{"id":2205,"date":"2017-11-09T09:59:01","date_gmt":"2017-11-09T14:59:01","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2205"},"modified":"2017-12-15T12:05:28","modified_gmt":"2017-12-15T17:05:28","slug":"roadway-carpet-fractal","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/09\/roadway-carpet-fractal\/","title":{"rendered":"Roadway Carpet Fractal"},"content":{"rendered":"

Definition<\/strong><\/p>\n

My carpet fractal was created by removing the following coordinates: (2,0), (2,1), and (2,2).<\/p>\n

Surface Area<\/strong><\/p>\n

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

\"\"<\/p>\n

In this case, I am simply removing each coordinate that I had removed earlier, which each represent, for the sake of this post, a 1\u00a0ft^2 size square. So (2,0), (2,1), and (2,2) amount to 3\u00a0ft^2 subtracted, and thus I happen on 13\u00a0ft^2.<\/p>\n

16 – (3)(1) = 13 ft^2<\/p>\n

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

\"\"<\/p>\n

This is where things get interesting. Here, we are going to take the square footage of level 1, and multiply it by 3. We do this because, with each fractal iteration, we are removing 3 big boxes (those 3 coordinates we mentioned earlier). And yet, we are, this time, going to multiply by 1\/16, because now, we are moving 3 scaled boxes in 16 subsections of the original fractal. Through these calculations, we got a total area of about 2.4375\u00a0ft^2 removed in level 2.<\/p>\n

13 x 3 x (1\/16) = 2.4375<\/p>\n

Then, we subtract the area of level 2 from level 1, and get 11.5625\u00a0ft^2.<\/p>\n

13 – 2.4375 = 11.5625 ft^2<\/p>\n

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

\"\"<\/p>\n

Finally, we are going to do a similar situation, but this time break up those pesky 16 subsections into 16 more subsections, resulting in 256 subsections.<\/p>\n

13 x 13 x 3 x (1\/256) = 1.9805<\/p>\n

And finally, we subtract level 3 from level 2, resulting in 9.5820 ft^2.<\/p>\n

11.5025 – 1.9805 = 9.5820 ft^2<\/p>\n

Dimension<\/b><\/p>\n

log4(13) = 1.8502<\/p>\n

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

x=1.8502<\/p>\n

Infinite Series<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

My carpet fractal was created by removing the following coordinates: (2,0), (2,1), and (2,2). In Level 1, I am simply removing each coordinate that I had removed earlier, which each represent, for the sake of this post, a 1\u00a0ft^2 size square. So (2,0), (2,1), and (2,2) amount to […]<\/a><\/p>\n<\/div>","protected":false},"author":20,"featured_media":2559,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[17],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2205"}],"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\/20"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2205"}],"version-history":[{"count":8,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2205\/revisions"}],"predecessor-version":[{"id":2887,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2205\/revisions\/2887"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2559"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2205"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2205"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2205"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}