 traditional origami figures, most depicting such natural forms as birds, flowers, and fish. An abstract, ceremonial form of origami called noshi, is a pleated paper ornament attache to gifts. Most origami is folded from an uncut square of paper.The most common sizes of square are 6-inches and 10-inches. The preferred paper is thin Japanese paper called washi , but foil-backed wrapping paper, heavy art paper, and typing paper can be used. Origami, like paper, originated in China but flourished in Japan. The purpose of this page is to introduce students to the art of origami and to the various methods of folding paper. 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. (*Before you begin add this site to your bookmark list.) To initiate the exploration of paper folding find out some basic facts about the history of origami at
Origami & Math
presentation on origami In Creasing geometry in the Classroom. great origami challenges that you might enjoy trying to solve. These puzzles involve folding a piece of paper so that
http://www.paperfolding.com/math

Extractions: So, you're interested in origami and mathematics...perhaps you are a high school or K-8 math teacher, or a math student doing a report on the subject, or maybe you've always been interested in both and never made the connection, or maybe you're just curious. Origami really does have many educational benefits . Whether you are a student, a teacher, or just a casual surfer, I have tried my best to answer your questions, so please read on. So exactly how do origami and math relate to each other? The connection with geometry is clear and yet multifaceted; a folded model is both a piece of art and a geometric figure. Just unfold it and take a look! You will see a complex geometric pattern, even if the model you folded was a simple one. A beginning geometry student might want to figure out the types of triangles on the paper. What angles can be seen? What shapes? How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself? For instance, when you fold the traditional waterbomb base, you have created a crease pattern with eight congruent right triangles. The traditional bird base produces a crease pattern with many more triangles, and every reverse fold (such as the one to create the bird's neck or tail) creates four more! Any basic fold has an associated geometric pattern. Take a squash fold - when you do this fold and look at the crease pattern, you will see that you have bisected an angle, twice! Can you come up with similar relationships between a fold and something you know in geometry? You can get even more ideas from this presentation on
Paper Folding Geometry
paper folding. The impetus for this page came from a visitor's letter Hi,. I gotto your knot page from a page called geometry Junkyard. I have an origami page
http://www.cut-the-knot.com/pythagoras/PaperFolding/index.shtml

Extractions: Recommend this site The impetus for this page came from a visitor's letter Hi, I got to your knot page from a page called Geometry Junkyard . I have an Origami page where I have instructions for making a star from a paper strip knot. I used a link to your page for the proof. Let me know if you have any objections and I'll remove it. Thanks, Meenakshi As I never mentioned the word Origami on my pages (i.e., until now), my interest was somewhat piqued. Origami is an ancient Chinese and Japanese art of paper folding. From the Brief History of the Ancient Art of Paperfolding I gather that Origami gained acceptance in the West in the early 1950s. Very comprehensive bibliographies are available online. A search of the online bookstore has resulted in several dozen books and albums. In my own library, I discovered several books wholly or in part devoted to the mathematics of paper folding (see bibliography below.) Unexpectedly, information available online proved to be scarce. (This may explain a reference to my logo page .) Following is my attempt to catch up with the development of the last 50 years of which, regretfully, I knew so little. My main source of information was a
Paper Folding Geometry
origami is an ancient Chinese and Japanese art of paper folding. From the Brief History of the Ancient In the geometry of paper folding, a straight line becomes a crease
http://www.cut-the-knot.com/pythagoras/PaperFolding

Extractions: Recommend this site The impetus for this page came from a visitor's letter Hi, I got to your knot page from a page called Geometry Junkyard . I have an Origami page where I have instructions for making a star from a paper strip knot. I used a link to your page for the proof. Let me know if you have any objections and I'll remove it. Thanks, Meenakshi As I never mentioned the word Origami on my pages (i.e., until now), my interest was somewhat piqued. Origami is an ancient Chinese and Japanese art of paper folding. From the Brief History of the Ancient Art of Paperfolding I gather that Origami gained acceptance in the West in the early 1950s. Very comprehensive bibliographies are available online. A search of the online bookstore has resulted in several dozen books and albums. In my own library, I discovered several books wholly or in part devoted to the mathematics of paper folding (see bibliography below.) Unexpectedly, information available online proved to be scarce. (This may explain a reference to my logo page .) Following is my attempt to catch up with the development of the last 50 years of which, regretfully, I knew so little. My main source of information was a
The Geometry Junkyard: Origami
Resource listing of links for information about the relationship between origami and geometry.Category Arts Crafts origami geometry and Modulars pentagon. Cutthe-knot Logo. folding geometry. Wheaton college courseproject on modular origami. Geometric paper folding. David Huffman.
http://www.ics.uci.edu/~eppstein/junkyard/origami.html

