{"id":294,"date":"2010-11-07T22:30:57","date_gmt":"2010-11-07T22:30:57","guid":{"rendered":"http:\/\/doodlesprocket.com\/z303\/?p=294"},"modified":"2025-03-17T22:01:30","modified_gmt":"2025-03-17T22:01:30","slug":"fractals-an-excuse-to-show-some-pretty-pictures","status":"publish","type":"post","link":"https:\/\/www.z303.org\/?p=294","title":{"rendered":"Fractals (An excuse to show some pretty pictures)"},"content":{"rendered":"<p>This is a blog post version of my talk at <a href=\"http:\/\/lanyrd.com\/2010\/bathcamp-barcamp-2010\/\">Bathcamp 2010<\/a>. &nbsp;I&#8217;ve added a few extra links and pointers&nbsp;for some extra background. As a warning this post is heavy on images and video.<\/p>\n<p><!--more--><\/p>\n<p>\nThe term <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fractal\">Fractal<\/a> was coined by this man:\n<\/p>\n<p style=\"text-align: center\"><a href=\"http:\/\/commons.wikimedia.org\/wiki\/File:Benoit_Mandelbrot_mg_1804.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-297\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/Benoit_Mandelbrot_mg_18041.jpg\" alt=\"Beno\u00eet Mandelbrot\" width=\"599\" height=\"600\" \/><\/a> <a href=\"http:\/\/en.wikipedia.org\/wiki\/Beno%C3%AEt_Mandelbrot\">Beno\u00eet Mandelbrot <\/a> (1924 &#8211; 2010)<\/p>\n<p>He&#8217;s most famous for a simple equation:<\/p>\n<p style=\"text-align: center\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Mandelbrot_set\">f(z) = z<sup>2<\/sup> + c<\/a><\/p>\n<p style=\"text-align: center\">where z and c are <a href=\"http:\/\/en.wikipedia.org\/wiki\/Complex_number\">complex numbers<\/a><\/p>\n<p>\nBut we are much more used to seeing the equation like this\n<\/p>\n<p>\n<a href=\"http:\/\/commons.wikimedia.org\/wiki\/File:Mandelbrot_set_with_coloured_environment.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-306\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/Mandelbrot_set_with_coloured_environment.png\" alt=\"Mandelbrot set\" width=\"600\" height=\"450\" \/><\/a>\n<\/p>\n<p>\nA definition of a Fractal: <\/p>\n<blockquote><p>A rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole<\/p><\/blockquote>\n<p style=\"text-align: right\"><a href=\"http:\/\/books.google.com\/books?id=KN9nSQAACAAJ&amp;dq=isbn:0716711869&amp;cd=1\">The fractal geometry of nature<\/a><\/p>\n<\/p>\n<p>\nExamples exist in natural, such as <a href=\"http:\/\/en.wikipedia.org\/wiki\/Romanesco_broccoli\">Romanesco broccoli<\/a>:\n<\/p>\n<p>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Romanesco_Brassica_oleracea_Richard_Bartz.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/Romanesco_Brassica_oleracea_Richard_Bartz.jpg\" alt=\"Romanesco broccoli\" width=\"600\" height=\"440\" class=\"aligncenter size-full wp-image-316\" \/><\/a>\n<\/p>\n<p>\nOne of the strange properties of Fractals is their dimension\n<\/p>\n<p>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/line.png\" alt=\"line\" width=\"408\" height=\"85\" class=\"aligncenter size-full wp-image-327\" \/>\n<\/p>\n<p>\nLines are one-dimensional,\n<\/p>\n<p>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/rect.png\" alt=\"Rectangle\" width=\"408\" height=\"288\" class=\"aligncenter size-full wp-image-330\" \/>\n<\/p>\n<p>\nBoxes are two-dimensional,\n<\/p>\n<p>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/koch.png\" alt=\"Koch Curve\" width=\"408\" height=\"226\" class=\"aligncenter size-full wp-image-332\" \/>\n<\/p>\n<p>\nWhile a Fractal (In this case a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Koch_curve\">Koch Curve<\/a>) is <a href=\"http:\/\/en.wikipedia.org\/wiki\/List_of_fractals_by_Hausdorff_dimension\">one and a bit<\/a>\n<\/p>\n<p>\nSo some other famous fractals:\n<\/p>\n<p>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Julia_set\">Julia sets<\/a>\n<\/p>\n<p>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/Julia_0.4_0.6.png\" alt=\"Julia Set\" width=\"600\" height=\"450\" class=\"aligncenter size-full wp-image-335\" \/>\n<\/p>\n<p>\nThis use the same equation as Mandelbrot Sets but instead of varying c for a fixed z, keeps c constant and varies z.\n<\/p>\n<p>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/L-system\">Lindenmayer systems<\/a>\n<\/p>\n<p>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/z303.wpengine.com\/wp-content\/uploads\/2010\/11\/Fractal_tree.jpg\" alt=\"Fractal tree\" width=\"600\" height=\"450\" class=\"aligncenter size-full wp-image-338\" \/>\n<\/p>\n<p>\nThese follow a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Formal_grammar\">grammar<\/a>, so are great at <a href=\"http:\/\/www.delicious.com\/z303\/lsystems\">generating plants and trees<\/a>.\n<\/p>\n<p>\nMandelbrot sets can also be extended into 3D to create a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Mandelbulb\">Mandelbulb<\/a>\n<\/p>\n<p>\n<iframe loading=\"lazy\" title=\"Mandelbulb (raymarched SDF)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/eKUh4nkmQbc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/p>\n<p>\nAlso Julia Sets can be similarly extend into 4D space producing <a href=\"http:\/\/local.wasp.uwa.edu.au\/~pbourke\/fractals\/quatjulia\/\">Quaternion Julia Sets<\/a>, which can be <a href=\"http:\/\/www.devmaster.net\/forums\/showthread.php?t=4448\">raytraced on a modern GPU<\/a>.