{"id":1389,"date":"2023-04-06T14:37:25","date_gmt":"2023-04-06T14:37:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1389"},"modified":"2025-08-26T23:27:40","modified_gmt":"2025-08-26T23:27:40","slug":"fractal-basics-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractal-basics-learn-it-1\/","title":{"raw":"Fractal Basics: Learn It 1","rendered":"Fractal Basics: Learn It 1"},"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<p>Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We\u2019ll explore what that sentence means throughout the rest of this section. For now, we can begin with the idea of <strong>self-similarity<\/strong>, a characteristic of most fractals.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>self-similarity<\/h3>\r\n<p>A shape is <strong>self-similar<\/strong> when it looks essentially the same from a distance as it does closer up.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>Self-similarity can often be found in nature. In the Romanesco broccoli pictured below[footnote]http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG[\/footnote], if we zoom in on part of the image, the piece remaining looks similar to the whole.<\/p>\r\n<center>\r\n[caption id=\"attachment_3718\" align=\"aligncenter\" width=\"300\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023515\/1073px-Cauliflower_Fractal_AVM.jpg\"><img class=\"wp-image-3718 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023515\/1073px-Cauliflower_Fractal_AVM-300x201.jpg\" alt=\"Romanesco broccoli\" width=\"300\" height=\"201\" \/><\/a> Figure 1. If you zoom in on one part of the Romanesco broccoli, it will look like the whole[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Likewise, in the fern frond below[footnote]http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/[\/footnote], one piece of the frond looks similar to the whole.<\/p>\r\n<center>\r\n[caption id=\"attachment_3719\" align=\"aligncenter\" width=\"300\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023630\/3261398909_23253b5a24_c.jpg\"><img class=\"wp-image-3719 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023630\/3261398909_23253b5a24_c-300x201.jpg\" alt=\"A fern leaf\" width=\"300\" height=\"201\" \/><\/a> Figure 2. One frond in this fern also looks like the whole[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>This self-similar behavior can be replicated through <strong>recursion<\/strong>: repeating a process over and over.<\/p>\r\n<p>Suppose that we start with a filled-in triangle. We connect the midpoints of each side and remove the middle triangle. We then repeat this process.<\/p>\r\n<center>\r\n[caption id=\"attachment_1702\" align=\"aligncenter\" width=\"450\"]<img class=\"wp-image-1702 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22224647\/triangles.png\" alt=\"Initial, black equilateral triangle is completely filled in. Step 1, the triangle has been divided into three black and one white equilateral triangles, with the white triangle in the center. In step 2, each black triangle has been further divided into into three black and one white equilateral triangles with the white triangle in the center. In step 3, each black triangle has once again been divided into three black and one white equilateral triangles with the white one in the center.\" width=\"450\" height=\"97\" \/> Figure 3. The process of recursion[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>If we repeat this process, the shape that emerges is called the Sierpinski gasket. Notice that it exhibits self-similarity\u2014any piece of the gasket will look identical to the whole. In fact, we can say that the Sierpinski gasket contains three copies of itself, each half as tall and wide as the original. Of course, each of those copies also contains three copies of itself.<\/p>\r\n<center>\r\n[caption id=\"attachment_1704\" align=\"aligncenter\" width=\"408\"]<img class=\"wp-image-1704 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22230213\/Screen-Shot-2017-02-22-at-3.01.08-PM.png\" alt=\"A zoomed-in view of the triangles from the previous picture.\" width=\"408\" height=\"199\" \/> Figure 4. One part of the Sierpinski gasket looks like the whole[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>","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<p>Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We\u2019ll explore what that sentence means throughout the rest of this section. For now, we can begin with the idea of <strong>self-similarity<\/strong>, a characteristic of most fractals.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>self-similarity<\/h3>\n<p>A shape is <strong>self-similar<\/strong> when it looks essentially the same from a distance as it does closer up.<\/p>\n<\/div>\n<\/section>\n<p>Self-similarity can often be found in nature. In the Romanesco broccoli pictured below<a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG\" id=\"return-footnote-1389-1\" href=\"#footnote-1389-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>, if we zoom in on part of the image, the piece remaining looks similar to the whole.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_3718\" aria-describedby=\"caption-attachment-3718\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023515\/1073px-Cauliflower_Fractal_AVM.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3718 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023515\/1073px-Cauliflower_Fractal_AVM-300x201.jpg\" alt=\"Romanesco broccoli\" width=\"300\" height=\"201\" \/><\/a><figcaption id=\"caption-attachment-3718\" class=\"wp-caption-text\">Figure 1. If you zoom in on one part of the Romanesco broccoli, it will look like the whole<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Likewise, in the fern frond below<a class=\"footnote\" title=\"http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/\" id=\"return-footnote-1389-2\" href=\"#footnote-1389-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>, one piece of the frond looks similar to the whole.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_3719\" aria-describedby=\"caption-attachment-3719\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023630\/3261398909_23253b5a24_c.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3719 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/04023630\/3261398909_23253b5a24_c-300x201.jpg\" alt=\"A fern leaf\" width=\"300\" height=\"201\" \/><\/a><figcaption id=\"caption-attachment-3719\" class=\"wp-caption-text\">Figure 2. One frond in this fern also looks like the whole<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>This self-similar behavior can be replicated through <strong>recursion<\/strong>: repeating a process over and over.<\/p>\n<p>Suppose that we start with a filled-in triangle. We connect the midpoints of each side and remove the middle triangle. We then repeat this process.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_1702\" aria-describedby=\"caption-attachment-1702\" style=\"width: 450px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1702 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22224647\/triangles.png\" alt=\"Initial, black equilateral triangle is completely filled in. Step 1, the triangle has been divided into three black and one white equilateral triangles, with the white triangle in the center. In step 2, each black triangle has been further divided into into three black and one white equilateral triangles with the white triangle in the center. In step 3, each black triangle has once again been divided into three black and one white equilateral triangles with the white one in the center.\" width=\"450\" height=\"97\" \/><figcaption id=\"caption-attachment-1702\" class=\"wp-caption-text\">Figure 3. The process of recursion<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>If we repeat this process, the shape that emerges is called the Sierpinski gasket. Notice that it exhibits self-similarity\u2014any piece of the gasket will look identical to the whole. In fact, we can say that the Sierpinski gasket contains three copies of itself, each half as tall and wide as the original. Of course, each of those copies also contains three copies of itself.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_1704\" aria-describedby=\"caption-attachment-1704\" style=\"width: 408px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1704 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22230213\/Screen-Shot-2017-02-22-at-3.01.08-PM.png\" alt=\"A zoomed-in view of the triangles from the previous picture.\" width=\"408\" height=\"199\" \/><figcaption id=\"caption-attachment-1704\" class=\"wp-caption-text\">Figure 4. One part of the Sierpinski gasket looks like the whole<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1389-1\">http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG <a href=\"#return-footnote-1389-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1389-2\">http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/ <a href=\"#return-footnote-1389-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":15,"menu_order":4,"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":"learn_it","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\/1389"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1389\/revisions"}],"predecessor-version":[{"id":15690,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1389\/revisions\/15690"}],"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\/1389\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1389"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1389"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1389"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}