{"id":1427,"date":"2023-04-06T15:16:18","date_gmt":"2023-04-06T15:16:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1427"},"modified":"2025-05-22T22:19:34","modified_gmt":"2025-05-22T22:19:34","slug":"fractals-generated-by-complex-numbers-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-fresh-take\/","title":{"raw":"Fractals Generated by Complex Numbers: Fresh Take","rendered":"Fractals Generated by Complex Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the difference between imaginary numbers and complex numbers<\/li>\r\n\t<li>Learn how to plot a complex number on a special graph called the complex plane<\/li>\r\n\t<li>Do math operations with complex numbers, and learn how these operations can be shown as scaling or rotation<\/li>\r\n\t<li>Create a series of numbers that are made by repeating a rule, and learn how to find specific terms in the series<\/li>\r\n\t<li>Determine whether a complex number is part of a special set of numbers called the Mandelbrot Set<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Complex Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Complex numbers<\/strong> are a type of number that expand the traditional notion of numbers by including imaginary numbers.<\/p>\r\n<p>The <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex]. Any real multiple of [latex]i[\/latex], like [latex]5i[\/latex], is also an imaginary number.<\/p>\r\n<p>A complex number is composed of two parts: a <strong>real part<\/strong> and an <strong>imaginary part<\/strong>, often written in the form [latex]a + bi[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/SP-YJe7Vldo\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Introduction+to+complex+numbers+_+Imaginary+and+complex+numbers+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Complex Plane<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>In the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.<\/p>\r\n<center><img class=\"aligncenter wp-image-1729 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"205\" height=\"143\" \/><\/center><\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Plotting+complex+numbers+on+the+complex+plane+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Arithmetic on Complex Numbers<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.<\/p>\r\n<p>When we add complex numbers, we can visualize the addition as a shift, or <strong>translation<\/strong>, of a point in the complex plane.<\/p>\r\n<p>We can also multiply complex numbers by a real number, or multiply two complex numbers.<\/p>\r\n<p>In general, multiplication by a complex number can be thought of as a <strong>scaling<\/strong>, changing the distance from the origin, combined with a rotation about the origin.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Plotting+complex+numbers+on+the+complex+plane+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vPZAW7Lhh1E\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Visualizing+complex+arithmetic.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVisualizing complex arithmetic\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Multiply: [latex]4\\left(2+5i\\right)[\/latex]<br \/>\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703380\"]<br \/>\r\n<p>To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials. Distribute and simplify.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}&amp;4(2+5i)\\\\&amp;=4\\cdot2+4\\cdot5i\\\\&amp;=8+20i\\end{array}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Visualize the product [latex]i\\left(1+2i\\right)[\/latex]<br \/>\r\n[reveal-answer q=\"703386\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703386\"]<br \/>\r\nMultiplying, we\u2019d get:\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;i\\cdot1+i\\cdot2i\\\\&amp;=i+2{{i}^{2}}\\\\&amp;=i+2(-1)\\\\&amp;=-2+i\\\\\\end{align}[\/latex]<\/p>\r\n<p>In this case, the distance from the origin has not changed, but the point has been rotated about the origin, [latex]90\u00b0[\/latex] counter-clockwise.<\/p>\r\n<center><img class=\"aligncenter size-full wp-image-1735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/><\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Complex Recursive Sequences<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\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>\r\n<p>The sequence of values produced is the <strong>recursive sequence<\/strong>.<\/p>\r\n<\/div>\r\n<p>In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lOyusyTsLTs\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Recursive+complex+sequences.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRecursive complex sequences\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Mandelbrot Set<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>The <strong>Mandelbrot Set<\/strong> is a set of numbers defined based on recursive sequences.<\/p>\r\n<p>For any complex number [latex]c[\/latex], define the sequence: [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]<\/p>\r\n<p>If this sequence always stays close to the origin (within [latex]2[\/latex] units), then the number [latex]c[\/latex] is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number [latex]c[\/latex] is not part of the set.<\/p>\r\n<\/div>\r\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>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ORqk5jAFpWg\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Mandelbrot+sequences.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMandelbrot sequences\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox linkToLearning\" aria-label=\"Link to Learning\">\r\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>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Benoit+Mandelbrot_+Fractals+and+the+art+of+roughness.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals and the art of roughness\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the difference between imaginary numbers and complex numbers<\/li>\n<li>Learn how to plot a complex number on a special graph called the complex plane<\/li>\n<li>Do math operations with complex numbers, and learn how these operations can be shown as scaling or rotation<\/li>\n<li>Create a series of numbers that are made by repeating a rule, and learn how to find specific terms in the series<\/li>\n<li>Determine whether a complex number is part of a special set of numbers called the Mandelbrot Set<\/li>\n<\/ul>\n<\/section>\n<h2>Complex Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Complex numbers<\/strong> are a type of number that expand the traditional notion of numbers by including imaginary numbers.