{"id":42885,"date":"2025-04-14T12:41:35","date_gmt":"2025-04-14T12:41:35","guid":{"rendered":"https:\/\/idea.membrane.com\/songwriter\/?p=42885"},"modified":"2025-04-14T13:49:03","modified_gmt":"2025-04-14T13:49:03","slug":"cracked-fractal","status":"publish","type":"post","link":"https:\/\/kingarthur.com\/songwriter\/2025\/04\/14\/cracked-fractal\/","title":{"rendered":"Cracked Fractal"},"content":{"rendered":"<p><a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XII\/Cracked-Fractal-I.mp3\">Cracked-Fractal-I.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XII\/Cracked-Fractal-I.mp4\">Cracked-Fractal-I.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XII\/Cracked-Fractal-II.mp3\">Cracked-Fractal-II.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XII\/Cracked-Fractal-II.mp4\">Cracked-Fractal-II.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XII\/Cracked-Fractal-intro.mp3\">Cracked-Fractal-intro.mp3<\/a><\/p>\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/idea.membrane.com\/songwriter\/wp-content\/uploads\/2025\/04\/branchin.gif\"><img decoding=\"async\" width=\"204\" height=\"48\" loading=\"lazy\" src=\"https:\/\/idea.membrane.com\/songwriter\/wp-content\/uploads\/2025\/04\/branchin.gif\" alt=\"\" class=\"wp-image-42902\" style=\"width:308px;height:auto\"\/><\/a><\/figure>\n\n\n<p>[Intro]<br \/>\nCracked fractal<\/p>\n<p>Took a branch<br \/>\n(By chance)<\/p>\n<p>[Bridge]<br \/>\nA system under stress (near collapse)<br \/>\nUnder duress (synapse relapse)<\/p>\n<p>[Verse 1]<br \/>\nAnd so we split apart<br \/>\n(Didn&#8217;t have the heart)<br \/>\nDashed! A new start<br \/>\n(Spreading further apart)<\/p>\n<p>[Bridge]<br \/>\nIt seems our seems&#8230; are<br \/>\n(Our splitting afar)<\/p>\n<p>[Chorus]<br \/>\nCracked (fractal)<br \/>\nTook a branch<br \/>\n(By chance)<br \/>\nCracked (factual)<br \/>\nSuch a stance<br \/>\n(Dirge dance)<\/p>\n<p>[Bridge]<br \/>\nA system under stress (near collapse)<br \/>\nUnder duress (synapse relapse)<\/p>\n<p>[Verse 2]<br \/>\nRocked! My windshield<br \/>\n(Visions had to yield)<br \/>\nSmashed! Frozen heart<br \/>\n(Splintering further apart)<\/p>\n<p>[Bridge]<br \/>\nIt seems our seems&#8230; are<br \/>\n(Our splitting afar)<\/p>\n<p>[Chorus]<br \/>\nCracked (fractal)<br \/>\nTook a branch<br \/>\n(By chance)<br \/>\nCracked (factual)<br \/>\nSuch a stance<br \/>\n(Dirge dance)<\/p>\n<p>[Outro]<br \/>\nA system under stress (near collapse)<br \/>\nUnder duress (synapse relapse)<\/p>\n<p>A SCIENCE NOTE: What\u2019s a \u201cCracked\u201d Fractal?<\/p>\n<article class=\"text-token-text-primary w-full\" dir=\"auto\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"true\">\n<div class=\"text-base my-auto mx-auto py-5 [--thread-content-margin:--spacing(4)] @[37rem]:[--thread-content-margin:--spacing(6)] @[70rem]:[--thread-content-margin:--spacing(12)] px-(--thread-content-margin)\">\n<div class=\"[--thread-content-max-width:32rem] @[34rem]:[--thread-content-max-width:40rem] @[64rem]:[--thread-content-max-width:48rem] mx-auto flex max-w-(--thread-content-max-width) flex-1 text-base gap-4 md:gap-5 lg:gap-6 group\/turn-messages focus-visible:outline-hidden\" tabindex=\"-1\">\n<div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\">\n<div class=\"relative flex-col gap-1 md:gap-3\">\n<div class=\"flex max-w-full flex-col grow\">\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-5\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"9660802a-9f6b-4f63-b519-26c9d6da2047\" data-message-model-slug=\"gpt-4o\">\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\">\n<div