{"id":1766,"date":"2017-03-14T00:20:23","date_gmt":"2017-03-14T00:20:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1766"},"modified":"2019-05-30T16:33:49","modified_gmt":"2019-05-30T16:33:49","slug":"fractal-basics","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/fractal-basics\/","title":{"raw":"Fractal Basics","rendered":"Fractal Basics"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define and identify self-similarity in geometric shapes, plants, and geological formations<\/li>\r\n \t<li>Generate a fractal shape given an initiator and a generator<\/li>\r\n \t<li>Scale a geometric object by a specific scaling\u00a0factor using the scaling dimension relation<\/li>\r\n \t<li>Determine the fractal dimension of a fractal object<\/li>\r\n<\/ul>\r\n<\/div>\r\nFractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We\u2019ll explore what that sentence means through the rest of the chapter. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.\r\n<div class=\"textbox\">\r\n<h3>Self-similarity<\/h3>\r\nA shape is <strong>self-similar<\/strong> when it looks essentially the same from a distance as it does closer up.\r\n\r\n<\/div>\r\nSelf-similarity can often be found in nature. In the Romanesco broccoli pictured below[footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG<\/a>[\/footnote], if we zoom in on part of the image, the piece remaining looks similar to the whole.\r\n\r\nLikewise, in the fern frond below[footnote]<a href=\"http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/\" target=\"_blank\" rel=\"noopener\">http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/<\/a>[\/footnote], one piece of the frond looks similar to the whole.\r\n\r\nSimilarly, if we zoom in on the coastline of Portugal[footnote]Openstreetmap.org, CC-BY-SA[\/footnote], each zoom reveals previously hidden detail, and the coastline, while not identical to the view from further way, does exhibit similar characteristics.\r\n<h2>Iterated Fractals<\/h2>\r\nThis self-similar behavior can be replicated through recursion: repeating a process over and over.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose 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.\r\n\r\n<img class=\"aligncenter 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\" \/>\r\n\r\nIf 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.\r\n\r\n<img class=\"aligncenter 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\" \/>\r\n\r\n<\/div>\r\nIn the following video, we present another explanation of how to generate a Sierpinski gasket using the idea of self-similarity.\r\n\r\nhttps:\/\/youtu.be\/vro9BUfJxTA\r\n\r\nWe can construct other fractals using a similar approach. To formalize this a bit, we\u2019re going to introduce the idea of initiators and generators.\r\n<div class=\"textbox\">\r\n<h3>Initiators and Generators<\/h3>\r\nAn <strong>initiator<\/strong> is a starting shape\r\n\r\nA <strong>generator<\/strong> is an arranged collection of scaled copies of the initiator\r\n\r\n<\/div>\r\nTo generate fractals from initiators and generators, we follow a simple rule:\r\n<div class=\"textbox\">\r\n<h3>Fractal Generation Rule<\/h3>\r\nAt each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary\r\n\r\n<\/div>\r\nThis process is easiest to understand through example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the initiator and generator shown to create the iterated fractal.\r\n\r\n<img class=\"aligncenter wp-image-1705 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22231540\/initiatorgenerator.png\" alt=\"A straight, horizontal line labeled initiator. And a horizontal line that forms a peak in the middle labeled generator.\" width=\"249\" height=\"68\" \/>\r\n\r\nThis tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of 4 copies of the initiator. In step 1, the single line segment in the initiator is replaced with the generator. For step 2, each of the four line segments of step 1 is replaced with a scaled copy of the generator:\r\n\r\n<img class=\"aligncenter wp-image-1706 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22232232\/1.png\" alt=\"Step 1, the generator. Next, a scaled copy of generator (smaller copy). Next, a scaled copy replaces each line segment of Step 1. In step 2, the fractal.\" width=\"450\" height=\"99\" \/>\r\n\r\nThis process is repeated to form Step 3. Again, each line segment is replaced with a scaled copy of the generator.\r\n\r\n<img class=\"aligncenter size-full wp-image-1707\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22233010\/Screen-Shot-2017-02-22-at-3.29.42-PM.png\" alt=\"Step 2, the fractal. Next, a scaled copy of generator. Step 3, a more complicated fractal.\" width=\"521\" height=\"124\" \/>\r\n\r\nNotice that since Step 0 only had 1 line segment, Step 1 only required one copy of Step 0.\r\n\r\nSince Step 1 had 4 line segments, Step 2 required 4 copies of the generator.\r\n\r\nStep 2 then had 16 line segments, so Step 3 required 16 copies of the generator.\r\n\r\nStep 4, then, would require [latex]16\\cdot4=64[\/latex] copies of the generator.\r\n\r\n<img class=\"size-full wp-image-1708 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22233242\/kochcurve.png\" alt=\"A fractal using the horizontal peaked line seen in previous examples.\" width=\"119\" height=\"66\" \/>\r\n\r\nThe shape resulting from iterating this process is called the Koch curve, named for Helge von Koch who first explored it in 1904.