{"id":1390,"date":"2023-04-06T14:37:23","date_gmt":"2023-04-06T14:37:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1390"},"modified":"2024-10-18T20:53:50","modified_gmt":"2024-10-18T20:53:50","slug":"fractal-basics-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractal-basics-fresh-take\/","title":{"raw":"Fractal Basics: Fresh Take","rendered":"Fractal Basics: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand self-similarity<\/li>\r\n\t<li>Create a fractal shape<\/li>\r\n\t<li>Change the size of a shape using a formula and measure how complex a shape is and how it changes as it gets bigger or smaller<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Self-Similarity<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Self-similarity<\/strong> refers to a characteristic where a fractal pattern retains its shape, irrespective of the level of magnification. In other words, no matter how much you zoom in or out, you will encounter the same pattern repeating over and over. This recursive, infinite detailing and scaling is a fundamental property that distinguishes fractals from other geometric shapes, serving as a visual manifestation of certain mathematical concepts and phenomena in nature.<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10350388&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DJdOH-l6X9g&amp;video_target=tpm-plugin-4zkb4t6g-DJdOH-l6X9g\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/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\/Fractals+and+Self+Similarity.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals and Self Similarity\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Constructing a Fractal<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>An <strong>initiator<\/strong> is a starting shape.<\/p>\r\n<p>A <strong>generator<\/strong> is an arranged collection of scaled copies of the initiator.<\/p>\r\n<p>To construct a fractal, at each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Use the initiator and generator below, however only iterate on the \u201cbranches.\u201d Sketch several steps of the iteration.<center><img class=\"aligncenter size-full wp-image-1709\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22234920\/branches1.png\" alt=\"Initiator is a vertical line. Generator is a vertical line with two smaller lines at an angle to form a Y shape.\" width=\"249\" height=\"112\" \/><\/center><br \/>\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"703380\"]<br \/>\r\nWe begin by replacing the initiator with the generator. We then replace each \u201cbranch\u201d of Step [latex]1[\/latex] with a scaled copy of the generator to create Step [latex]2[\/latex].<center><img class=\"size-full wp-image-1710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22235633\/branches2.png\" alt=\"\" width=\"249\" height=\"111\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n\r\nStep [latex]1[\/latex], the generator. Step [latex]2[\/latex], one iteration of the generator. We can repeat this process to create later steps. Repeating this process can create intricate tree shapes.[footnote]http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/[\/footnote]<center><img class=\"aligncenter size-full wp-image-1711\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23000509\/branches3.png\" alt=\"Step 3 and Step 4, each with another iteration of the fractal. The final shape resembles a tree.\" width=\"470\" height=\"140\" \/><\/center>[\/hidden-answer]<\/section>\r\n<p>The following video provides another view of branching fractals, and randomness.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OyAL-66GkJY\" 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\/Iterated+tree+and+twisted+gasket.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIterated tree and twisted gasket\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Fractal Dimension<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Fractal dimension<\/strong> is a statistical quantity that provides a measure of complexity in a fractal, representing how detail in a pattern changes with the scale at which it is measured. It goes beyond the traditional dimensions (1D, 2D, 3D) by incorporating the scaling properties of fractals. For instance, a fractal line could have a dimension between [latex]1[\/latex] and [latex]2[\/latex], depending on how much space it takes up as it twists and curves.<\/p>\r\n<p><strong>Scaling-dimension relation<\/strong>: To scale a [latex]D[\/latex]-dimensional shape by a scaling factor [latex]S[\/latex], the number of copies [latex]C[\/latex] of the original shape needed will be given by:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Copies}=\\text{Scale}^{\\text{Dimension}}[\/latex], or [latex]C=S^{D}[\/latex]<\/p>\r\n<p><strong>Scaling-dimension relation to find dimension<\/strong>: To find the dimension [latex]D[\/latex] of a fractal, determine the scaling factor [latex]S[\/latex] and the number of copies [latex]C[\/latex] of the original shape needed, then use the formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]D=\\frac{\\log\\left(C\\right)}{\\log(S)}[\/latex]<\/p>\r\n<\/div>\r\n<p>In the following video, we present a worked example of how to determine the dimension of the Sierpinski gasket<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/B1WTSsuDvWc\" 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\/Fractal+dimension.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractal dimension\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Here's a longer video summarizing all of the major concepts.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/gB9n2gHsHN4\" 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\/Fractals+are+typically+not+self-similar.