{"id":2064,"date":"2017-11-10T19:42:37","date_gmt":"2017-11-11T00:42:37","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2064"},"modified":"2017-12-16T08:39:28","modified_gmt":"2017-12-16T13:39:28","slug":"2064","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/10\/2064\/","title":{"rendered":"Computer RAM Carpet Fractal"},"content":{"rendered":"

Definition<\/h3>\n

In my fractal, I removed 3 coordinates, [2,1], [0,2], and [2,2].<\/p>\n

Surface Area Calculations<\/strong><\/h3>\n

Level 1<\/h4>\n

16 – (13)<\/p>\n

Before I removed squares in my fractal, there were 16 squares in total. For my level 1, I removed 3 squares, leaving 13 in the fractal. The area for level one is 13.<\/p>\n

Level 2<\/h4>\n

16 – (13) – ((13 * 3) (1\/16))<\/p>\n

For level 2, I multiplied the 13 remaining squares I had by the 3 squares I removed. Then, I multiplied that number by 1\/16 because that is the area of each remaining square. The equation above equals 2.4375 but then we must subtract that from the area of level 1, 13-2.4375 = 11.5625 ft^2.<\/p>\n

Level 3<\/h4>\n

16 – (13) – ((13 * 3) (1\/16)) – (13(13 * 3) (1\/256))<\/p>\n

For level 3, I multiplied 13 by 3 again but this time I multiplied that number by 13 because for level 3, 13 times more squares were added. Then, I multiplied that amount by 1\/256, which is 1\/16^2, because that is the area of each remaining square. The equation above equals 1.9805 but we must subtract the area of level 2 from that, 11.5625 – 1.9805 = 9.5820 ft^2<\/p>\n

\"\" \"\" \"\"<\/p>\n

Surface Area of “Infinite” Levels<\/strong><\/p>\n

16 – (3(1)) – (3(1\/16) (12)) – (3(1\/16\u00b2) (13)\u00b2) – (3(1\/16\u00b3) (13)\u00b3) – (3(1\/16^4) (13)^4)<\/p>\n

16 [ (3) + 3(13\/16) + 3(13\/16)\u00b2 + 3(13\/16)\u00b3 + 3(13\/16)^4 ]<\/p>\n

16 – (3 x 1\/1-13\/16)<\/p>\n

I started out my sum of an infinite geometric series with the number of squares I started with, 16. Then, I plugged my values into A (3) x 1\/1-r (13\/16). In order to solve what numbers I would need to plug in for A and r, I set out my equation for the surface area up to the fourth power. After level 1, each level was the same except its power was increased by 1.<\/p>\n

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

(1\/4)^D = 1\/13<\/p>\n

4^D = 13<\/p>\n

log4 13 = D<\/p>\n

D\u00a0\u2248 1.8502<\/p>\n

As the fractal carpet grows, it is scaled down by 1\/4 to create an identical square. After locating a square that looked identical to the level 1 fractal, I can clearly see that the area is 13. As the video presented to us, my formula will be (1\/4)^D = 1\/13. Then, I plugged in my values to the formula A^D = B, which translates to logA B = D. My formula was then 4^D = 13 and then log4 13 = D. When I plugged that equation into my calculator, I received the number 1.8502.<\/p>\n

You can download my carpet fractal on Tinkercad<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In my fractal, I removed 3 coordinates, [2,1], [0,2], and [2,2]. Before I removed squares in my fractal, there were 16 squares in total. For my level 1, I removed 3 squares, leaving 13 in the fractal. The area for level one is 13. For level 2, I multiplied the 13 remaining squares […]<\/a><\/p>\n<\/div>","protected":false},"author":25,"featured_media":2065,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[34],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2064"}],"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\/25"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2064"}],"version-history":[{"count":17,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2064\/revisions"}],"predecessor-version":[{"id":2922,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2064\/revisions\/2922"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2065"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2064"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2064"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2064"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2064"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}