Extractions: Origami Krystyna Burczyk's Origami Gallery - regular polyhedra The business card Menger sponge project . Jeannine Mosely wants to build a fractal cube out of 66048 business cards. The MIT Origami Club has already made a smaller version of the same shape. Cranes, planes, and cuckoo clocks . Announcement for a talk on mathematical origami by Robert Lang. Crumpling paper: states of an inextensible sheet Cut-the-knot logo . With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon. Folding geometry . Wheaton college course project on modular origami. Geometric paper folding . David Huffman. Rona Gurkewitz' Modular Origami Polyhedra Systems Page . With many nice images from two modular origami books by Gurkewitz, Simon, and Arnstein. How to fold a piece of paper in half twelve times . Britney Gallivan took on this previously-thought-impossible task as a high school science project, worked out an accurate mathematical model of the requirements, and used that model to complete the task. Knotology . How to form regular polyhedra from folded strips of paper?
Sy's Paper Folding Page
Festival. Flower. Function. geometry. Heart. Instrument. Love. Money fold all about my paper folding related works. It does not contain any general origami information, history, or
http://users.erols.com/sychen1/pprfld.html
Origami-Math Bibliography
by origami (paper folding), Symmetry Culture and Science, Vol. 5, No. 1 (1994),6984. Huzita, Humiaki, The trisection of a given angle solved by the geometry
http://web.merrimack.edu/hullt/Geombib.html

Extractions: These articles deal with pure geometrical aspects of folding a piece of paper. Since so many of these articles are also about using origami as an educational tool, I'm listing both of these categories together. Alperin, R.C., A mathematical theory of origami constructions and numbers, New York Journal of Mathematics , No. 6 (2000), 119-133. Bruckheimer, M. and R. Hershkowitz, Constructing the parabola without calculus, Mathematics Teacher , Vol. 70, No. 8 (Nov. 1997), 658-662. Faulkner, J., Paper folding as a technique in visualizing a certain class of transformations, Mathematics Teacher , Vol. 68. No. 5 (May 1975), 376-377. Fehlen, J., Paper folds and proofs, Mathematics Teacher , Vol. 68, No. 8 (Nov. 1975), 608-611. Frigerio, Emma, New relations in origami geometry proposed by J. Justin, Proceedings of the First International Meeting of Origami Science and Technology , H. Huzita ed. (1989), 125-130. Frigerio, Emma, Origami geometry: old and new, Proceedings of the First International Meeting of Origami Science and Technology , H. Huzita ed. (1989), 379-386.
Folding And Unfolding (Erik Demaine)
folding and unfolding is an exciting area of geometry. It is attractive in the way paper folding origami Mathematics and Computational origami. origami (paper folding) has lead to
http://www.db.uwaterloo.ca/~eddemain/folding

Extractions: 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
Paper Folding - Books
list of folding symbols, plus stepby-step folding diagrams for origami for Beginners$5.95 By Vincent Palacios. Patty paper geometry $21.95 By Michael Serra.
http://www.mathartfun.com/shopsite_sc/store/html/OrigamiBooks.html

Extractions: By Betsy Franco. A book of blackline activity masters designed for algebra and geometry students in high schoool or middle school. Contains 16 activities arranged in order of increasing difficulty. The book's primary purpose is to teach mathematics, but it also introduces students to the art of origami. 136 pages. By Michael Serra. Patty Paper - the stuff that separates hamburger patties - can also be used for geometric investigations! These square of paper can be written on, they hold creases well, and they are semi-transparent like tracing paper. "Patty Paper Geometry" contains dozens of activities that motivate kids to read, write, and talk about geometry. This book is a blackline master book with 12 chapters of guided and open investigations. Grades 6-10. 272 pages, paperback.
Paper Folding - Models
paper folding Models. Patty paper geometry Student Workbook $3.95 This companionstudent workbook contains origami paper $4.95 24 7 x 7 sheets of origami
http://www.mathartfun.com/shopsite_sc/store/html/OrigamiManips.html