class=\"markdown prose dark:prose-invert w-full break-words light\">\n<p class=\"\" data-start=\"0\" data-end=\"187\">A <strong data-start=\"64\" data-end=\"102\">&#8220;cracked glass&#8221; look and branching fractal<\/strong>, ties into deep ideas in <strong data-start=\"128\" data-end=\"144\">chaos theory<\/strong>, <strong data-start=\"146\" data-end=\"158\">fractals<\/strong>, and <strong data-start=\"164\" data-end=\"186\">nonlinear dynamics<\/strong>.<\/p>\n<h3 class=\"\" data-start=\"216\" data-end=\"247\">Chaos Theory: The Basics<\/h3>\n<ul data-start=\"249\" data-end=\"470\">\n<li class=\"\" data-start=\"249\" data-end=\"382\">\n<p class=\"\" data-start=\"251\" data-end=\"382\"><strong data-start=\"251\" data-end=\"267\">Chaos theory<\/strong> studies systems that appear random, but are actually <strong data-start=\"321\" data-end=\"381\">deterministic and highly sensitive to initial conditions<\/strong>.<\/p>\n<\/li>\n<li class=\"\" data-start=\"383\" data-end=\"470\">\n<p class=\"\" data-start=\"385\" data-end=\"470\">Small changes lead to <strong data-start=\"407\" data-end=\"436\">vastly different outcomes<\/strong> \u2014 this is the &#8220;butterfly effect.&#8221;<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"477\" data-end=\"501\">Fractals in Chaos<\/h3>\n<ul data-start=\"503\" data-end=\"716\">\n<li class=\"\" data-start=\"503\" data-end=\"593\">\n<p class=\"\" data-start=\"505\" data-end=\"593\">A <strong data-start=\"507\" data-end=\"518\">fractal<\/strong> is a self-similar geometric shape \u2014 it looks the same at different scales.<\/p>\n<\/li>\n<li class=\"\" data-start=\"594\" data-end=\"716\">\n<p class=\"\" data-start=\"596\" data-end=\"716\">In chaotic systems, fractals often describe the <strong data-start=\"644\" data-end=\"661\">&#8220;state space&#8221;<\/strong> \u2014 the map of all possible behaviors a system can take.<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"723\" data-end=\"757\">\u00a0What\u2019s a \u201cCracked\u201d Fractal?<\/h3>\n<p class=\"\" data-start=\"759\" data-end=\"889\">A <strong data-start=\"761\" data-end=\"782\">&#8220;cracked fractal&#8221;<\/strong> \u2014 especially one that looks like shattered glass with <strong data-start=\"837\" data-end=\"856\">branching paths<\/strong> \u2014 often arises in systems where:<\/p>\n<ol data-start=\"891\" data-end=\"1138\">\n<li class=\"\" data-start=\"891\" data-end=\"934\">\n<p class=\"\" data-start=\"894\" data-end=\"934\"><strong data-start=\"894\" data-end=\"933\">The attractor is broken or unstable<\/strong>.<\/p>\n<\/li>\n<li class=\"\" data-start=\"935\" data-end=\"1022\">\n<p class=\"\" data-start=\"938\" data-end=\"1022\"><strong data-start=\"938\" data-end=\"955\">Singularities<\/strong> (discontinuities, infinite gradients, or undefined regions) occur.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1023\" data-end=\"1138\">\n<p class=\"\" data-start=\"1026\" data-end=\"1138\">The system is near a <strong data-start=\"1047\" data-end=\"1077\">critical bifurcation point<\/strong> \u2014 where a qualitative change in behavior is about to happen.