{"id":1768,"date":"2017-03-14T00:20:44","date_gmt":"2017-03-14T00:20:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1768"},"modified":"2019-05-30T16:33:58","modified_gmt":"2019-05-30T16:33:58","slug":"fractal-dimension","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/fractal-dimension\/","title":{"raw":"Fractal Dimension","rendered":"Fractal Dimension"},"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\nIn addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.\r\n\r\nTo explore this idea, we need to discuss dimension. Something like a line is 1-dimensional; it only has length. Any curve is 1-dimensional. Things like boxes and circles are 2-dimensional, since they have length and width, describing an area. Objects like boxes and cylinders have length, width, and height, describing a volume, and are 3-dimensional.\r\n\r\n<img class=\"aligncenter size-full wp-image-1721\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181657\/Screen-Shot-2017-02-23-at-10.16.26-AM.png\" alt=\"1-dimensional objects include a straight line and an irregular wavy line. 2-dimensional objects include a rectangle, a triangle, and a circle. 3-dimensional objects include a cube and a cylinder.\" width=\"559\" height=\"172\" \/>\r\n\r\n&nbsp;\r\n\r\nCertain rules apply for scaling objects, related to their dimension.\r\n\r\nIf I had a line with length 1, and wanted to scale its length by 2, I would need two copies of the original line. If I had a line of length 1, and wanted to scale its length by 3, I would need three copies of the original.\r\n\r\n<img class=\"aligncenter size-full wp-image-1722\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181754\/Screen-Shot-2017-02-23-at-10.17.31-AM.png\" alt=\"1, a horizontal line. 2, a horizontal line twice as long. 3, a horizontal line three times as long.\" width=\"518\" height=\"46\" \/>\r\n\r\nIf I had a rectangle with length 2 and height 1, and wanted to scale its length and width by 2, I would need four copies of the original rectangle. If I wanted to scale the length and width by 3, I would need nine copies of the original rectangle.\r\n\r\n<img class=\"aligncenter size-full wp-image-1723\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181900\/Screen-Shot-2017-02-23-at-10.18.34-AM.png\" alt=\"A rectangle with sides measuring 1 and 2. Next, four copies of that rectangle to form a larger rectangle with sides measuring 2 and 4. Next, nine copies of the first rectangle to form a larger rectangle with sides measuring 3 and 6.\" width=\"474\" height=\"108\" \/>\r\n\r\nIf I had a cubical box with sides of length 1, and wanted to scale its length and width by 2, I would need eight copies of the original cube. If I wanted to scale the length and width by 3, I would need 27 copies of the original cube.\r\n\r\n<img class=\"aligncenter size-full wp-image-1724\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23182239\/Screen-Shot-2017-02-23-at-10.22.06-AM.png\" alt=\"A cube with sides measuring 1 by 1 by 1. Next, eight copies of that cube to form a larger cube with sides measuring 2 by 2 by 2. Next, 27 copies of the original cube to form a larger cube measuring 3 by 3 by 3.\" width=\"342\" height=\"91\" \/>\r\n\r\nNotice that in the 1-dimensional case, copies needed = scale.\r\n\r\nIn the 2-dimensional case, copies needed = scale[latex]^{2}[\/latex].\r\n\r\nIn the 3-dimensional case, copies needed = scale[latex]^{3}[\/latex].\r\n\r\nFrom these examples, we might infer a pattern.\r\n<div class=\"textbox\">\r\n<h3>Scaling-Dimension Relation<\/h3>\r\nTo scale a <em>D<\/em>-dimensional shape by a scaling factor <em>S<\/em>, the number of copies <em>C<\/em> of the original shape needed will be given by:\r\n<p style=\"text-align: center;\">[latex]\\text{Copies}=\\text{Scale}^{\\text{Dimension}}[\/latex], or [latex]C=S^{D}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the scaling-dimension relation to determine the dimension of the Sierpinski gasket.\r\n\r\nSuppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2.\r\n\r\n<img class=\"aligncenter wp-image-1725 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23182647\/triangles1-2.png\" alt=\"Sierpinski gasket triangle. Next, a bigger triangle made of three Sierpinski gasket triangles and a white triangle in the center.\" width=\"182\" height=\"91\" \/>\r\n\r\nNotice that to construct the larger gasket, 3 copies of the original gasket are needed.\r\n\r\nUsing the scaling-dimension relation [latex]C=S^{D}[\/latex], we obtain the equation [latex]3=2^{D}[\/latex].\r\n\r\nSince [latex]2^{1}=2[\/latex] and [latex]2^{2}=4[\/latex], we can immediately see that <em>D<\/em> is somewhere between 1 and 2; the gasket is more than a 1-dimensional shape, but we\u2019ve taken away so much area its now less than 2-dimensional.\r\n\r\nSolving the equation [latex]3=2^{D}[\/latex] requires logarithms. If you studied logarithms earlier, you may recall how to solve this equation (if not, just skip to the box below and use that formula with the log key on a calculator):\r\n\r\nTake the logarithm of both sides.