{"id":1969,"date":"2016-04-20T07:00:58","date_gmt":"2016-04-20T12:00:58","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=1969"},"modified":"2016-04-20T02:10:42","modified_gmt":"2016-04-20T07:10:42","slug":"fold-your-way-to-glory","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2016\/04\/20\/fold-your-way-to-glory\/","title":{"rendered":"Fold Your Way to Glory"},"content":{"rendered":"<p class=\"p1\"><span class=\"s1\">Yesterday, I led a meeting of a Teachers\u2019 Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. <a href=\"http:\/\/blogs.scientificamerican.com\/roots-of-unity\/make-your-own-font-1-cut-at-a-time\/\">I learned about<\/a> the theorem last year when Numberphile posted this excellent video featuring a virtuoso performance from Katie Steckles, who folds and cuts every letter of the alphabet from memory.<\/span><br \/>\n<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/ZREp1mAPKTM?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=en-US&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">I was a little nervous about leading the program\u00a0because I had prepared almost nothing to say. Everything I thought about saying was boring, so I decided\u00a0the best way to approach the activity was to just get people started on it. Luckily for me, the group was ready to jump right in. I dumped a bunch of paper into the middle of the table, and people started folding.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">I encouraged people to try the most symmetric shapes first, but other than that, I didn\u2019t have to give them many suggestions. I was <a href=\"https:\/\/twitter.com\/MrHonner\/status\/722226085132419073\">prepared for some frustration<\/a> when they started trying the scalene triangle because it\u2019s a big step up\u00a0in difficulty, but several of them got the scalene pretty quickly,\u00a0and no one seemed to give up. In general, the strange shapes people got\u00a0when you mess up were amusing rather than frustrating.<\/span><\/p>\n<div id=\"attachment_1971\" style=\"width: 235px\" class=\"wp-caption alignright\"><a href=\"https:\/\/twitter.com\/evelynjlamb\/status\/649310029783273473\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1971\" class=\"wp-image-1971 size-medium\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/CQLQCCSUEAA4MOJ.jpg?resize=225%2C300\" alt=\"My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.\" width=\"225\" height=\"300\" srcset=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/CQLQCCSUEAA4MOJ.jpg?resize=225%2C300&amp;ssl=1 225w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/CQLQCCSUEAA4MOJ.jpg?w=600&amp;ssl=1 600w\" sizes=\"auto, (max-width: 225px) 100vw, 225px\" \/><\/a><p id=\"caption-attachment-1971\" class=\"wp-caption-text\">My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.<\/p><\/div>\n<p class=\"p1\"><span class=\"s1\">Participants almost immediately started asking mathematical questions and trying to extend the activity: do we have an existence theorem? Must we always fold along every angle bisector? Is there a general theory of folding? I liked <a href=\"https:\/\/twitter.com\/AnnaWeltman\/status\/722190221559734272\"><span class=\"s2\">Anna Weltman\u2019s suggestion<\/span><\/a> of trying to make things without drawing on the paper, and I spent some time trying to fold stars without drawing them, but the teachers didn\u2019t really bite on that. Instead, some of them started thinking about minimal folding numbers for different shapes, and some of them worked on developing a folding algorithm.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">Erik Demaine is one of the pioneers of fold-and-cut theory and the mathematics of paper folding in general. <a href=\"http:\/\/erikdemaine.org\/foldcut\/\"><span class=\"s2\">His page about folding and cutting<\/span><\/a> has links to all the gory mathematical details as well as some templates. I ended up bringing copies of\u00a0his\u00a0swan to the teachers\u2019 circle. They are beautiful, but I had mixed feelings about bringing them because they have the fold lines marked on them already. I didn\u2019t hand them out until one group had started talking about how to use angle bisectors and perpendiculars in their folding algorithm, and I thought the swan template might give them some ideas. Because I gave them only the template, not any explanation of how it was made, I think it didn\u2019t take away too much of their fun.<\/span><\/p>\n<div id=\"attachment_1970\" style=\"width: 310px\" class=\"wp-caption alignleft\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1970\" class=\"wp-image-1970 size-medium\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?resize=300%2C225\" alt=\"My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb.\" width=\"300\" height=\"225\" srcset=\"https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?resize=768%2C576&amp;ssl=1 768w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?resize=1024%2C768&amp;ssl=1 1024w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?w=1280 1280w, https:\/\/i0.wp.com\/blogs.ams.org\/blogonmathblogs\/files\/2016\/04\/IMG_1815.jpg?w=1920 1920w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><p id=\"caption-attachment-1970\" class=\"wp-caption-text\">My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb. Swan template: Erik Demaine.<\/p><\/div>\n<p class=\"p1\"><span class=\"s1\">In addition to Demaine\u2019s swan, I brought templates for lots of different shapes from <a href=\"http:\/\/mrhonner.com\/fun-with-folding\"><span class=\"s2\">Patrick Honner<\/span><\/a> and <span class=\"s2\"><a href=\"http:\/\/jdh.hamkins.org\/math-for-nine-year-olds-fold-punch-cut\/\">Joel David Hamkins<\/a>, who uses hole punching symmetry activities as a warm-up for cutting<\/span>. I also got ideas from <a href=\"https:\/\/mikesmathpage.wordpress.com\/2015\/09\/28\/fold-and-cut-part-3\/\"><span class=\"s2\">Mike Lawler<\/span><\/a>, who has done fold and cut activities with kids, and <a href=\"http:\/\/blogs.cofc.edu\/owensks\/2016\/03\/23\/paper-folding\/\"><span class=\"s2\">Kate Owens<\/span><\/a>, who\u00a0ran\u00a0a fold-and-cut workshop for teachers.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">I&#8217;ve done a little bit of origami, but I\u2019ve never gotten good enough to feel like I had geometric intuition for doing it. I\u2019m still at the level where I follow directions and get what the book says I should. Making these fold-and-cut shapes, though, is an easy way to start\u00a0thinking about paper folding mathematically and creatively. Thanks to the resources I mentioned above,\u00a0you too can easily introduce people to\u00a0the joys of mathematical paper folding.<\/span><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>Yesterday, I led a meeting of a Teachers\u2019 Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. I &hellip; <a href=\"https:\/\/blogs.ams.org\/blogonmathblogs\/2016\/04\/20\/fold-your-way-to-glory\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/blogonmathblogs\/2016\/04\/20\/fold-your-way-to-glory\/><\/div>\n","protected":false},"author":61,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3,4],"tags":[560,559,563,162,561,564,512,181,562,220],"class_list":["post-1969","post","type-post","status-publish","format-standard","hentry","category-math-education","category-mathematics-and-the-arts","tag-erik-demaine","tag-fold-and-cut-theorem","tag-joel-david-hamkins","tag-kate-owens","tag-katie-steckles","tag-math-teachers-circle","tag-mike-lawler","tag-mikesmathpage","tag-numberphile","tag-origami"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3tW3N-vL","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/1969","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/users\/61"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/comments?post=1969"}],"version-history":[{"count":4,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/1969\/revisions"}],"predecessor-version":[{"id":1975,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/posts\/1969\/revisions\/1975"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/media?parent=1969"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/categories?post=1969"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/blogonmathblogs\/wp-json\/wp\/v2\/tags?post=1969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}