<\/p>\n<\/li>\n<\/ol>\n<p class=\"\" data-start=\"1140\" data-end=\"1185\">This kind of structure typically shows up in:<\/p>\n<h4 class=\"\" data-start=\"1187\" data-end=\"1242\">\ud83d\udd39 1. <strong data-start=\"1198\" data-end=\"1242\">Fractured Attractors \/ Broken Symmetries<\/strong><\/h4>\n<ul data-start=\"1246\" data-end=\"1453\">\n<li class=\"\" data-start=\"1246\" data-end=\"1348\">\n<p class=\"\" data-start=\"1248\" data-end=\"1348\">Normally smooth chaotic attractors become <strong data-start=\"1290\" data-end=\"1304\">fragmented<\/strong> when the system is pushed past a threshold.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1352\" data-end=\"1453\">\n<p class=\"\" data-start=\"1354\" data-end=\"1453\">You get <strong data-start=\"1362\" data-end=\"1389\">fractal discontinuities<\/strong> where the structure literally &#8220;breaks apart&#8221; \u2014 like <strong data-start=\"1442\" data-end=\"1452\">cracks<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1455\" data-end=\"1490\">\ud83d\udd39 2. <strong data-start=\"1466\" data-end=\"1490\">Escape-Time Fractals<\/strong><\/h4>\n<ul data-start=\"1494\" data-end=\"1737\">\n<li class=\"\" data-start=\"1494\" data-end=\"1553\">\n<p class=\"\" data-start=\"1496\" data-end=\"1553\">Generated by iterating a function (e.g., Mandelbrot set).<\/p>\n<\/li>\n<li class=\"\" data-start=\"1557\" data-end=\"1649\">\n<p class=\"\" data-start=\"1559\" data-end=\"1649\">The &#8220;cracks&#8221; often represent <strong data-start=\"1588\" data-end=\"1602\">boundaries<\/strong> between regions of vastly different behaviors.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1653\" data-end=\"1737\">\n<p class=\"\" data-start=\"1655\" data-end=\"1737\">Similar structures: <strong data-start=\"1675\" data-end=\"1689\">Julia Sets<\/strong>, <strong data-start=\"1691\" data-end=\"1715\">Burning Ship fractal<\/strong>, <strong data-start=\"1717\" data-end=\"1736\">Newton fractals<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1739\" data-end=\"1774\">\ud83d\udd39 3. <strong data-start=\"1750\" data-end=\"1774\">Bifurcation Diagrams<\/strong><\/h4>\n<ul data-start=\"1778\" data-end=\"1899\">\n<li class=\"\" data-start=\"1778\" data-end=\"1899\">\n<p class=\"\" data-start=\"1780\" data-end=\"1899\">When zoomed in, the branches from a bifurcation tree can resemble <strong data-start=\"1846\" data-end=\"1865\">shattered glass<\/strong>, especially near chaotic regimes.<\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1901\" data-end=\"1940\">\ud83d\udd39 4. <strong data-start=\"1912\" data-end=\"1940\">Fractal Basin Boundaries<\/strong><\/h4>\n<ul data-start=\"1944\" data-end=\"2257\">\n<li class=\"\" data-start=\"1944\" data-end=\"2091\">\n<p class=\"\" data-start=\"1946\" data-end=\"2091\">Imagine you&#8217;re dropping a ball into a landscape \u2014 depending on the tiniest change in the start point, the ball might roll into different valleys.<\/p>\n<\/li>\n<li class=\"\" data-start=\"2095\" data-end=\"2257\">\n<p class=\"\" data-start=\"2097\" data-end=\"2257\">The dividing lines (basins of attraction) between outcomes can have <strong data-start=\"2165\" data-end=\"2216\">extremely fine, cracked, branch-like boundaries<\/strong> \u2014 an expression of sensitive dependence.<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"2264\" data-end=\"2319\">\u00a0Mathematical Sources of the Cracked Fractal Form<\/h3>\n<ul data-start=\"2321\" data-end=\"2582\">\n<li class=\"\" data-start=\"2321\" data-end=\"2404\">\n<p class=\"\" data-start=\"2323\" data-end=\"2404\"><strong data-start=\"2323\" data-end=\"2354\">Nonlinear complex functions<\/strong> \u2014 e.g., Newton\u2019s method applied to complex roots.