\r\n<p style=\"text-align: center;\">[latex]3={{2}^{D}}[\/latex]<\/p>\r\nUse the exponent property of logs.\r\n<p style=\"text-align: center;\">[latex]\\log(3)=\\log\\left({{2}^{D}}\\right)[\/latex]<\/p>\r\nDivide by log(2).\r\n<p style=\"text-align: center;\">[latex]\\log(3)=D\\log\\left(2\\right)[\/latex]<\/p>\r\nThe dimension of the gasket is about 1.585.\r\n<p style=\"text-align: center;\">[latex]D=\\frac{\\log\\left(3\\right)}{\\log(2)}\\approx1.585[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Scaling-Dimension Relation, to find Dimension<\/h3>\r\nTo find the dimension <em>D<\/em> of a fractal, determine the scaling factor <em>S<\/em> and the number of copies <em>C<\/em> of the original shape needed, then use the formula\r\n<p style=\"text-align: center;\">[latex]D=\\frac{\\log\\left(C\\right)}{\\log(S)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDetermine the fractal dimension of the fractal produced using the initiator and generator.\r\n\r\n<img class=\"aligncenter size-full wp-image-1728\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23183509\/Screen-Shot-2017-02-23-at-10.34.29-AM.png\" alt=\"Initiator is a square. Generator is five squares forming a checkered pattern.\" width=\"209\" height=\"105\" \/>\r\n[reveal-answer q=\"355720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"355720\"]\r\n\r\n<img class=\"aligncenter size-full wp-image-1739\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23230845\/tryitnow2.png\" alt=\"\" width=\"158\" height=\"78\" \/>\r\n\r\nScaling the fractal by a factor of 3 requires 5 copies of the original.\r\n<p style=\"text-align: center;\">[latex]D=\\frac{\\text{log}\\left(5\\right)}{\\text{log}\\left(3\\right)}\\approx1.465[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131177&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video, we present a worked example of how to determine the dimension of the Sierpinski gasket\r\n\r\nhttps:\/\/youtu.be\/B1WTSsuDvWc","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>In addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.<\/p>\n<p>To explore this idea, we need to discuss dimension. Something like a line is 1-dimensional; it only has length. Any curve is 1-dimensional. Things like boxes and circles are 2-dimensional, since they have length and width, describing an area. Objects like boxes and cylinders have length, width, and height, describing a volume, and are 3-dimensional.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1721\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181657\/Screen-Shot-2017-02-23-at-10.16.26-AM.png\" alt=\"1-dimensional objects include a straight line and an irregular wavy line. 2-dimensional objects include a rectangle, a triangle, and a circle. 3-dimensional objects include a cube and a cylinder.\" width=\"559\" height=\"172\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Certain rules apply for scaling objects, related to their dimension.<\/p>\n<p>If I had a line with length 1, and wanted to scale its length by 2, I would need two copies of the original line. If I had a line of length 1, and wanted to scale its length by 3, I would need three copies of the original.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1722\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181754\/Screen-Shot-2017-02-23-at-10.17.31-AM.png\" alt=\"1, a horizontal line. 2, a horizontal line twice as long. 3, a horizontal line three times as long.\" width=\"518\" height=\"46\" \/><\/p>\n<p>If I had a rectangle with length 2 and height 1, and wanted to scale its length and width by 2, I would need four copies of the original rectangle. If I wanted to scale the length and width by 3, I would need nine copies of the original rectangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1723\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23181900\/Screen-Shot-2017-02-23-at-10.18.34-AM.png\" alt=\"A rectangle with sides measuring 1 and 2. Next, four copies of that rectangle to form a larger rectangle with sides measuring 2 and 4. Next, nine copies of the first rectangle to form a larger rectangle with sides measuring 3 and 6.\" width=\"474\" height=\"108\" \/><\/p>\n<p>If I had a cubical box with sides of length 1, and wanted to scale its length and width by 2, I would need eight copies of the original cube. If I wanted to scale the length and width by 3, I would need 27 copies of the original cube.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1724\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23182239\/Screen-Shot-2017-02-23-at-10.22.06-AM.png\" alt=\"A cube with sides measuring 1 by 1 by 1. Next, eight copies of that cube to form a larger cube with sides measuring 2 by 2 by 2. Next, 27 copies of the original cube to form a larger cube measuring 3 by 3 by 3.\" width=\"342\" height=\"91\" \/><\/p>\n<p>Notice that in the 1-dimensional case, copies needed = scale.