{"id":3984,"date":"2024-06-01T12:14:39","date_gmt":"2024-06-01T16:14:39","guid":{"rendered":"https:\/\/josmfs.net\/?p=3984"},"modified":"2025-02-15T16:11:17","modified_gmt":"2025-02-15T21:11:17","slug":"classic-geometry-paradox","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2024\/06\/01\/classic-geometry-paradox\/","title":{"rendered":"Classic Geometry Paradox"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-3982\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Classic-Geometry-Paradox-Fig2.jpg\" alt=\"\" width=\"201\" height=\"182\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Classic-Geometry-Paradox-Fig2.jpg 493w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Classic-Geometry-Paradox-Fig2-300x272.jpg 300w\" sizes=\"auto, (max-width: 201px) 100vw, 201px\" \/>Coming across this <a href=\"https:\/\/www.futilitycloset.com\/2024\/04\/17\/constitutional-crisis\/\">classic geometric paradox<\/a> recently in <em>Futility Closet<\/em> motivated me to write down its solution in detail.<\/p>\n<p>\u201cWhere did the empty square come from?\u201d<\/p>\n<p>In any case, this is the canonical example for why I avoid visual geometric proofs\u2014you can be so easily fooled.\u00a0 Real proofs require plane or analytic geometry arguments.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Classic-Geometry-Paradox-240424.pdf\">Classic Geometry Paradox<\/a><\/p>\n<p><strong>(Update 9\/14\/2024) Penn &amp; Teller \u2013 Fool Us &#8211; Magic Trick<br \/>\n<\/strong><!--more--><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=RyMA1CY2MYY&amp;t=2730s\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-4277\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/02\/Classic-Geo-Paradox-Penn-and-Teller-Fig2.jpg\" alt=\"\" width=\"222\" height=\"131\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/02\/Classic-Geo-Paradox-Penn-and-Teller-Fig2.jpg 450w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2025\/02\/Classic-Geo-Paradox-Penn-and-Teller-Fig2-300x177.jpg 300w\" sizes=\"auto, (max-width: 222px) 100vw, 222px\" \/><\/a>I just saw a rerun of a Penn &amp; Teller \u2013 Fool Us <a href=\"https:\/\/www.youtube.com\/watch?v=RyMA1CY2MYY7&amp;t=2730s\">episode<\/a> (Season 10 Episode 17) in which they performed a magic trick (@45:30) that amazingly used a version of this classic geometric paradox.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/09\/Classic-Geo-Paradox-Penn-and-Teller.pdf\">Penn and Teller Magic Trick<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Coming across this classic geometric paradox recently in Futility Closet motivated me to write down its solution in detail. \u201cWhere did the empty square come from?\u201d In any case, this is the canonical example for why I avoid visual geometric proofs\u2014you can be so easily fooled.\u00a0 Real proofs require plane or analytic geometry arguments. See [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[16,279,266,278,13],"class_list":["post-3984","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-futility-closet","tag-magic","tag-paradoxes","tag-penn-and-teller","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3984","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=3984"}],"version-history":[{"count":4,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3984\/revisions"}],"predecessor-version":[{"id":4278,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3984\/revisions\/4278"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=3984"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=3984"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=3984"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}