Math on the Street: Where's the math?

<div id="SideBar"> <p align="center"> <a href="http://math.serenevy.net/?page=Home" target="_top"><img src="http://math.serenevy.net/Images/SmallLogo.gif" alt="Math on the Street" border=0 width=166 height=70></a> </p> <p><img src="http://math.serenevy.net/Images/RedDot.gif" alt="" height=8 width=8> <a href="http://math.serenevy.net/?page=About" target="_top">About the Project</a></p> <p><img src="http://math.serenevy.net/Images/RedDot.gif" alt="" height=8 width=8> <a href="http://math.serenevy.net/?page=Acknowledgments" target="_top">Acknowledgments</a></p> <p><img src="http://math.serenevy.net/Images/RedDot.gif" alt="" height=8 width=8> <a href="http://math.serenevy.net/?page=FunMath" target="_top">Fun Math Activities</a> <ul> <li><a href="http://math.serenevy.net/?page=OrigamiHome" target="_top">Origami</a></li> </ul> </p> </div>      <div id="Content"> <a name="Top"></a> <h1>Origami</h1> <center><table width="85%"><tr><td width="33%" align="center"><a href="http://math.serenevy.net/?page=OrigamiHome" class="button" target="_top">Polyhedra</a></td><td width="33%" align="center"><a href="http://math.serenevy.net/?page=Origami-Figures" class="button" target="_top">Magic Pinwheel</a></td><td width="33%" align="center"><a href="http://math.serenevy.net/?page=Origami-WhereMath" class="button" target="_top">Where's The Math?</a></td></tr></table><table width="85%"><tr><td width="33%" align="center"><a href="http://math.serenevy.net/?page=Origami-Challenge" class="button" target="_top">Challenge Problems</a></td><td width="33%" align="center"><a href="http://math.serenevy.net/?page=Origami-Handouts" class="button" target="_top">Handouts</a></td><td width="33%" align="center"><a href="http://math.serenevy.net/?page=Origami-Links" class="button" target="_top">Links</a></td></tr></table></center><center><hr width=80% /></center> <br> <br><br> <h2>Where's the math in origami?</h2> <p>Origami may not seem like it involves very much mathematics. Yes, origami involves symmetry. If we build a polyhedron then, sure, we encounter a shape from geometry. Is that as far as it goes? Do any interesting mathematical questions arise from the process of folding paper? Can origami help someone learn arithmetic, algebra, geometry, and other standard math topics? Is there any deep mathematics in origami? Is the mathematics behind origami useful for anything other than making pretty decorations?</p> <a href="#Art">Mathematics in Origami</a><br> <a href="#Teach">Using Origami to Teach Standard Mathematics Topics</a><br> <a href="#Math">Origami as a Field of Mathematics</a><br> <a href="#Applications">Applications of Mathematical Origami</a><br> <a href="#Handouts">Printable, Condensed Version of This Page</a><br> <br><br><hr width="50%"><br><br> <a name="Art"><h3>Mathematics in Origami</h3></a> <p>People who spend time folding paper often ask themselves questions that are ultimately mathematical in nature. Is there a simpler procedure for folding a certain figure? Where on the original square paper do the wings of a crane come from? Why do so many origami figures start with square paper? What size paper should I use to make a chair to sit at the origami table I already made? What words should I use to teach people to make a jumping frog? Is it possible to make an origami beetle that has six legs and two antennae from a single square sheet of paper? Is there a precise procedure for folding a paper into 5 rectangular strips? Which polyhedra can be constructed using Sonobe modules and what do they have in common?</p> <p>The <a href="http://math.serenevy.net/?page=Origami-Challenge" target="_top">Challenge Problems</a> section gives some more mathematical origami questions that you might like to think about.</p> <p>In the last few decades, folders inspired by questions like these have revolutionized origami by bringing mathematical techniques to their art. In the early 1990s, Robert Lang proved that for any number of appendages, there is an origami base that can produce the desired effect from a single square sheet of paper. Robert has created a computer program that can design a somewhat optimized base for any stick figure outline. This has enabled many folders to create origami animals that were considered impossible years ago. You can see some of Robert's creations in these <a href="http://www1.zetosa.com.pl/~burczyk/origami/e3-21-en.htm#RL" target="_top">photos</a> on Krystyna Burczyk's web page.</p> <p>More information about the recent <a href="http://math.serenevy.net/?page=Origami-ArtLinks" target="_top">origami revolution</a> and links to several sites showing how mathematical questions come up in paper folding are available in the references. </p> <a href="#Top">Return to the top</a><br> <br><br><hr width="50%"><br><br> <a name="Teach"><h3>Using Origami to Teach Standard Mathematics Topics</h3></a> <p>Many teachers have developed hands-on lessons that use origami to make math come to life for their students. Topics taught in this way range across the entire curriculum: problem solving; precise use of mathematical terminology; ratios, fractions, and percents; angles; area and volume; congruence; tessellations; combinatorics; properties of parallel lines; products and factors; conic sections; Euler's formula; logic; proofs; even concepts from calculus such as tangent line approximations to curves or negative curvature in a hyperbolic paraboloid. Origami also abounds with accessible open problems that give students a chance to contribute original ideas.