<\/p>\n<p>In the 2-dimensional case, copies needed = scale[latex]^{2}[\/latex].<\/p>\n<p>In the 3-dimensional case, copies needed = scale[latex]^{3}[\/latex].<\/p>\n<p>From these examples, we might infer a pattern.<\/p>\n<div class=\"textbox\">\n<h3>Scaling-Dimension Relation<\/h3>\n<p>To scale a <em>D<\/em>-dimensional shape by a scaling factor <em>S<\/em>, the number of copies <em>C<\/em> 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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket.<\/p>\n<p>Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1725 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23182647\/triangles1-2.png\" alt=\"Sierpinski gasket triangle. Next, a bigger triangle made of three Sierpinski gasket triangles and a white triangle in the center.\" width=\"182\" height=\"91\" \/><\/p>\n<p>Notice that to construct the larger gasket, 3 copies of the original gasket are needed.<\/p>\n<p>Using the scaling-dimension relation [latex]C=S^{D}[\/latex], we obtain the equation [latex]3=2^{D}[\/latex].<\/p>\n<p>Since [latex]2^{1}=2[\/latex] and [latex]2^{2}=4[\/latex], we can immediately see that <em>D<\/em> is somewhere between 1 and 2; the gasket is more than a 1-dimensional shape, but we\u2019ve taken away so much area its now less than 2-dimensional.<\/p>\n<p>Solving the equation [latex]3=2^{D}[\/latex] requires logarithms. If you studied logarithms earlier, you may recall how to solve this equation (if not, just skip to the box below and use that formula with the log key on a calculator):<\/p>\n<p>Take the logarithm of both sides.<\/p>\n<p style=\"text-align: center;\">[latex]3={{2}^{D}}[\/latex]<\/p>\n<p>Use the exponent property of logs.<\/p>\n<p style=\"text-align: center;\">[latex]\\log(3)=\\log\\left({{2}^{D}}\\right)[\/latex]<\/p>\n<p>Divide by log(2).<\/p>\n<p style=\"text-align: center;\">[latex]\\log(3)=D\\log\\left(2\\right)[\/latex]<\/p>\n<p>The dimension of the gasket is about 1.585.<\/p>\n<p style=\"text-align: center;\">[latex]D=\\frac{\\log\\left(3\\right)}{\\log(2)}\\approx1.585[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Scaling-Dimension Relation, to find Dimension<\/h3>\n<p>To find the dimension <em>D<\/em> of a fractal, determine the scaling factor <em>S<\/em> and the number of copies <em>C<\/em> 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Determine the fractal dimension of the fractal produced using the initiator and generator.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1728\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23183509\/Screen-Shot-2017-02-23-at-10.34.29-AM.png\" alt=\"Initiator is a square. Generator is five squares forming a checkered pattern.\" width=\"209\" height=\"105\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q355720\">Show Solution<\/span><\/p>\n<div id=\"q355720\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1739\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23230845\/tryitnow2.png\" alt=\"\" width=\"158\" height=\"78\" \/><\/p>\n<p>Scaling the fractal by a factor of 3 requires 5 copies of the original.<\/p>\n<p style=\"text-align: center;\">[latex]D=\\frac{\\text{log}\\left(5\\right)}{\\text{log}\\left(3\\right)}\\approx1.465[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=131177&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/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<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Fractal dimension\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B1WTSsuDvWc?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-1768\">\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>Question ID 131177. <strong>Authored 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><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>Fractal dimension . <strong>Authored by<\/strong>: OCLPhase2. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/B1WTSsuDvWc\">https:\/\/youtu.be\/B1WTSsuDvWc<\/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>","protected":false},"author":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Fractal dimension 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Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"2a9b6968-9194-4583-bbea-36ec8658b31b","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1768","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1768","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1768\/revisions"}],"predecessor-version":[{"id":2971,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1768\/revisions\/2971"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/1768\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/media?parent=1768"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapter-type?post=1768"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/contributor?post=1768"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/license?post=1768"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}