\n<\/p>\n<p>\n<iframe loading=\"lazy\" title=\"4D Julia Fractal - Constant cycle\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/GtrO6jPbbUo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/p>\n<p>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Menger_sponge\">Menger sponge<\/a> is another type of space-filling curve like the Koch Curve above.\n<\/p>\n<p>\n<iframe loading=\"lazy\" title=\"tbc - spongy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/36BPql6Nl_U?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/p>\n<p>\nThis video is of the 128 byte <a href=\"http:\/\/en.wikipedia.org\/wiki\/Demoscene\">demo<\/a> called <a href=\"http:\/\/pouet.net\/prod.php?which=53871\">Spongy<\/a> by <a href=\"http:\/\/pouet.net\/groups.php?which=1623\">TBC<\/a>, which has also been ported to a <a href=\"http:\/\/www.p01.org\/releases\/512b_jspongy\/\">512 byte JavaScript version<\/a>.<\/div>\n<\/p>\n<p>\nAfter the Mandelbrot set, other famous use of fractals is in <a href=\"http:\/\/www.delicious.com\/z303\/terrain\">generating terrain<\/a>. An early example is Peak from 1980. They are also used in the genesis sequence in Star Trek II: The Wrath of Khan\n<\/p>\n<div><\/div>\n<p>\n<iframe loading=\"lazy\" title=\"Peak (~1980)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/YRy3J-_u_Jg?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/p>\n<p>Fractals have not just found use in generating pretty pictures. The Space filling curves mentioned earlier have found a use in creating <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fractal_antenna\">compact aerials<\/a> in mobile devices. <a href=\"http:\/\/en.wikipedia.org\/wiki\/Iterated_function_system\">Iterated function systems<\/a> like the previously mentioned Lindenmayer systems have been used for <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fractal_compression\">image compression<\/a> but due to some early patents work has slowed because of the legal risks. <a href=\"http:\/\/www.scientificamerican.com\/article.cfm?id=multifractals-explain-wall-street\">Modelling financial markets<\/a> is another application, originally proposed by Beno\u00eet Mandelbrot , which over the last few years has been enjoying renewed interest.\n<\/p>\n<p>\nRendering of fractal landscapes has also not stood still, with the power of modern hardware, much more realistic terrain can now be rendered in real-time instead of to a film recorder as in the past,  producing result like, <a href=\"http:\/\/pouet.net\/prod.php?which=52938\">Elevated<\/a> by <a href=\"http:\/\/www.rgba.org\/\">RGBA<\/a> &amp; <a href=\"http:\/\/pouet.net\/groups.php?which=1623\">TBC<\/a>, a 4 kilobyte demo running on a normal PC.\n<\/p>\n<p>\n<iframe loading=\"lazy\" title=\"Elevated HQ by RGBA and TBC\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/I5CTFMuFvb0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/p>\n<p>\nFor any technical people, one of the programmers gave a presentation on how the terrain is <a href=\"http:\/\/iquilezles.org\/www\/material\/function2009\/function2009.htm\">generated and rendered<\/a> (<a href=\"http:\/\/iquilezles.org\/www\/material\/function2009\/function2009.pdf\">PDF<\/a>) if you are interested.\n<\/p>\n<p>\nI have more links on the <a href=\"http:\/\/pinboard.in\/u:Z303\/t:fractals\/\">fractals<\/a> tag on pinboard is you want to explore some more.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a blog post version of my talk at Bathcamp 2010. &nbsp;I&#8217;ve added a few extra links and pointers&nbsp;for some extra background. As a warning this post is heavy on images and video.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_genesis_block_theme_hide_title":false,"footnotes":""},"categories":[11,21],"tags":[],"class_list":["post-294","post","type-post","status-publish","format-standard","hentry","category-maths","category-talks","without-featured-image"],"featured_image_src":null,"featured_image_src_square":null,"author_info":{"display_name":"Zoe","author_link":"https:\/\/www.z303.org\/?author=1"},"_links":{"self":[{"href":"https:\/\/www.z303.org\/index.php?rest_route=\/wp\/v2\/posts\/294","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.z303.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.z303.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.z303.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.z303.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=294"}],"version-history":[{"count":0,"href":"https:\/\/www.z303.org\/index.php?rest_route=\/wp\/v2\/posts\/294\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.z303.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=294"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.z303.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=294"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.z303.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}