{"id":2079,"date":"2017-11-14T05:38:52","date_gmt":"2017-11-14T10:38:52","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2079"},"modified":"2017-12-16T08:46:59","modified_gmt":"2017-12-16T13:46:59","slug":"window-fractal-carpet","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/14\/window-fractal-carpet\/","title":{"rendered":"Window Fractal Carpet"},"content":{"rendered":"

For this fractal, I removed 3 squares out of the 16 that we started with. I removed a corner square, a square in the middle and one on the bottom row. The coordinates removed are (0,1), (1,2), and (3,3) \u00a0To be able to calculate the area, we had to use a specific formula and apply it to the number of boxes removed in the fractal. The formula was 16- total # of big boxes- total #of a medium box- total # of a small box. The first level had 3 boxes taken out so I subtracted 16 minus 3 which gave me 13. In the second level, the medium box area was 1\/16, so I multiplied the total of level one (13) by 3 and then by 1\/16 and got 2.4375. Then applying the formula, I subtracted 16 by 3 and by 2.4375 and got the medium box total of 10.5625. For the third level, the small box area was 1\/16^2 which is equal to 1\/256. For this calculation, I had to multiply 13 (13)(3)(1\/16^2). The answer I received was 1.98046875. When I applied the formula and did 16-3-2.4375-1.98046875, the total of the small box came out to be 8.58203125. These are the calculations of my fractal’s surface area:<\/p>\n

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As more boxes are added to each level, the lesser amount of surface area there will be after each iteration.<\/p>\n

Infinite Geometric Series Calculations<\/strong><\/p>\n

These calculations were very difficult to calculate but when I figured it out, the calculations made sense. With the geometric series formula, a was equal to 3 and r is equal to 13\/16. In my calculations below I learned that the answer to the formula a(1\/1-r) with my calculations was equal to 16. By applying the formula, I multiplied 3 by 1\/(1-13\/16), which was then simplified to 3 times 1\/(3\/16) which gave me an answer of 16.\u00a0 I later subtracted 16 by 16 and got zero, which proves that my series is infinite through my calculations below.<\/p>\n

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Fractal Dimension Calculations<\/strong><\/p>\n

For the equation of the fractal dimensions, I had to solve a logarithm. The squares in my fractal were scaled down to 1\/4. When looking at a smaller copy of the fractal, the larger one is 13 copies of the original fractal so the little square is 1\/13 of the whole fractal. My calculations below explain the dimensions of my fractal. I used a logarithm to calculate the dimensions of the fractal which resulted in the answer of 1.85022. It starts out with 1\/4^D equals 1\/13. I then simplify it to 4^D equals 13. To make it a logarithm, the equation then becomes log4^13 equals D, resulting in D being 1.85022<\/p>\n

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 <\/p>\n

Level 1 Fractal<\/strong><\/p>\n

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The surface area of the fractal is 13.<\/p>\n

Level 2 Fractal<\/strong><\/p>\n

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The surface area for the second level of this fractal is 10.5625.<\/p>\n

Level 3 Fractal<\/strong><\/p>\n

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The surface area of the third level of this fractal is 8.58203125.<\/p>\n

Come check out my page on Thingiverse<\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"

For this fractal, I removed 3 squares out of the 16 that we started with. I removed a corner square, a square in the middle and one on the bottom row. The coordinates removed are (0,1), (1,2), and (3,3). To be able to calculate the area, we had to use a specific formula and apply it to […]<\/a><\/p>\n<\/div>","protected":false},"author":27,"featured_media":2135,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[16],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2079"}],"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\/27"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2079"}],"version-history":[{"count":8,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2079\/revisions"}],"predecessor-version":[{"id":2926,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2079\/revisions\/2926"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2135"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2079"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2079"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2079"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}