{"id":249,"date":"2013-05-27T21:12:59","date_gmt":"2013-05-28T04:12:59","guid":{"rendered":"http:\/\/puzzlezapper.com\/blog\/?p=249"},"modified":"2013-05-27T21:12:59","modified_gmt":"2013-05-28T04:12:59","slug":"3-8-24","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2013\/05\/3-8-24\/","title":{"rendered":"3.8.24"},"content":{"rendered":"<p>A regular 24-gon, octagon, and triangle together form one of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Tiling_by_regular_polygons#Combinations_of_regular_polygons_that_can_meet_at_a_vertex\">the 17 ways<\/a> (called vertex figures) to surround a vertex with regular polygons. This vertex figure can&#8217;t do anything nice like tile the plane. You can surround the 24-gon with octagons and triangles, but then you have gaps that can&#8217;t be filled with regular polygons, and you have to stop.<\/p>\n<p><a href=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-rose.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-rose.png\" alt=\"24-8-3-rose\" width=\"313\" height=\"313\" class=\"aligncenter size-full wp-image-252\" srcset=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-rose.png 313w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-rose-150x150.png 150w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-rose-300x300.png 300w\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" \/><\/a><\/p>\n<p>Or, instead of stopping, you could take that 24-gon surrounded by octagons and triangles, and surround it with 12 more 24-gons surrounded by octagons and triangles, overlapping in an elegant sort of way:<\/p>\n<p><a href=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-doodle.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-doodle.png\" alt=\"24-8-3-doodle\" width=\"450\" height=\"450\" class=\"aligncenter size-full wp-image-250\" srcset=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-doodle.png 450w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-doodle-150x150.png 150w, https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2013\/05\/24-8-3-doodle-300x300.png 300w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/a><\/p>\n<p>And then, instead of stopping there, you could surround that whole thing with more copies of that whole thing&#8230;<\/p>\n<p>But sometimes art is where you stop.<\/p>\n<p>Hat tip to John Baez, who brought up the 3.7.42 vertex figure in a <a href=\"https:\/\/plus.google.com\/u\/0\/117663015413546257905\/posts\/VebrAE5kMNN\">Google+ post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A regular 24-gon, octagon, and triangle together form one of the 17 ways (called vertex figures) to surround a vertex with regular polygons. This vertex figure can&#8217;t do anything nice like tile the plane. You can surround the 24-gon with octagons and triangles, but then you have gaps that can&#8217;t be filled with regular polygons, &hellip; <a href=\"https:\/\/puzzlezapper.com\/blog\/2013\/05\/3-8-24\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">3.8.24<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[112,110,116,113,115,114,111],"class_list":["post-249","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-24-gons","tag-3-8-24","tag-art","tag-octagons","tag-tilings","tag-triangles","tag-vertex-figures"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/249","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/comments?post=249"}],"version-history":[{"count":4,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/249\/revisions"}],"predecessor-version":[{"id":255,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/249\/revisions\/255"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=249"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=249"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}