<\/p>\n<p>The <strong>imaginary number<\/strong> [latex]i[\/latex] is defined to be [latex]i=\\sqrt{-1}[\/latex]. Any real multiple of [latex]i[\/latex], like [latex]5i[\/latex], is also an imaginary number.<\/p>\n<p>A complex number is composed of two parts: a <strong>real part<\/strong> and an <strong>imaginary part<\/strong>, often written in the form [latex]a + bi[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/SP-YJe7Vldo\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Introduction+to+complex+numbers+_+Imaginary+and+complex+numbers+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Complex Plane<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>In the <strong>complex plane<\/strong>, the horizontal axis is the real axis and the vertical axis is the imaginary axis.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1729\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23184017\/axisofimaginaryreal.png\" alt=\"The vertical axis is imaginary, and the horizontal axis is real.\" width=\"205\" height=\"143\" \/><\/div>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Plotting+complex+numbers+on+the+complex+plane+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Arithmetic on Complex Numbers<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.<\/p>\n<p>When we add complex numbers, we can visualize the addition as a shift, or <strong>translation<\/strong>, of a point in the complex plane.<\/p>\n<p>We can also multiply complex numbers by a real number, or multiply two complex numbers.<\/p>\n<p>In general, multiplication by a complex number can be thought of as a <strong>scaling<\/strong>, changing the distance from the origin, combined with a rotation about the origin.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/kGzXIbauGQk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Plotting+complex+numbers+on+the+complex+plane+_+Precalculus+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPlotting complex numbers on the complex plane | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vPZAW7Lhh1E\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Visualizing+complex+arithmetic.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVisualizing complex arithmetic\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Multiply: [latex]4\\left(2+5i\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703380\">Show Solution<\/button><\/p>\n<div id=\"q703380\" class=\"hidden-answer\" style=\"display: none\"><\/p>\n<p>To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials. Distribute and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}&4(2+5i)\\\\&=4\\cdot2+4\\cdot5i\\\\&=8+20i\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Visualize the product [latex]i\\left(1+2i\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703386\">Show Solution<\/button><\/p>\n<div id=\"q703386\" class=\"hidden-answer\" style=\"display: none\">\nMultiplying, we\u2019d get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&i\\cdot1+i\\cdot2i\\\\&=i+2{{i}^{2}}\\\\&=i+2(-1)\\\\&=-2+i\\\\\\end{align}[\/latex]<\/p>\n<p>In this case, the distance from the origin has not changed, but the point has been rotated about the origin, [latex]90\u00b0[\/latex] counter-clockwise.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23220204\/Screen-Shot-2017-02-23-at-2.00.11-PM.png\" alt=\"The imaginary-real graph with the point 1,2, which is labeled 1 plus 2i, and the point negative 2, 1, which is labeled negative 2 plus i. A dotted red line extends from the origin to 1 plus 2i. A red arrow indicates this dotted line moves so that it extends from the origin to negative 2 plus i.\" width=\"333\" height=\"250\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Complex Recursive Sequences<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\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<p>The sequence of values produced is the <strong>recursive sequence<\/strong>.<\/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<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/lOyusyTsLTs\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Recursive+complex+sequences.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRecursive complex sequences\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Mandelbrot Set<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The <strong>Mandelbrot Set<\/strong> is a set of numbers defined based on recursive sequences.<\/p>\n<p>For any complex number [latex]c[\/latex], 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 [latex]2[\/latex] units), then the number [latex]c[\/latex] is part of the <strong>Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number [latex]c[\/latex] is not part of the set.<\/p>\n<\/div>\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<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ORqk5jAFpWg\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Mandelbrot+sequences.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMandelbrot sequences\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox linkToLearning\" aria-label=\"Link to Learning\">\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<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Benoit+Mandelbrot_+Fractals+and+the+art+of+roughness.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals and the art of roughness\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1427"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":44,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1427\/revisions"}],"predecessor-version":[{"id":15524,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1427\/revisions\/15524"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1427\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1427"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1427"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1427"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}