Extractions: Captured Worlds is a set of five pop-out, glue, and fold-up polyhedra. The five used in Captured Worlds are known as the Platonic Solids. They are decorated with fanciful scenes rendered in six-point perspective by internationally-known artist Dick Termes. This unique perspective system allows an entire three-dimensional surrounding to be projected onto the polyhedra. By Michael Serra. Patty Paper - the stuff that separates hamburger patties - can also be used for geometric investigations! These square of paper can be written on, they hold creases well, and they are semi-transparent like tracing paper. "Patty Paper Geometry" contains dozens of activities that motivate kids to read, write, and talk about geometry. This book is a blackline master book with 12 chapters of guided and open investigations. Grades 6-10. 272 pages, paperback.
Google Directory - Arts > Crafts > Origami > Geometry And Modulars
origami Mathematics http//web.merrimack.edu/~thull/ origamiMath.html. Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry
http://directory.google.fr/Top/Arts/Crafts/Origami/Geometry_and_Modulars
Origami -- From MathWorld
Huzita, H. Understanding geometry through origami Axioms. In Proceedings ofthe First Kasahara, K. origami Omnibus paperfolding for Everyone.
http://mathworld.wolfram.com/Origami.html

Extractions: The Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami constructed by E. K. Herrstrom, together with an animated crane constructed in Mathematica by L. Zamiatina. Cube duplication and trisection of an angle can be solved using origami, although they cannot be solved using the traditional rules for geometric constructions . There are a number of recent very powerful results in origami mathematics. A very general result states that any planar straight-line drawing may be cut out of one sheet of paper by a single straight cut, after appropriate folding (Demaine, Demaine, and Lubiw 1998, 1999; O'Rourke 1999). Another result is that any polyhedron may be wrapped with a sufficiently large square sheet of paper. This implies that any connected, planar, polygonal region may be covered by a flat origami folded from a single square of paper. Moreover, any 2-coloring of the faces may be realized with paper whose two sides are those colors (Demaine, Demaine, and Mitchell 1999; O'Rourke 1999).
Icosahedron -- From MathWorld
geometry Technologies. Icosahedron. http//www.scienceu.com/geometry/facts/solids/icosa.html. Kasahara,K. origami Omnibus paperfolding for Everyone.
http://mathworld.wolfram.com/Icosahedron.html

Extractions: A Platonic solid having 12 polyhedron vertices polyhedron edges , and 20 equivalent equilateral triangle faces, . It is also uniform polyhedron and Wenninger model . It is described by the and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included). Two icosahedra constructed by E. K. Herrstrom in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper. The icosahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the icosahedral graph The dual polyhedron of an icosahedron with unit edge lengths is the dodecahedron with edge lengths , where is the golden Ratio . As a result, the centers of the faces of an icosahedron form a dodecahedron , and vice versa, illustrated above (Steinhaus 1999, pp. 199-201). There are 59 distinct icosahedra when each triangle is colored differently (Coxeter 1969).
Origami
Because of this, the practice of paper folding was originally origami appealed tothe same aesthetic which created as life it is moral geometry, inasmuch as
http://www.tuvy.com/resource/origami.htm

