{"id":2121,"date":"2021-05-01T08:03:48","date_gmt":"2021-05-01T12:03:48","guid":{"rendered":"https:\/\/josmfs.net\/?p=2121"},"modified":"2021-05-01T08:03:48","modified_gmt":"2021-05-01T12:03:48","slug":"a-self-characterizing-figure","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2021\/05\/01\/a-self-characterizing-figure\/","title":{"rendered":"A Self-Characterizing Figure"},"content":{"rendered":"<p><em><a href=\"https:\/\/www.futilitycloset.com\/2021\/03\/17\/a-self-characterizing-figure\/\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2117\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Self-Characterizing-Fig-Fig2.jpg\" alt=\"\" width=\"201\" height=\"205\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Self-Characterizing-Fig-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Self-Characterizing-Fig-Fig2-294x300.jpg 294w\" sizes=\"auto, (max-width: 201px) 100vw, 201px\" \/>Futility Closet<\/a><\/em> describes a result that is startling, amazing, and mysterious.<\/p>\n<p>\u201cThis is pretty: If you choose <em>n<\/em> &gt; 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals <em>n<\/em>.\u201d<\/p>\n<p>The <em>Futility Closet<\/em> posting includes an interactive display using Wolfram Technology by Jay Warendorff that let\u2019s you select different <em>n<\/em> and see the results.\u00a0 It also includes a reference to a paper that proves the result; only the paper uses residue theory from complex variables, which seems a bit over-kill, though slick, for such a problem.\u00a0 I found a simpler route.<\/p>\n<p>See a <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Self-Characterizing-Figure-210421.pdf\">Self-Characterizing Figure<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Futility Closet describes a result that is startling, amazing, and mysterious. \u201cThis is pretty: If you choose n &gt; 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals n.\u201d The Futility Closet posting includes an interactive display [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[51,16],"class_list":["post-2121","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-complex-numbers","tag-futility-closet"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2121","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=2121"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2121\/revisions"}],"predecessor-version":[{"id":2122,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2121\/revisions\/2122"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}