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Topic: Mathematics of origami

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	Origami
		Origami only uses a small number of different folds, but they can be combined in an infinite variety of ways to make extremely intricate designs.
		An origami design can be as simple as a party hat[?] or paper airplane[?], or as complex as a model of the Eiffel Tower or a leaping gazelle.
		Sadako was exposed to the radiation of the atomic bombing of Hiroshima as an infant, and it took its inevitable toll on her health.
	www.ebroadcast.com.au /lookup/encyclopedia/or/Origami.html (502 words)

	Math on the Street: Where's the math? (Site not responding. Last check: 2007-10-18)
		Origami provides a nice link to topics in other parts of the curriculum and can give students a reason to want to learn the mathematical ideas in their texts.
		Many of today's most prominent practitioners of origami are also mathematicians, and so it is not surprising that the recent origami revolution has coincided with the development of origami as a field of mathematics.
		Although mathematical origami is a fairly new field, mathematicians are already investigating a wide range of questions relating to paper folding.
	math.serenevy.net /?page=Origami-WhereMath (787 words)

	Origami: Complexity in Crease (Again)
		Origami geometric constructions are part of a family of pure mathematical problems in which the object is to fold an arbitrary geometric shape or a pure number represented as a distance proportional to the edge of the paper.
		In mathematical terms, this shadow forms a “tree graph,” which is a fancy term for a “stick figure.” The tree graph consists of “edges,” or line segments, and “nodes,” which are points where edges either come together or terminate.
		Origami art and origami science have sometimes been at odds in the past, but now the origami designer can focus on the art of folding, secure in the knowledge that the science will take care of itself.
	pr.caltech.edu /periodicals/EandS/articles/LXVII1/origami.html (4202 words)

	Origami
		Origami is an art of paper folding (折り紙, Japanese 'ori', to fold, and 'kami', paper).
		That of European origami, represented by a little bird (Pajarita in Spanish or Cocotte in French), is probably the baptismal certificate of 16th century.
		An origami design can be as simple as a party hat or paper airplane, or as complex as a model of the Eiffel Tower, a leaping gazelle or a stegosaurus that takes an hour and a half to fold.
	www.knowledgefun.com /book/o/or/origami.html (774 words)

	Origami Mathematics Page (Site not responding. Last check: 2007-10-18)
		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.
		Such geometry, this mathematics of origami, has been studied extensively by origamists, mathematicians, scientists and artists.
		In March of 2001 the 3rd International Meeting of Origami Science and Technology (3OSME) was held.
	web.merrimack.edu /~thull/OrigamiMath.html (442 words)

	Dynamic Miscellany : Origami Fold Language
		This subset of general origami is useful for generating mathematical figures such as the golden section, regular polygons, or any number of others.
		Mathematics is a science of exploration, wherein the same concepts can easily be discovered by completely independent researchers.
		All the work I found regarding the axiomatic nature of origami was very interesting and helped me to organize my thoughts on the subject, but they were not works on written forms for origami and as such were not alternate solutions to my own work so far.
	www.ganymeta.org /~darren/origami2.php (2750 words)

	Origami enthusiasts joining the fold \| www.azstarnet.com ®
		Showcasing a renaissance in the ancient Japanese art of origami, Wu and some of the best paper-folders in the world descended on Tokyo Friday for a three-day competition and convention to celebrate the artistic possibilities hidden in the humble sheet of paper.
		Origami, which dates back at least 1,600 years, flows from a simple set of rules.
		Wu said the focus on creativity and complexity is breaking with the past, when origami was limited largely to a few well-known and relatively straightforward patterns, and is inspiring a growing number of admirers.
	www.azstarnet.com /sn/printDS/35283 (433 words)

	Mathematics of paper folding - Wikipedia, the free encyclopedia
		The art of paper folding or origami has received a considerable amount of mathematical study.
		As a result of origami study through the application of geometric principles, methods such as the Haga Theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths.
		The problem of rigid origami, treating the folds as hinges joining two flat, rigid surfaces such as sheet metal, has great practical importance.
	en.wikipedia.org /wiki/Mathematics_of_origami (403 words)

	Origami & Math
		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.
		But back to origami construction...origami construction is defined as those geometric operations that can be formed by folding a piece of paper, using the raw edges and points of the paper, as well as any subsequent crease lines and points created while folding.
		Thomas Hull, an assistant professor of mathematics at Merrimack College in North Andover, Massachusetts, is the expert in the field of origami and topology.
	www.paperfolding.com /math (2393 words)

