{"id":361,"date":"2004-04-18T01:40:09","date_gmt":"2004-04-18T08:40:09","guid":{"rendered":"http:\/\/stefangeens.com\/?p=361"},"modified":"2004-04-18T01:40:09","modified_gmt":"2004-04-18T08:40:09","slug":"stockholm-random-observations","status":"publish","type":"post","link":"https:\/\/stefangeens.com\/2001-2013\/2004\/04\/stockholm-random-observations\/","title":{"rendered":"Stockholm: Random observations"},"content":{"rendered":"<p><span class=\"posted\">Today was the first<\/span> perfect spring day, as bright as a day can get, and Stockholmers walked about their city as if they had just been released from a dark room, squinting, and a little dazed. Tourists were easy to spot &mdash; they were the ones dressed rationally, whereas the locals were shedding layers like eskimos in a sauna. And not a leaf on the trees yet, though that should change within hours.<\/p>\n<p><span class=\"sg-marginalia-250\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" alt=\"make23.gif\" src=\"https:\/\/i0.wp.com\/www.stefangeens.com\/make23.gif?resize=208%2C202\" width=\"208\" height=\"202\" border=\"0\" \/><\/span><span class=\"posted\">Waiting for a train<\/span> at Stockholm S\u00f6dra Friday night, I started staring at the  platform pavement. <a href=\"http:\/\/www.scienceu.com\/geometry\/articles\/tiling\/penrose.html\">Penrose tiles<\/a>! These were invented (or discovered, depending on your take on mathematics) by <a href=\"http:\/\/en.wikipedia.org\/wiki\/Roger_Penrose\">Roger Penrose<\/a>: He figured out how to cover a surface with only two different kinds of tiles in a way that never repeats. It sounds impossible, but it&#8217;s not. Penrose used two kinds of rhombi, and then added some specific instructions as to how they are supposed to be placed next to each other. The result is a complex mathematical structure made from simple rules, putting it in the same seductive league as the Mandelbrot set and Wolfram&#8217;s <a href=\"http:\/\/www.wolframscience.com\/nksonline\/page-29\">Rule 30<\/a>.<\/p>\n<p>In S\u00f6dra, the rhombi are disguised, but cleverly so: By curving the edges in specific ways, Penrose&#8217;s rules are embodied in the shapes: It was impossible for the workmen building the platform to accidentally create a repeating pattern (unless they were to use only one shape exclusively, I think, but I am not sure about that, and need to go have another look).<\/p>\n<p>I started to wonder whether for each separate segment of pavement covered in tiles, the ratio of the frequency of the two shapes was similar. Luckily, the train arrived. Just now, <a href=\"http:\/\/ccins.camosun.bc.ca\/~jbritton\/goldengeom\/goldenpenrose.html\">I found out<\/a> that for large enough areas, this ratio converges on the golden mean. That is deep in ways I have not yet begun to fathom.<\/p>\n<p>I love public architecture that strives for obscure details that hide underfoot, literally. Next time, I will take my camera, take a picture, and post it here.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Today was the first perfect spring day, as bright as a day can get, and Stockholmers walked about their city as if they had just been released from a dark room, squinting, and a little dazed. Tourists were easy to &hellip; <a href=\"https:\/\/stefangeens.com\/2001-2013\/2004\/04\/stockholm-random-observations\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[8,10],"tags":[],"class_list":["post-361","post","type-post","status-publish","format-standard","hentry","category-science-technology","category-sweden"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7eNhC-5P","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/posts\/361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/comments?post=361"}],"version-history":[{"count":0,"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/posts\/361\/revisions"}],"wp:attachment":[{"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/media?parent=361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/categories?post=361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/stefangeens.com\/2001-2013\/wp-json\/wp\/v2\/tags?post=361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}