\r\n\r\n<\/div>\r\nNotice that the Sierpinski gasket can also be described using the initiator-generator approach.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the initiator and generator below, however only iterate on the \u201cbranches.\u201d Sketch several steps of the iteration.\r\n\r\n<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\" \/>\r\n\r\nWe begin by replacing the initiator with the generator. We then replace each \u201cbranch\u201d of Step 1 with a scaled copy of the generator to create Step 2.\r\n\r\n[caption id=\"attachment_1710\" align=\"aligncenter\" width=\"249\"]<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\" \/> Step 1, the generator. Step 2, one iteration of the generator.[\/caption]\r\n\r\nWe can repeat this process to create later steps. Repeating this process can create intricate tree shapes.[footnote]<a href=\"http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/\" target=\"_blank\" rel=\"noopener\">http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/<\/a>[\/footnote]\r\n\r\n<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\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the initiator and generator shown to produce the next two stages.\r\n\r\n<img class=\"aligncenter size-full wp-image-1740\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23231215\/Screen-Shot-2017-02-23-at-3.05.42-PM.png\" alt=\"Initiator is a pentagon. Generator is five pentagons arranged to form a larger pentagon.\" width=\"306\" height=\"144\" \/>\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703380\"]\r\n\r\n<img class=\"size-full wp-image-1738 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23230424\/tryitnow1.png\" alt=\"\" width=\"264\" height=\"85\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><img class=\"aligncenter size-full wp-image-1713\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23001817\/fern1.png\" alt=\"\" width=\"184\" height=\"138\" \/><\/td>\r\n<td><img class=\"aligncenter size-full wp-image-1714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23001941\/fern2.png\" alt=\"a fern shape formed by fractals\" width=\"144\" height=\"138\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing iteration processes like those above can create a variety of beautiful images evocative of nature.[footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Fractal_tree_%28Plate_b_-_2%29.jpg\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Fractal_tree_%28Plate_b_-_2%29.jpg<\/a>[\/footnote][footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Barnsley_Fern_fractals_-_4_states.PNG\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Barnsley_Fern_fractals_-_4_states.PNG<\/a>[\/footnote]\r\n\r\nMore natural shapes can be created by adding in randomness to the steps.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCreate a variation on the Sierpinski gasket by randomly skewing the corner points each time an iteration is made.\r\n\r\nSuppose we start with the triangle below. We begin, as before, by removing the middle triangle. We then add in some randomness.\r\n\r\n<img class=\"aligncenter size-full wp-image-1715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23002708\/random1.png\" alt=\"Step 0, an obtuse triangle. Step 1, that triangle divided into four triangles. Step 1 with randomness, The triangle divided into four triangles, but the big triangle is now irregular and no longer a true triangle.\" width=\"439\" height=\"105\" \/>\r\n\r\nWe then repeat this process.\r\n\r\n<img class=\"aligncenter size-full wp-image-1716\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23003207\/random2.png\" alt=\"Step 1 with randomness from the last image. Next is Step 2 without randomness. Next is Step 2 with randomness.\" width=\"439\" height=\"105\" \/>\r\n\r\nContinuing this process can create mountain-like structures. This landscape[footnote]<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg<\/a>[\/footnote]\u00a0was created using fractals, then colored and textured.\r\n\r\n<img class=\"aligncenter size-full wp-image-1717\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23004553\/FractalLandscape-1.jpg\" alt=\"A digitally created landscape\" width=\"640\" height=\"427\" \/>\r\n\r\n<\/div>\r\nThe following video provides another view of branching fractals, and randomness.\r\n\r\nhttps:\/\/youtu.be\/OyAL-66GkJY","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define and identify self-similarity in geometric shapes, plants, and geological formations<\/li>\n<li>Generate a fractal shape given an initiator and a generator<\/li>\n<li>Scale a geometric object by a specific scaling\u00a0factor using the scaling dimension relation<\/li>\n<li>Determine the fractal dimension of a fractal object<\/li>\n<\/ul>\n<\/div>\n<p>Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We\u2019ll explore what that sentence means through the rest of the chapter. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.<\/p>\n<div class=\"textbox\">\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<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-1766-1\" href=\"#footnote-1766-1\" 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<p>Likewise, in the fern frond below<a class=\"footnote\" title=\"http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/\" id=\"return-footnote-1766-2\" href=\"#footnote-1766-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>, one piece of the frond looks similar to the whole.