{"id":4246,"date":"2016-06-22T07:49:38","date_gmt":"2016-06-22T14:49:38","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=4246"},"modified":"2017-09-02T07:56:05","modified_gmt":"2017-09-02T14:56:05","slug":"hofstadters-quantum-mechanical-butterfly","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2016\/06\/22\/hofstadters-quantum-mechanical-butterfly\/","title":{"rendered":"Hofstadter&#8217;s Quantum-Mechanical Butterfly"},"content":{"rendered":"<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hofstadter%27s_butterfly\">Hofstadter&#8217;s Quantum-Mechanical Butterfly<\/a> relates to the fractional <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quantum_Hall_effect\">Quantum Hall Effect<\/a>, which is (IMHO) at the heart of understanding (interpreting) how QM really works!<\/p>\n<p>I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the <a href=\"http:\/\/demonstrations.wolfram.com\/HofstadtersQuantumMechanicalButterfly\/\">Hofstadter&#8217;s Quantum-Mechanical Butterfly<\/a><\/p>\n<p>I found an interesting pattern. By modifying the integer used in the solution i=12, the Golden Ratio \u0424=(1+Sqrt[5])\/2=1.618 &#8230; emerges within the butterfly! Can you find it (or 1\/\u0424=.618&#8230;)?<br \/>\nHint: Use the interactive version and mouse-over the red or green dots in the white space of the wings of butterfly.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/UEv3C9WFCUQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A snapshot of the code and last frame@n=50:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/06\/HofstadterButterfly-1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4249\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/06\/HofstadterButterfly-1.png\" alt=\"HofstadterButterfly\" width=\"684\" height=\"1001\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/06\/HofstadterButterfly-1.png 684w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/06\/HofstadterButterfly-1-342x500.png 342w\" sizes=\"auto, (max-width: 684px) 100vw, 684px\" \/><\/a><\/p>\n<p>If you have the <a title=\"free Mathematica  version of the CDF plugin\" href=\"http:\/\/www.wolfram.com\/cdf-player\/\" target=\"_blank\" rel=\"noopener\">free <em>Mathematica<\/em> CDF plugin<\/a> on a non-Chrome browser, you can interactively analyze this with n&lt;11 being presented with symbolic solutions in a mouse-over Tooltip (below).<\/p>\n<p><script type=\"text\/javascript\" src=\"http:\/\/www.wolfram.com\/cdf-player\/plugin\/v2.1\/cdfplugin.js\"><\/script><script type=\"text\/javascript\">\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[ var cdf = new cdfplugin(); cdf.embed('http:\/\/theoryofeverything.org\/TOE\/JGM\/HofstadterButterfly.cdf',1000,1000); \/\/ ]]><\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hofstadter&#8217;s Quantum-Mechanical Butterfly relates to the fractional Quantum Hall Effect, which is (IMHO) at the heart of understanding (interpreting) how QM really works! I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter&#8217;s Quantum-Mechanical Butterfly I found an interesting pattern. By modifying the integer used in &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2016\/06\/22\/hofstadters-quantum-mechanical-butterfly\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Hofstadter&#8217;s Quantum-Mechanical Butterfly<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[14,16],"class_list":["post-4246","post","type-post","status-publish","format-standard","hentry","category-physics","tag-math","tag-physics"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Hofstadter&#039;s Quantum-Mechanical Butterfly relates to the fractional Quantum Hall Effect, which is (IMHO) at the heart of understanding (interpreting) how QM really works! I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter&#039;s Quantum-Mechanical Butterfly I found an interesting pattern. By modifying the integer used in\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"jgmoxness\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/theoryofeverything.org\/theToE\/2016\/06\/22\/hofstadters-quantum-mechanical-butterfly\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 4.9.8\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Visualizing a Theory of Everything! - Dedicated to the pursuit of beauty and Truth in Nature!\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Hofstadter\u2019s Quantum-Mechanical Butterfly - Visualizing a Theory of Everything!\" \/>\n\t\t<meta property=\"og:description\" content=\"Hofstadter&#039;s Quantum-Mechanical Butterfly relates to the fractional Quantum Hall Effect, which is (IMHO) at the heart of understanding (interpreting) how QM really works! 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I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter&#039;s Quantum-Mechanical Butterfly I found an interesting pattern. 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I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter's Quantum-Mechanical Butterfly I found an interesting pattern. 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By modifying the integer used in"},"aioseo_meta_data":{"post_id":"4246","title":null,"description":null,"keywords":null,"keyphrases":null,"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":null,"og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"","isEnabled":true},"graphs":[],"defaultGraph":"","defaultPostTypeGraph":""},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":null,"robots_max_videopreview":null,"robots_max_imagepreview":"large","priority":null,"frequency":null,"local_seo":null,"limit_modified_date":false,"created":"2022-04-06 04:21:09","updated":"2025-06-10 15:47:16"},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/\" title=\"Physics\">Physics<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tHofstadter\u2019s Quantum-Mechanical Butterfly\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/theoryofeverything.org\/theToE"},{"label":"Physics","link":"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/"},{"label":"Hofstadter&#8217;s Quantum-Mechanical Butterfly","link":"https:\/\/theoryofeverything.org\/theToE\/2016\/06\/22\/hofstadters-quantum-mechanical-butterfly\/"}],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4246","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=4246"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4246\/revisions"}],"predecessor-version":[{"id":4407,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4246\/revisions\/4407"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=4246"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=4246"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=4246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}