{"id":4652,"date":"2012-10-09T08:20:43","date_gmt":"2012-10-09T08:20:43","guid":{"rendered":"http:\/\/meeseeks:5080\/blog\/?p=4652"},"modified":"2021-12-20T23:03:35","modified_gmt":"2021-12-20T23:03:35","slug":"polygon-construction","status":"publish","type":"post","link":"https:\/\/vukutu.com\/blog\/2012\/10\/polygon-construction\/","title":{"rendered":"Polygon construction"},"content":{"rendered":"<p>Mathematician <a href=\"http:\/\/seaneberhard.blogspot.co.uk\/\" target=\"_blank\" rel=\"noopener\">Sean Eberhard<\/a> has a nice post about <a href=\"https:\/\/randompermutations.com\/2012\/09\/13\/constructible-regular-polygons\/\" target=\"_blank\" rel=\"noopener\">constructible regular polygons<\/a>, giving a proof of a characterization of the n-sided polygons (aka n-gons) which are constructible only with a ruler and a compass. Those which are so constructible correspond to <em>n<\/em> being decomposable into a power of 2 and a product of primes of a certain form:<\/p>\n<p><b>Theorem<\/b> The regular n-gon is constructible by ruler and compass if and only if n has the form  p_1  * . . . . * p_l * 2^k, where p_1, . . . , p_l are distinct primes of the form 2^{2^m} + 1.<\/p>\n<p>That physical geometric actions should map to &#8211; and from &#8211; certain prime numbers is a good example of some of the deep interactions that exist between different parts of mathematics, interactions that often take us by surprise and usually compel our wonder.<br \/>\nOne question that immediately occurs to me is whether there are other instruments besides ruler and compass which, jointly with those two instruments, would enable n-gon construction for other values of n.&nbsp;&nbsp; Indeed, is there a collection of instruments (presumably some of them &#8220;non-constructible&#8221; or infinite in themselves) which would eventually garner all n, or at least other interesting subsets of the natural numbers?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mathematician Sean Eberhard has a nice post about constructible regular polygons, giving a proof of a characterization of the n-sided polygons (aka n-gons) which are constructible only with a ruler and a compass. Those which are so constructible correspond to n being decomposable into a power of 2 and a product of primes of a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[50],"tags":[],"class_list":["post-4652","post","type-post","status-publish","format-standard","hentry","category-mathematics","p1","y2012","m10","d09","h08"],"_links":{"self":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/4652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/comments?post=4652"}],"version-history":[{"count":4,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/4652\/revisions"}],"predecessor-version":[{"id":10132,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/4652\/revisions\/10132"}],"wp:attachment":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/media?parent=4652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/categories?post=4652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/tags?post=4652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}