	Origami - Deafhomeschool
		Origami is an essential tool in the exploration of mathematics and science.
		Origami demonstrates, in a very concrete way, the fact that simple shapes develop into complex patterns.
		Background is given on origami, including the development and critical aspects of the practice of origami in the United States.
	www.deafhomeschool.com /study/crafts/origami.html (524 words)

	Origami - Wikipedia, the free encyclopedia
		Origami was mostly a traditional art for the amusement of children in China until Akira Yoshizawa spurred a renaissance of the art form with his new advancements, including wet-folding and the Yoshizawa-Randlett system of diagramming.
		A paper art star on a window, an example of a form of origami known as modular origami, in which the model is constructed from multiple pieces of paper.
		Special origami paper, often also referred to as "kami" (Just for reference kami is Japanese for paper.), is sold in prepackaged squares of various sizes ranging from 2.5 cm to 25 cm or more.
	en.wikipedia.org /wiki/Origami (1469 words)

	The Science, Art, and Mathematics of Origami - Asymptotia
		It was swiftly overtaken by other arts and crafts, and these were hobbies long before I started taking apart cameras and radios, and the like, to see how they worked, and collecting pondwater and pressing leaves and….
		It is “The Fourth International Conference on Origami in Science, Mathematics, and Education”, and the web information is here.
		“Origami aficionados agree that Kamiya’s most extraordinary sculpture is a dragon he created in 2004, at age 20.
	asymptotia.com /2006/09/09/the-science-art-and-mathematics-of-origami (692 words)

	The Geometry Junkyard: Origami
		Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon.
		The MIT Origami Club has already made a smaller version of the same shape.
		Origami Menger Sponge built from Sonobe modules by K. and W. Burczyk.
	www.ics.uci.edu /~eppstein/junkyard/origami.html (649 words)

	Mathematical Imagery Presented by the American Mathematical Society - Thomas Hull :: The mathematics of origami
		Mathematical Imagery Presented by the American Mathematical Society - Thomas Hull :: The mathematics of origami
		His dissertation was on list coloring bipartite graphs, now he mostly studies the mathematics of origami (paper folding).
		Tom Hull is an associate professor in the Department of Mathematics at Merrimack College in North Andover, MA.
	www.ams.org /mathimagery/thumbnails.php?album=1 (220 words)

	Folding and Unfolding (Erik Demaine)
		Origami (paper folding) has lead to an intriguing collection of problems to study mathematically and computationally.
		Origami mathematics is a recent branch of mathematics (whose major study started circa 1980) that studies the properties of origami, such as what patterns you might get when you unfold a flat origami.
		Tom Hull's page on origami mathematics includes an origami-math bibliography, a list of other origami-math pages, a nice tutorial on the axiomatization of geometric constructions with origami, and instructions for a few models of polyhedra.
	theory.lcs.mit.edu /~edemaine/folding (1403 words)

	Origami Summary
		Origami is ingenious in its use of simple creases to capture the essence of a subject.
		Technical origami, also known as origami sekkei, is a field of origami that has developed almost hand-in-hand with the field of mathematical origami.
		This is not a unique mathematical process, hence it is possible for two designs to have the same circle-packing, and yet different crease pattern structures.
	www.bookrags.com /Origami (2048 words)

	Math on the Street: Mathematics in Origami Links (Site not responding. Last check: 2007-10-18)
		Check out his origami gallery and his musings about designing an origami compound of three octahedra.
		Joseph Wu's origami page includes a discussion of how Joseph went about designing several of his origami animals.
		Exploring Paper, discusses many facets of the revolution that has occured in the origami world in the last few decades as a result of new mathematical techniques.
	math.serenevy.net /?page=Origami-ArtLinks (336 words)

	Origami paper background
		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.
		The mathematics reform movement is consistent with the general educational theory called constructivism.
		Origami is reflective of this, with new models being formed based on old ones, modelers learning from one another, and communication as a key part of the process.
	rachel.ns.purchase.edu /~Jeanine/origami/oriback.htm (704 words)

	Origami (reflection)
		Peter, whose mathematically virtuosic constructions put him at the forefront of origami, I met for the first time at this event.
		Since origami is the art of paper folding, I created a design that illustrates the symmetry of folding.
		The Folding Universe is a unique combination of origami instruction and mind-expanding philosophy...sort of the Goedel, Escher, Bach of paper folding.
	www.scottkim.com /inversions/gallery/origami.html (701 words)

	Origami Sightings
		"Origami is regarded as a geometric mathematical recreation, but little has been done to analyze mathematically how the crease patterns are related to the resulting forms.
		Gerestschla"ger is also the author of a paper on the geometry of origami, “Euclidean Constructions and the Geometry of Origami”, in Mathematics Magazine, volume 68, number 5 (December 1995, pp.
		While this idea is not new, origami and the mira share the novel feature of constructing the tangents from the parabolas' foci and directrices rather than from the parabolas themselves.
	www.origamisources.com /origami_sightings_math.htm (1762 words)