</p> <p>Many students who have not previously experienced success in math flourish after a well-crafted unit involving origami. 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.</p> <p>There are a number of <a href="http://math.serenevy.net/?page=Origami-TeachingLinks" target="_top">links for teachers</a> in the references. Included are links to <a href="http://math.serenevy.net/?page=Origami-TeachingLinks&layout=frame#Lessons">on-line lesson plans</a>, <a href="http://math.serenevy.net/?page=Origami-TeachingLinks&layout=frame#Ideas">ideas for using origami to teach mathematics</a>, information about the <a href="http://math.serenevy.net/?page=Origami-TeachingLinks&layout=frame#Benefits">benefits of using origami in the classroom</a>, and relevant <a href="http://math.serenevy.net/?page=Origami-TeachingLinks&layout=frame#Resources">books and other resources for teachers</a>.</p> <a href="#Top">Return to the top</a><br> <br><br><hr width="50%"><br><br> <a name="Math"><h3>Origami as a Field of Mathematics</h3></a> <p>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. Some of these investigations try to design, classify, or find colorings for origami polyhedra, tessellations, fractal models, or other paper representations of mathematical objects. Others unfold origami figures and try to understand what the crease patterns from different figures have in common. One popular approach is to develop origami geometry as an axiomatic system and compare it to Euclid's ruler and compass geometry. Still another sub-field takes advantage of modern technology to design computer algorithms that solve mathematical origami questions -- sometimes just determining whether it is mathematically possible for an efficient algorithm to be designed is the focus of research. Another approach is to consider folding in unfamiliar contexts -- folding spherical paper or doing theoretical origami in 4 dimensions with 3-dimensional paper.</p> <p>To get a beter feel for the scope of the field of mathematical origami, take a look at Erik Demaine's <a href="http://theory.lcs.mit.edu/~edemaine/photos/OSME_March2001/models/" target="_top">photos</a> and some of the <a href="http://www.merrimack.edu/~thull/osm/abstracts.html" target="_top">abstracts</a> from the 3rd International Meeting of Origami Science, Math, and Education. The references include <a href="http://math.serenevy.net/?page=Origami-MathLinks" target="_top">links to a variety of mathematical origami pages</a>, ranging from accessible descriptions with nice pictures to highly technical work with lots of mathematical notation.</p> <a href="#Top">Return to the top</a><br> <br><br><hr width="50%"><br><br> <a name="Applications"><h3>Applications of Mathematical Origami</h3></a> <p>Recently, mathematical origami theory has been applied to produce an amazing range of practical applications. New technologies being developed include: paper product designs involving no adhesives, better ways of folding maps, unfolding space telescopes and solar sails, software systems that test the safety of airbag packings for car manufacturers, and self-organizing artificial intelligence systems.</p> <p><a href="http://math.serenevy.net/?page=Origami-ApplicationLinks" target="_top">Details</a> about the innovations mentioned above and information about other applications of mathematical origami are included in the references.</p> <a href="#Top">Return to the top</a><br> <br><br><hr width="50%"><br><br> <a name="Handouts"><h3>Printable, Condensed Version of This Page</h3></a> <p>A <a href="http://math.serenevy.net/?page=Origami-WhereMathOrigami" target="_top">single-page version</a> of the information on this page is also available. <div class="PDFPS"> <table> <tr><td>Printer friendly<br>format:</td> <td><a href="http://math.serenevy.net/Origami/WheresTheMathInOrigami.pdf"><img src="http://math.serenevy.net/Images/acrobat.gif" alt="" border=0 width=32 height=32></a></td></tr> <tr><td></td> <td><a href="http://math.serenevy.net/Origami/WheresTheMathInOrigami.pdf">PDF</a></td></tr> <tr><td>Software:</td> <td><a href="http://www.acrobat.com">acrobat</a></td></td> </tr> </table> </div> </p> <p style="clear: both"> <br><br> <a href="#Top">Return to the top</a><br> </p> <div id="BottomBar"> <p /><hr> <img class="left" src="http://math.serenevy.net/Images/BWLogo.gif" alt="Math on the Street!" border=0 width="150" height="57"> <p class="about"> Last Updated: February 2003<br> <a href="http://math.serenevy.net/?page=Home" target="_top">http://math.serenevy.net/</a><br> Amanda Katharine Serenevy<br> <a href="mailto:viajera@bu.edu" target="_top">viajera@bu.edu</a><br> </p> </div> </div>