{"id":3990,"date":"2024-06-08T11:02:05","date_gmt":"2024-06-08T15:02:05","guid":{"rendered":"https:\/\/josmfs.net\/?p=3990"},"modified":"2024-06-08T12:14:11","modified_gmt":"2024-06-08T16:14:11","slug":"hjelmslevs-theorem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2024\/06\/08\/hjelmslevs-theorem\/","title":{"rendered":"Hjelmslev\u2019s Theorem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-3988\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Hjelmslevs-Theorem-Fig2.jpg\" alt=\"\" width=\"201\" height=\"158\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Hjelmslevs-Theorem-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Hjelmslevs-Theorem-Fig2-300x236.jpg 300w\" sizes=\"auto, (max-width: 201px) 100vw, 201px\" \/>I came across this remarkable <a href=\"https:\/\/www.futilitycloset.com\/2024\/05\/28\/hjelmslevs-theorem\/\">result<\/a> in Futility Closet:<\/p>\n<p>\u201cOn each of these two black lines is a trio of red points marked by the same distances.\u00a0 The midpoints of segments drawn between corresponding points are collinear.<\/p>\n<p>(Discovered by Danish mathematician Johannes Hjelmslev.)\u201d<\/p>\n<p>This result seems amazing and mysterious.\u00a0 I wondered if I could think of a proof.\u00a0 I found a simple approach that did not use plane geometry.\u00a0 And suddenly, like a magic trick exposed, the result seemed obvious.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/06\/Hjelmslevs-Theorem-240603.pdf\">Hjelmslev\u2019s Theorem<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I came across this remarkable result in Futility Closet: \u201cOn each of these two black lines is a trio of red points marked by the same distances.\u00a0 The midpoints of segments drawn between corresponding points are collinear. (Discovered by Danish mathematician Johannes Hjelmslev.)\u201d This result seems amazing and mysterious.\u00a0 I wondered if I could think [&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":[40,32,16,31],"class_list":["post-3990","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-analytic-geometry","tag-calculus","tag-futility-closet","tag-vectors"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3990","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=3990"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3990\/revisions"}],"predecessor-version":[{"id":3995,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3990\/revisions\/3995"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=3990"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=3990"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=3990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}