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals are typically not self-similar\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand self-similarity<\/li>\n<li>Create a fractal shape<\/li>\n<li>Change the size of a shape using a formula and measure how complex a shape is and how it changes as it gets bigger or smaller<\/li>\n<\/ul>\n<\/section>\n<h2>Self-Similarity<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Self-similarity<\/strong> refers to a characteristic where a fractal pattern retains its shape, irrespective of the level of magnification. In other words, no matter how much you zoom in or out, you will encounter the same pattern repeating over and over. This recursive, infinite detailing and scaling is a fundamental property that distinguishes fractals from other geometric shapes, serving as a visual manifestation of certain mathematical concepts and phenomena in nature.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10350388&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=DJdOH-l6X9g&amp;video_target=tpm-plugin-4zkb4t6g-DJdOH-l6X9g\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/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\/Fractals+and+Self+Similarity.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals and Self Similarity\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Constructing a Fractal<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>An <strong>initiator<\/strong> is a starting shape.<\/p>\n<p>A <strong>generator<\/strong> is an arranged collection of scaled copies of the initiator.<\/p>\n<p>To construct a fractal, at each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.<\/p>\n<\/div>\n<section class=\"textbox example\">Use the initiator and generator below, however only iterate on the \u201cbranches.\u201d Sketch several steps of the iteration.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1709\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22234920\/branches1.png\" alt=\"Initiator is a vertical line. Generator is a vertical line with two smaller lines at an angle to form a Y shape.\" width=\"249\" height=\"112\" \/><\/div>\n<div class=\"wp-nocaption \"><\/div>\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\">\nWe begin by replacing the initiator with the generator. We then replace each \u201cbranch\u201d of Step [latex]1[\/latex] with a scaled copy of the generator to create Step [latex]2[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22235633\/branches2.png\" alt=\"\" width=\"249\" height=\"111\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Step [latex]1[\/latex], the generator. Step [latex]2[\/latex], one iteration of the generator. We can repeat this process to create later steps. Repeating this process can create intricate tree shapes.<a class=\"footnote\" title=\"http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/\" id=\"return-footnote-1390-1\" href=\"#footnote-1390-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1711\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23000509\/branches3.png\" alt=\"Step 3 and Step 4, each with another iteration of the fractal. The final shape resembles a tree.\" width=\"470\" height=\"140\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video provides another view of branching fractals, and randomness.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OyAL-66GkJY\" 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\/Iterated+tree+and+twisted+gasket.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIterated tree and twisted gasket\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Fractal Dimension<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Fractal dimension<\/strong> is a statistical quantity that provides a measure of complexity in a fractal, representing how detail in a pattern changes with the scale at which it is measured. It goes beyond the traditional dimensions (1D, 2D, 3D) by incorporating the scaling properties of fractals. For instance, a fractal line could have a dimension between [latex]1[\/latex] and [latex]2[\/latex], depending on how much space it takes up as it twists and curves.<\/p>\n<p><strong>Scaling-dimension relation<\/strong>: To scale a [latex]D[\/latex]-dimensional shape by a scaling factor [latex]S[\/latex], the number of copies [latex]C[\/latex] of the original shape needed will be given by:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Copies}=\\text{Scale}^{\\text{Dimension}}[\/latex], or [latex]C=S^{D}[\/latex]<\/p>\n<p><strong>Scaling-dimension relation to find dimension<\/strong>: To find the dimension [latex]D[\/latex] of a fractal, determine the scaling factor [latex]S[\/latex] and the number of copies [latex]C[\/latex] of the original shape needed, then use the formula:<\/p>\n<p style=\"text-align: center;\">[latex]D=\\frac{\\log\\left(C\\right)}{\\log(S)}[\/latex]<\/p>\n<\/div>\n<p>In the following video, we present a worked example of how to determine the dimension of the Sierpinski gasket<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/B1WTSsuDvWc\" 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\/Fractal+dimension.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractal dimension\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Here&#8217;s a longer video summarizing all of the major concepts.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/gB9n2gHsHN4\" 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\/Fractals+are+typically+not+self-similar.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFractals are typically not self-similar\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1390-1\">http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/ <a href=\"#return-footnote-1390-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":15,"menu_order":9,"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\/1390"}],"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":19,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1390\/revisions"}],"predecessor-version":[{"id":15362,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1390\/revisions\/15362"}],"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\/1390\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1390"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1390"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1390"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}