{"id":407,"date":"2013-08-21T16:01:57","date_gmt":"2013-08-21T20:01:57","guid":{"rendered":"https:\/\/jasoncantarella.com\/wordpress\/?p=407"},"modified":"2013-08-21T16:01:57","modified_gmt":"2013-08-21T20:01:57","slug":"folding-movies-illustrate-theorem-about-flat-surfaces","status":"publish","type":"post","link":"https:\/\/jasoncantarella.com\/wordpress\/folding-movies-illustrate-theorem-about-flat-surfaces\/","title":{"rendered":"Folding movies illustrate theorem about flat surfaces"},"content":{"rendered":"<p style=\"text-align: center;\"><a href=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/FoldingSplash.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-408 aligncenter\" alt=\"FoldingSplash\" src=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/FoldingSplash-300x161.png\" width=\"300\" height=\"161\" srcset=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/FoldingSplash-300x161.png 300w, https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/FoldingSplash-150x80.png 150w, https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/FoldingSplash.png 808w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>I learned the really interesting theorem from Joe Fu that a flat polyhedral surface always has the property that the spherical polygon enclosed by all the surface normals at a vertex always encloses area zero. As a favor to Joe, I made some neat Mathematica movies showing the folding process taking a flat sheet with three folds at 120 degree angles to a square pyramid that you get by folding a pair of faces into the center of the pyramid. The resulting animations are\u00a0<a href=\"https:\/\/jasoncantarella.com\/downloads\/FoldingMovie.mov\" rel=\"nofollow\">here<\/a>\u00a0and\u00a0<a href=\"https:\/\/jasoncantarella.com\/downloads\/TopViewFoldingMovie.mov\" rel=\"nofollow\">in a top view here<\/a>. The Mathematica notebook that I used to create them is<a href=\"https:\/\/jasoncantarella.com\/downloads\/FoldingDemonstration.nb\" rel=\"nofollow\">here<\/a>; it contains some code which computes the folded shape in terms of the first n-3 bending angles (the last three are determined by the intrinsic geometry of the surface).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I learned the really interesting theorem from Joe Fu that a flat polyhedral surface always has the property that the spherical polygon enclosed by all the surface normals at a vertex always encloses area zero. As a favor to Joe, I made some neat Mathematica movies showing the folding process taking a flat sheet with [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-407","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/comments?post=407"}],"version-history":[{"count":1,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/407\/revisions"}],"predecessor-version":[{"id":409,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/407\/revisions\/409"}],"wp:attachment":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/media?parent=407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/categories?post=407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/tags?post=407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}