	Ma Baker's Origami and Paper Folding Web Quest Page
		Origami is the art of folding paper into decorative objects.
		Most origami is folded from an uncut square of paper.The most common sizes of square are 6-inches and 10-inches.
		The purpose of this page is to introduce students to the art of origami and to the various methods of folding paper.
	education.nmsu.edu /webquest/wq/origami/index.htm (822 words)

	Mathematical Imagery Presented by the American Mathematical Society - Home
		The connection between mathematics and art goes back thousands of years.
		Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings.
		In this sense, I like to feel that theory (mathematics), art (outcome), software (algorithms) and engineering (hardware) are integrated and interdependent and that no part survives without the others.
	www.ams.org /mathimagery (430 words)

	Bern's Origami Page (Site not responding. Last check: 2007-10-18)
		Technical origami includes mathematical models such as David Huffman's folded curves, and also representational models, such as Robert Lang's little creatures, that pose and solve difficult folding problems while keeping to more traditional subject matter.
		Origami is not usually presented as high art-- artists do not publish books showing how to reproduce their designs!-- and yet origami artists have quite recognizable styles.
		For example, Hojyo is a Ph.D. candidate in biology, Hull an assistant professor of mathematics, Lang a laser physicist, and Huffman a computer scientist and the inventor of Huffman coding.
	www2.parc.com /csl/members/bern/origami.html (372 words)

	A Photo Gallery of Origami Notables
		Mathematics professor, creator of the famous rose and master of origami math.
		A prolific origami author whose models seem to form the line that a folder must cross before being able claim to be a folder of complex models.
		The acknowledged grandmaster of origami, the father of modern creative origami.
	www.origami.as /Gallery/people.htm (480 words)

	Key College Publishing: Unfolding Mathematics with Origami Boxes
		Unfolding Mathematics with Origami Boxes combines the ancient art of paper folding with high school mathematics to help students discover important concepts in geometry.
		Unfolding Mathematics with Origami Boxes was designed with the classroom in mind.
		Explorations help students discover the mathematics in their creations, beginning with the mathematical patterns found in each box.
	www.keycollege.com /catalog/titles/unfolding_origami_boxes.html (165 words)

	Lafayette Professor Ethan Berkove Will Talk about “Mathematics and Origami” Wednesday
		One might be surprised to learn that over the last couple of decades the connections between origami and mathematics have been extensively studied,” says Berkove.
		MAAD is a series of talks on mathematical topics and applications that are often not encountered in mathematics courses.
		They are open to the Lafayette community and assume no special mathematical preparation on the part of the audience.
	www.lafayette.edu /news.php/view/4546 (239 words)

	K's Origami : Origami Construction
		We use folds in the origami construction, of course.
		Then, the origami construction would be a finite sequence of either "to fold a line at a time without any tool" or "given some creases, to locate points of intersection" on a plane with some lines placed arbitrarily.
		In addition, we can solve with origami some problems which cannot be solved with a straight edge and a compass, such as doubling a cube or trisecting an angle.
	www.jade.dti.ne.jp /~hatori/library/conste.html (810 words)

	Roger Alperin's Home Page
		I am a professor on the faculty of the Department of Mathematics,
		The Mathematics of Origami, New York Journal of Mathematics, July 2000
		Mathematical Origami and Alhazen's Problem and Origami Heptagon
	www.math.sjsu.edu /~alperin (110 words)

	A K Peters, Ltd. - Project Origami (Site not responding. Last check: 2007-10-18)
		Origami, the art and science of paper folding, can be used to explain concepts and solve problems in mathematics-and not just in the field of geometry.
		The origami activities collected here also relate to topics in calculus, abstract algebra, discrete mathematics, topology, and more.
		Overall, this book is an excellent resource for mathematics educators who would like to include some hands-on experimentation in their teaching.
	www.akpeters.com /product.asp?ProdCode=2582 (200 words)

	Origami :A Peace of Paper
		Mathematical origami Helena Verrill's site includes constructions of a shape with greater perimeter than the original square, tesselations, hyperbolic paraboloids, and more.
		Origami Mathematics Tom Hull's page contains: Origami math bibliography; tutorial in origami geometric constructions, links to other origami mathematics sites.
		Cranes for Peace began as a project to collect paper cranes to be sent to Hiroshima for the 50th anniversary of the bombing.
	ww2.madonna.edu /kathleen/Interests/origami.htm (97 words)

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