{"id":1774,"date":"2017-03-14T00:24:40","date_gmt":"2017-03-14T00:24:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1774"},"modified":"2019-05-30T16:35:07","modified_gmt":"2019-05-30T16:35:07","slug":"generating-fractals-with-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/generating-fractals-with-complex-numbers\/","title":{"raw":"Generating Fractals With Complex Numbers","rendered":"Generating Fractals With Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the difference between an imaginary number and a complex number<\/li>\r\n \t<li>Identify the real and imaginary parts of a complex number<\/li>\r\n \t<li>Plot a complex number on the complex plane<\/li>\r\n \t<li>Perform arithmetic operations on complex numbers<\/li>\r\n \t<li>Graph physical representations of arithmetic operations on complex numbers as scaling or rotation<\/li>\r\n \t<li>Generate several terms of a recursive relation<\/li>\r\n \t<li>Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Complex Recursive Sequences<\/h2>\r\nSome fractals are generated with complex numbers. \u00a0The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.\r\n<div class=\"textbox\">\r\n<h3>Recursive Sequence<\/h3>\r\nA <strong>recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].\r\n\r\n<\/div>\r\nThe sequence of values produced is the recursive sequence.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2,\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.\r\n\r\nWe are given the starting value, [latex]{{z}_{0}}=4[\/latex]. The recursive formula holds for any value of <em>n<\/em>, so if [latex]n = 0[\/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[\/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[\/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[\/latex].\r\n\r\nNotice this defines [latex]{{z}_{1}}[\/latex] in terms of the known [latex]{{z}_{0}}[\/latex], so we can compute the value:\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[\/latex].<\/p>\r\nNow letting [latex]n = 1[\/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[\/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[\/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.\r\n<p style=\"text-align: center;\">[latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[\/latex]<\/p>\r\nContinuing,\r\n<p style=\"text-align: center;\">[latex]{{z}_{3}}={{z}_{2}}+2=8+2=10\\\\{{z}_{4}}={{z}_{3}}+2=10+2=12[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5816&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\\cdot{i}+(1-i),\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.\r\n\r\nWe are given [latex]{{z}_{0}}=4[\/latex]. Using the recursive formula:\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i\\\\{{z}_{2}}={{z}_{1}}\\cdot{i}+(1-i)=(1+3i)\\cdot{i}+(1-i)=i+3{{i}^{2}}+(1-i)=i-3+(1-i)=-2\\\\{{z}_{3}}={{z}_{2}}\\cdot{i}+(1-i)=(-2)\\cdot{i}+(1-i)=-2i+(1-i)=1-3i\\\\{{z}_{4}}={{z}_{3}}\\cdot{i}+(1-i)=(1-3i)\\cdot{i}+(1-i)=i-3{{i}^{2}}+(1-i)=i+3+(1-i)=4\\\\{{z}_{5}}={{z}_{4}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i[\/latex]<\/p>\r\nNotice this sequence is exhibiting an interesting pattern\u2014it began to repeat itself.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131230&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.\r\n\r\nhttps:\/\/youtu.be\/lOyusyTsLTs\r\n<h2>Mandelbrot Set<\/h2>\r\nThe Mandelbrot Set is a set of numbers defined based on recursive sequences.\r\n<div class=\"textbox\">\r\n<h3>Mandelbrot Set<\/h3>\r\nFor any complex number <em>c<\/em>, define the sequence [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]\r\n\r\nIf this sequence always stays close to the origin (within 2 units), then the number <em>c<\/em> is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number <em>c<\/em> is not part of the set.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine if [latex]c=1+i[\/latex] is part of the Mandelbrot set.\r\n\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+1+i=0+1+i=1+i\\\\{{z}_{2}}={{z}_{1}}^{2}+1+i={{(1+i)}^{2}}+1+i=1+3i\\\\{{z}_{3}}={{z}_{2}}^{2}+1+i={{(1+3i)}^{2}}+1+i=-7+7i\\\\{{z}_{4}}={{z}_{3}}^{2}+1+i={{(-7+7i)}^{2}}+1+i=1-97i[\/latex]<\/p>\r\nWe can already see that these values are getting quite large. It does not appear that [latex]c=1+i[\/latex] is part of the Mandelbrot set.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine if [latex]c=0.5i[\/latex] is part of the Mandelbrot set.\r\n\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations\r\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+0.5i=0+0.5i=0.5i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.5i={{(0.5i)}^{2}}+0.5i=-0.25+0.5i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]<\/p>\r\nWhile not definitive with this few iterations, it does appear that this value is remaining small, suggesting that 0.5<em>i<\/em> is part of the Mandelbrot set.