<\/p>\n<p>Similarly, if we zoom in on the coastline of Portugal<a class=\"footnote\" title=\"Openstreetmap.org, CC-BY-SA\" id=\"return-footnote-1766-3\" href=\"#footnote-1766-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>, each zoom reveals previously hidden detail, and the coastline, while not identical to the view from further way, does exhibit similar characteristics.<\/p>\n<h2>Iterated Fractals<\/h2>\n<p>This self-similar behavior can be replicated through recursion: repeating a process over and over.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter 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\" \/><\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter 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\" \/><\/p>\n<\/div>\n<p>In the following video, we present another explanation of how to generate a Sierpinski gasket using the idea of self-similarity.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Self similarity\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vro9BUfJxTA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We can construct other fractals using a similar approach. To formalize this a bit, we\u2019re going to introduce the idea of initiators and generators.<\/p>\n<div class=\"textbox\">\n<h3>Initiators and Generators<\/h3>\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<\/div>\n<p>To generate fractals from initiators and generators, we follow a simple rule:<\/p>\n<div class=\"textbox\">\n<h3>Fractal Generation Rule<\/h3>\n<p>At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary<\/p>\n<\/div>\n<p>This process is easiest to understand through example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the initiator and generator shown to create the iterated fractal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1705 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22231540\/initiatorgenerator.png\" alt=\"A straight, horizontal line labeled initiator. And a horizontal line that forms a peak in the middle labeled generator.\" width=\"249\" height=\"68\" \/><\/p>\n<p>This tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of 4 copies of the initiator. In step 1, the single line segment in the initiator is replaced with the generator. For step 2, each of the four line segments of step 1 is replaced with a scaled copy of the generator:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1706 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22232232\/1.png\" alt=\"Step 1, the generator. Next, a scaled copy of generator (smaller copy). Next, a scaled copy replaces each line segment of Step 1. In step 2, the fractal.\" width=\"450\" height=\"99\" \/><\/p>\n<p>This process is repeated to form Step 3. Again, each line segment is replaced with a scaled copy of the generator.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1707\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22233010\/Screen-Shot-2017-02-22-at-3.29.42-PM.png\" alt=\"Step 2, the fractal. Next, a scaled copy of generator. Step 3, a more complicated fractal.\" width=\"521\" height=\"124\" \/><\/p>\n<p>Notice that since Step 0 only had 1 line segment, Step 1 only required one copy of Step 0.<\/p>\n<p>Since Step 1 had 4 line segments, Step 2 required 4 copies of the generator.<\/p>\n<p>Step 2 then had 16 line segments, so Step 3 required 16 copies of the generator.<\/p>\n<p>Step 4, then, would require [latex]16\\cdot4=64[\/latex] copies of the generator.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1708 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/22233242\/kochcurve.png\" alt=\"A fractal using the horizontal peaked line seen in previous examples.\" width=\"119\" height=\"66\" \/><\/p>\n<p>The shape resulting from iterating this process is called the Koch curve, named for Helge von Koch who first explored it in 1904.<\/p>\n<\/div>\n<p>Notice that the Sierpinski gasket can also be described using the initiator-generator approach.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the initiator and generator below, however only iterate on the \u201cbranches.\u201d Sketch several steps of the iteration.<\/p>\n<p><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\" \/><\/p>\n<p>We begin by replacing the initiator with the generator. We then replace each \u201cbranch\u201d of Step 1 with a scaled copy of the generator to create Step 2.<\/p>\n<div id=\"attachment_1710\" style=\"width: 259px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1710\" 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\" \/><\/p>\n<p id=\"caption-attachment-1710\" class=\"wp-caption-text\">Step 1, the generator. Step 2, one iteration of the generator.<\/p>\n<\/div>\n<p>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-1766-4\" href=\"#footnote-1766-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><\/p>\n<p><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\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the initiator and generator shown to produce the next two stages.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1740\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23231215\/Screen-Shot-2017-02-23-at-3.05.42-PM.png\" alt=\"Initiator is a pentagon. Generator is five pentagons arranged to form a larger pentagon.