<\/p>\n<\/li>\n<li class=\"\" data-start=\"2405\" data-end=\"2494\">\n<p class=\"\" data-start=\"2407\" data-end=\"2494\"><strong data-start=\"2407\" data-end=\"2433\">Piecewise chaotic maps<\/strong> \u2014 systems that abruptly switch rules, causing fragmentation.<\/p>\n<\/li>\n<li class=\"\" data-start=\"2495\" data-end=\"2582\">\n<p class=\"\" data-start=\"2497\" data-end=\"2582\"><strong data-start=\"2497\" data-end=\"2523\">Singular perturbations<\/strong> \u2014 when small smoothing is removed, the system can &#8220;crack.&#8221;<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"2589\" data-end=\"2616\">\u00a0Real-World Analogies<\/h3>\n<ul data-start=\"2618\" data-end=\"3004\">\n<li class=\"\" data-start=\"2618\" data-end=\"2689\">\n<p class=\"\" data-start=\"2620\" data-end=\"2689\"><strong data-start=\"2620\" data-end=\"2639\">Cracks in glass<\/strong> follow fractal patterns, especially under stress.<\/p>\n<\/li>\n<li class=\"\" data-start=\"2690\" data-end=\"2819\">\n<p class=\"\" data-start=\"2692\" data-end=\"2819\"><strong data-start=\"2692\" data-end=\"2710\">River networks<\/strong> and <strong data-start=\"2715\" data-end=\"2734\">lightning bolts<\/strong> also exhibit branching fractals \u2014 reflecting energy dispersal through complex media.<\/p>\n<\/li>\n<li class=\"\" data-start=\"2820\" data-end=\"3004\">\n<p class=\"\" data-start=\"2822\" data-end=\"3004\"><strong data-start=\"2822\" data-end=\"2843\">Financial crashes<\/strong>, <strong data-start=\"2845\" data-end=\"2866\">neural breakdowns<\/strong>, and <strong data-start=\"2872\" data-end=\"2898\">climate tipping points<\/strong> sometimes exhibit this \u201ccracked\u201d structure in models \u2014 suggesting a system under stress or near collapse.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n<h3>From the album &#8220;<a href=\"\/songwriter\/tag\/deviation\/\">Deviation<\/a>&#8220;<\/h3>\n<h2><a href=\"http:\/\/membrane.com\/global_warming\/\">The Human Induced Climate Change Experiment<\/a><\/h2>\n\n","protected":false},"excerpt":{"rendered":"<p>Cracked-Fractal-I.mp3 Cracked-Fractal-I.mp4 Cracked-Fractal-II.mp3 Cracked-Fractal-II.mp4 Cracked-Fractal-intro.mp3 [Intro] Cracked fractal Took a branch (By chance) [Bridge] A system under stress (near collapse) Under duress (synapse relapse) [Verse 1] And so we split apart (Didn&#8217;t have the heart) Dashed! A new start (Spreading further apart) [Bridge] It seems our seems&#8230; are (Our splitting afar) [Chorus] Cracked (fractal) Took [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[3,5,9,14],"tags":[163,4118],"class_list":["post-42885","post","type-post","status-publish","format-standard","hentry","category-4d-music","category-chaos-theory","category-daniel","category-lyrics","tag-climate-change","tag-deviation"],"_links":{"self":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/42885","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/comments?post=42885"}],"version-history":[{"count":9,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/42885\/revisions"}],"predecessor-version":[{"id":42907,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/42885\/revisions\/42907"}],"wp:attachment":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/media?parent=42885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/categories?post=42885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/tags?post=42885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}