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDetermine if [latex]c=0.4+0.3i[\/latex] is part of the Mandelbrot set.\r\n\r\n[reveal-answer q=\"31678\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"31678\"]\r\n\r\n[latex]{{z}_{1}}={{z}_{0}}^{2}+0.4+0.3i=0+0.4+0.3i=0.4+0.3i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.4+0.3i={{(0.4+0.3i)}^{2}}+0.4+0.3i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-1737 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"187\" height=\"127\" \/>If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.[footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png<\/a>[\/footnote]\r\n\r\nThe boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.\r\n\r\nWatch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.\r\n\r\nhttps:\/\/youtu.be\/ORqk5jAFpWg\r\n\r\nIn addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using [latex]c=1+i [\/latex] above, the sequence was distance 2 from the origin after only two recursions.\r\n\r\nFor some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below[footnote]This series was generated using Scott\u2019s Mandelbrot Set Explorer[\/footnote], light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in\u2014those are used for numbers that grow very slowly.\r\n\r\nThe Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<h2>Additional Resources<\/h2>\r\nA much more extensive coverage of fractals can be found on the <a href=\"http:\/\/classes.yale.edu\/fractals\/\">Fractal Geometry site<\/a>. This site includes links to several Java software programs for exploring fractals.\r\n\r\nIf you are impressed with the Mandelbrot set, <a href=\"http:\/\/www.ted.com\/talks\/benoit_mandelbrot_fractals_the_art_of_roughness\">check out this TED talk from 2010<\/a>\u00a0given by Benoit Mandelbrot on fractals and the art of roughness.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the difference between an imaginary number and a complex number<\/li>\n<li>Identify the real and imaginary parts of a complex number<\/li>\n<li>Plot a complex number on the complex plane<\/li>\n<li>Perform arithmetic operations on complex numbers<\/li>\n<li>Graph physical representations of arithmetic operations on complex numbers as scaling or rotation<\/li>\n<li>Generate several terms of a recursive relation<\/li>\n<li>Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set<\/li>\n<\/ul>\n<\/div>\n<h2>Complex Recursive Sequences<\/h2>\n<p>Some fractals are generated with complex numbers. \u00a0The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.<\/p>\n<div class=\"textbox\">\n<h3>Recursive Sequence<\/h3>\n<p>A <strong>recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].<\/p>\n<\/div>\n<p>The sequence of values produced is the recursive sequence.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2,\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.<\/p>\n<p>We are given the starting value, [latex]{{z}_{0}}=4[\/latex]. The recursive formula holds for any value of <em>n<\/em>, so if [latex]n = 0[\/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[\/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[\/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[\/latex].<\/p>\n<p>Notice this defines [latex]{{z}_{1}}[\/latex] in terms of the known [latex]{{z}_{0}}[\/latex], so we can compute the value:<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[\/latex].<\/p>\n<p>Now letting [latex]n = 1[\/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[\/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[\/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[\/latex]<\/p>\n<p>Continuing,<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{3}}={{z}_{2}}+2=8+2=10\\\\{{z}_{4}}={{z}_{3}}+2=10+2=12[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5816&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\\cdot{i}+(1-i),\\quad{{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.<\/p>\n<p>We are given [latex]{{z}_{0}}=4[\/latex]. Using the recursive formula:<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i\\\\{{z}_{2}}={{z}_{1}}\\cdot{i}+(1-i)=(1+3i)\\cdot{i}+(1-i)=i+3{{i}^{2}}+(1-i)=i-3+(1-i)=-2\\\\{{z}_{3}}={{z}_{2}}\\cdot{i}+(1-i)=(-2)\\cdot{i}+(1-i)=-2i+(1-i)=1-3i\\\\{{z}_{4}}={{z}_{3}}\\cdot{i}+(1-i)=(1-3i)\\cdot{i}+(1-i)=i-3{{i}^{2}}+(1-i)=i+3+(1-i)=4\\\\{{z}_{5}}={{z}_{4}}\\cdot{i}+(1-i)=4\\cdot{i}+(1-i)=1+3i[\/latex]<\/p>\n<p>Notice this sequence is exhibiting an interesting pattern\u2014it began to repeat itself.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131230&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Recursive complex sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lOyusyTsLTs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Mandelbrot Set<\/h2>\n<p>The Mandelbrot Set is a set of numbers defined based on recursive sequences.