Extractions: Home About Books Authors ... Household Special thanks to Origami Garden for this info. In the East The art of paper folding is thought to have had its beginnings in China during the first or second century A.D. By the sixth century, it was being practiced in Japan. In this small island country, paper was a scarce and treasured material. Because of this, the practice of paper folding was originally confined to the wealthy nobility. Origami appealed to the same aesthetic which created the tea ceremony, which one scholar has described as "essentially a worship of the Imperfect, as it is a tender attempt to accomplish something possible in this impossible thing we know as life...it is moral geometry, inasmuch as it defines our sense of proportion to the universe." (Kakuzo Okakura, The Book of Tea) Increasing trade eventually led to the widespread availability of affordable paper, and origami grew into a popular pastime among rich and poor alike. Because of their culture which emphasizes respect for the economy of nature, however, Asian practitioners of this art have never lost the impulse to save even the tiniest scraps of paper to fold into miniature origami models. Hiden Senbazuru Orikata ("How to Fold One Thousand Cranes") was published in 1797, and is the oldest origami publication which survives. Kan no modo ("Window on Midwinter"), the first published collection of origami models, appeared in 1845. In the West The Moors, who were Muslims from West Africa, brought paper folding with them to Spain when they invaded in the eighth century. Although Islam proscribed the making of representational figures, Islamic mathematicians and astronomers were fascinated with pattern, symmetry, and space. Their explorations included studies on the geometry of tessellation and on the folding patterns hidden within the square. These investigations of pattern were often given form in architecture.
Citations: Euclidean Constructions And The Geometry Of Origami - Geretschlager (
We conclude in Section 6. 2 Background origami mathematics is the study of thegeometry and other properties of origami (paper folding) The area of origami
http://citeseer.nj.nec.com/context/471244/0

Extractions: This paper is cited in the following contexts: Folding and Cutting Paper - Demaine, Demaine, Lubiw (1998) (2 citations) (Correct) ....of the algorithm. We conclude in Section 6. 2 Background Origami mathematics is the study of the geometry and other properties of origami (paper folding) The area of origami mathematics is still in its infancy, having only been seriously studied for the past twenty years. Geretschlager and Huzita and Scimemi [13] examined the geometric constructions possible with origami, and compared them to a ruler and compass. Bern and Hayes [4] showed that it is NP hard to determine whether a crease pattern is flat foldable, as is computing a flat folding (overlap order) given a suitable ....
ORIGAMI For The Dummies
origami, the art of paper folding, is a materials; artistry which transcends boundariesof age and nationality; mysteries of geometry and metaphysics to
http://www.santel.lu/SANTEL/contact/vm/origami.html

Extractions: Ori (to fold) + kami (paper) = origami, the art of paper-folding. Origami, the art of paper folding, is a multidimensional metaphor for life which offers many pleasures simple, readily available materials; artistry which transcends boundaries of age and nationality; mysteries of geometry and metaphysics to ponder; each simple square of paper contains a universe of possibility. East The art of paper folding arose in China during the first or second century A.D. By the sixth century, it had spread to Japan. In this small island country, paper was a scarce, treasured material. Initially confined to the wealthy nobility, origami appealed to the same aesthetic sense which created the tea ceremony, which one scholar has described as "essentially a worship of the Imperfect, as it is a tender attempt to accomplish something possible in this impossible thing we know as life...it is moral geometry, inasmuch as it defines our sense of proportion to the universe." (Kakuzo Okakura, The Book of Tea).
Origami Links From Debbie's Educational Link Page
Books origami geometry Design origami folding Patterns **The Garden oforigami GateSuper site Ori (to fold) + kami (paper) = origami, the art of
http://www.venus.net/~dschweit/edsites/origami/origami2.htm
Origami Links With Visual Arts Standards
Plus inks to other geometric shapes. The geometry Junkyard origami Projectsand questions that revolve around the math of paper folding.
http://www.artteacherconnection.com/pages/origami.html

Extractions: Math and Art Origami The Japanese word "Origami" is now an internationally recognized word and is synonymous with the art and craft of paper folding.Origami is a traditional Japanese pastime where a single square of paper is folded in different ways to create shapes like cute animals and beautiful plants.Since it only takes a sheet of paper, the hobby can easily be enjoyed anywhere; many people in Japan enjoy it at home and at school. The practice of origami began in the early 700s, when paper was first introduced to Japan. At first paper was folded to make decorations for use in religious ceremonies at shrines, but gradually people came to use it in their regular lives as well. During the Heian period(794-1185), it was popular to fold valuable paper and use it to beautifully wrap letters and presents. Later, origami continued to be used in traditional ceremonies, but the women of the imperial court began to fold dolls and other shapes for their amusement. you can find more historical information at Origami - the ancient art of paper folding.

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