\" width=\"306\" height=\"144\" \/><\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703380\">Show Solution<\/span><\/p>\n<div id=\"q703380\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1738 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23230424\/tryitnow1.png\" alt=\"\" width=\"264\" height=\"85\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1713\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23001817\/fern1.png\" alt=\"\" width=\"184\" height=\"138\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23001941\/fern2.png\" alt=\"a fern shape formed by fractals\" width=\"144\" height=\"138\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using iteration processes like those above can create a variety of beautiful images evocative of nature.<a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:Fractal_tree_%28Plate_b_-_2%29.jpg\" id=\"return-footnote-1766-5\" href=\"#footnote-1766-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a><a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:Barnsley_Fern_fractals_-_4_states.PNG\" id=\"return-footnote-1766-6\" href=\"#footnote-1766-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/sup><\/a><\/p>\n<p>More natural shapes can be created by adding in randomness to the steps.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Create a variation on the Sierpinski gasket by randomly skewing the corner points each time an iteration is made.<\/p>\n<p>Suppose we start with the triangle below. We begin, as before, by removing the middle triangle. We then add in some randomness.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23002708\/random1.png\" alt=\"Step 0, an obtuse triangle. Step 1, that triangle divided into four triangles. Step 1 with randomness, The triangle divided into four triangles, but the big triangle is now irregular and no longer a true triangle.\" width=\"439\" height=\"105\" \/><\/p>\n<p>We then repeat this process.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1716\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23003207\/random2.png\" alt=\"Step 1 with randomness from the last image. Next is Step 2 without randomness. Next is Step 2 with randomness.\" width=\"439\" height=\"105\" \/><\/p>\n<p>Continuing this process can create mountain-like structures. This landscape<a class=\"footnote\" title=\"http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg\" id=\"return-footnote-1766-7\" href=\"#footnote-1766-7\" aria-label=\"Footnote 7\"><sup class=\"footnote\">[7]<\/sup><\/a>\u00a0was created using fractals, then colored and textured.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1717\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23004553\/FractalLandscape-1.jpg\" alt=\"A digitally created landscape\" width=\"640\" height=\"427\" \/><\/p>\n<\/div>\n<p>The following video provides another view of branching fractals, and randomness.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Iterated tree and twisted gasket\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OyAL-66GkJY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-1766\">\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>Self similarity. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vro9BUfJxTA\">https:\/\/youtu.be\/vro9BUfJxTA<\/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>Iterated tree and twisted gasket. <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OyAL-66GkJY\">https:\/\/youtu.be\/OyAL-66GkJY<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/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-1766-1\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Cauliflower_Fractal_AVM.JPG<\/a> <a href=\"#return-footnote-1766-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1766-2\"><a href=\"http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/\" target=\"_blank\" rel=\"noopener\">http:\/\/www.flickr.com\/photos\/cjewel\/3261398909\/<\/a> <a href=\"#return-footnote-1766-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1766-3\">Openstreetmap.org, CC-BY-SA <a href=\"#return-footnote-1766-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-1766-4\"><a href=\"http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/\" target=\"_blank\" rel=\"noopener\">http:\/\/www.flickr.com\/photos\/visualarts\/5436068969\/<\/a> <a href=\"#return-footnote-1766-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-1766-5\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Fractal_tree_%28Plate_b_-_2%29.jpg\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Fractal_tree_%28Plate_b_-_2%29.jpg<\/a> <a href=\"#return-footnote-1766-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-1766-6\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Barnsley_Fern_fractals_-_4_states.PNG\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:Barnsley_Fern_fractals_-_4_states.PNG<\/a> <a href=\"#return-footnote-1766-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-1766-7\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/File:FractalLandscape.jpg<\/a> <a href=\"#return-footnote-1766-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Self similarity\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vro9BUfJxTA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Iterated tree and twisted gasket\",\"author\":\"OCLPhase2\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/OyAL-66GkJY\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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