{"id":2221,"date":"2017-11-15T01:26:29","date_gmt":"2017-11-15T06:26:29","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2221"},"modified":"2017-12-16T12:20:31","modified_gmt":"2017-12-16T17:20:31","slug":"polka-dot-carpet-fractal","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/15\/polka-dot-carpet-fractal\/","title":{"rendered":"Polka Dot Carpet Fractal"},"content":{"rendered":"

Fractal Carpet<\/h3>\n

The Polka Dot Carpet Fractal was made by removing only one block from level one, which was coordinate (2,2). I only took out one block because I wanted to see how a simple design become more complicated going up each level.<\/p>\n

Below is the equation used to calculate surface area of level 1, 2 and 3 of the Polka Dot Carpet Fractal.<\/p>\n

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Below are pictures of my level 1, 2 and 3 prints.<\/p>\n

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Surface Area Calculations<\/h3>\n

Below is my surface area calculations for levels 1, 2 and 3 using the formal that was first pictured.<\/p>\n

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Sum of Infinite Series<\/h3>\n

Below is the sum of an infinite series. To calculate this, I had to find that a=1 and r =15\/16 and then follow the equation of a + ar + ar^2 + ar^3 + ar^4… if |r| < 1, and then I had to plug in ‘a’ and ‘r’ in the formula of a x 1\/(1- r) which ended up equally 16 so that cancelled out the initial area of 16, which equals to zero.<\/p>\n

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Dimension<\/h3>\n

The formula to find dimension is (scaled down factor)^(dimension)= 1\/(number of copies) so S^D = 1\/N.
\nIn this case, the fractal is scaled down by 1\/4. There are 15 copies within the whole fractal so the top left square drawn is 1\/15 of the whole fractal. S = 1\/4 and N = 15, which makes the equation look like (1\/4)^D= 1\/15. This also can equate to 4^D= 15 or log_4 (15)= D, which equals to 1.953.<\/p>\n

 <\/p>\n

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You can view the Polka Dot Carpet Fractal on Thingiverse<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

The Polka Dot Carpet Fractal was made by removing only one block from level one, which was coordinate (2,2). I only took out one block because I wanted to see how a simple design become more complicated going up each level. […]<\/a><\/p>\n<\/div>","protected":false},"author":21,"featured_media":2234,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[15],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2221"}],"collection":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/users\/21"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2221"}],"version-history":[{"count":8,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2221\/revisions"}],"predecessor-version":[{"id":2905,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2221\/revisions\/2905"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2234"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2221"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2221"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2221"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}