<\/p>\n<div class=\"textbox\">\n<h3>Mandelbrot Set<\/h3>\n<p>For any complex number <em>c<\/em>, define the sequence [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]<\/p>\n<p>If this sequence always stays close to the origin (within 2 units), then the number <em>c<\/em> is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number <em>c<\/em> is not part of the set.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine if [latex]c=1+i[\/latex] is part of the Mandelbrot set.<\/p>\n<p>We start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+1+i=0+1+i=1+i\\\\{{z}_{2}}={{z}_{1}}^{2}+1+i={{(1+i)}^{2}}+1+i=1+3i\\\\{{z}_{3}}={{z}_{2}}^{2}+1+i={{(1+3i)}^{2}}+1+i=-7+7i\\\\{{z}_{4}}={{z}_{3}}^{2}+1+i={{(-7+7i)}^{2}}+1+i=1-97i[\/latex]<\/p>\n<p>We can already see that these values are getting quite large. It does not appear that [latex]c=1+i[\/latex] is part of the Mandelbrot set.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine if [latex]c=0.5i[\/latex] is part of the Mandelbrot set.<\/p>\n<p>We start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations<\/p>\n<p style=\"text-align: center;\">[latex]{{z}_{1}}={{z}_{0}}^{2}+0.5i=0+0.5i=0.5i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.5i={{(0.5i)}^{2}}+0.5i=-0.25+0.5i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]<\/p>\n<p>While not definitive with this few iterations, it does appear that this value is remaining small, suggesting that 0.5<em>i<\/em> is part of the Mandelbrot set.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Determine if [latex]c=0.4+0.3i[\/latex] is part of the Mandelbrot set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q31678\">Show Solution<\/span><\/p>\n<div id=\"q31678\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{{z}_{1}}={{z}_{0}}^{2}+0.4+0.3i=0+0.4+0.3i=0.4+0.3i\\\\{{z}_{2}}={{z}_{1}}^{2}+0.4+0.3i={{(0.4+0.3i)}^{2}}+0.4+0.3i\\\\{{z}_{3}}={{z}_{2}}^{2}+0.5i={{(-0.25+0.5i)}^{2}}+0.5i=-0.1875+0.25i\\\\{{z}_{4}}={{z}_{3}}^{2}+0.5i={{(-0.1875+0.25i)}^{2}}+0.5i=-0.02734+0.40625i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1737 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\" alt=\"A shaded fractal shown on a graph.\" width=\"187\" height=\"127\" \/>If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.<a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" id=\"return-footnote-1774-1\" href=\"#footnote-1774-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.<\/p>\n<p>Watch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Mandelbrot sequences\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ORqk5jAFpWg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using [latex]c=1+i[\/latex] above, the sequence was distance 2 from the origin after only two recursions.<\/p>\n<p>For some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below<a class=\"footnote\" title=\"This series was generated using Scott\u2019s Mandelbrot Set Explorer\" id=\"return-footnote-1774-2\" href=\"#footnote-1774-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>, light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in\u2014those are used for numbers that grow very slowly.<\/p>\n<p>The Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h2>Additional Resources<\/h2>\n<p>A much more extensive coverage of fractals can be found on the <a href=\"http:\/\/classes.yale.edu\/fractals\/\">Fractal Geometry site<\/a>. This site includes links to several Java software programs for exploring fractals.<\/p>\n<p>If you are impressed with the Mandelbrot set, <a href=\"http:\/\/www.ted.com\/talks\/benoit_mandelbrot_fractals_the_art_of_roughness\">check out this TED talk from 2010<\/a>\u00a0given by Benoit Mandelbrot on fractals and the art of roughness.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1774\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and ADaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Recursive complex sequences. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lOyusyTsLTs\">https:\/\/youtu.be\/lOyusyTsLTs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Mandelbrot sequences. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ORqk5jAFpWg\">https:\/\/youtu.be\/ORqk5jAFpWg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Visualizing mandelbrot sequences and set . <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UJjForMx6-0\">https:\/\/youtu.be\/UJjForMx6-0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1774-1\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Mandelset_hires.png<\/a> <a href=\"#return-footnote-1774-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1774-2\">This series was generated using Scott\u2019s Mandelbrot Set Explorer <a href=\"#return-footnote-1774-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Recursive complex 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