{"id":2395,"date":"2017-11-09T00:28:41","date_gmt":"2017-11-09T05:28:41","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2395"},"modified":"2017-12-15T12:58:43","modified_gmt":"2017-12-15T17:58:43","slug":"bird-fractal","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/09\/bird-fractal\/","title":{"rendered":"Bird Fractal"},"content":{"rendered":"

Project Idea<\/h2>\n

I named my fractal the bird Fractal because I think the removed squares resemble little tiny bird. To create this , I removed (0,0), (1,1), (3,1), (1,3), and (3,3). The surface area for level one is 11 because we started out with 16 pieces total, and I removed 5 pieces. The area for level two would be 10.31, and the level three would be 7.95.<\/p>\n

Below images are my prints of four levels.<\/strong><\/p>\n

\"\"\u00a0Level 1<\/p>\n

\"\"\u00a0Level 2<\/p>\n

\"\"\u00a0Level 3<\/p>\n

\"\"Level 4<\/p>\n

Sum Calculations<\/h2>\n

Below is the equation used to calculate surface area of level 1, 2, and 3 of the bird fractal.<\/p>\n

\"\"<\/p>\n

For each level, I just use total area before moving minus the area that removed. However, for next level, five squares have\u00a0been moved. So we need to multiply\u00a0by 11 with area each time.<\/p>\n

Like the formula, I write down on the paper.<\/p>\n

Level 1: area of removed= 3*(1*1)<\/p>\n

Level 2:\u00a0 area of removed=3*11*(1\/4*1\/4)<\/p>\n

Level 3: area of removed=3*11*11*(1\/16*1\/16)<\/p>\n

Dimension<\/h2>\n

Finally, the sum of an infinite series. to calculate this, I had to find a and r. Then,\u00a0follow the equation of a + ar + ar^2 + ar^3 + ar^4\u2026 if |r| < 1, and then I had to plug in \u2018a\u2019 and \u2018r\u2019 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

\"\"<\/p>\n

 <\/p>\n

It is scaled down by 1\/4 from Level 1 to Level 2, and the same as Level 2 to Level 3. As I marked, the smaller copy of the whole fractal and the larger one is 11 copies of the big one. For my fractal, I cut down three squares in each box. It means the little square is 1\/11 of the whole fractal.<\/p>\n

Above, I use the formula to calculate dimension, which is (1\/4) ^D=1\/11.<\/p>\n

As I write down on the paper, I use log formula to get Dimension. At last, it came out the dimension is D=1.729.<\/p>\n

Thingiverse<\/h3>\n

Make it yourself by finding the files in\u00a0https:\/\/www.thingiverse.com\/thing:2712626<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

I named my fractal the bird Fractal because I think the removed squares resemble little tiny bird. To create this , I removed (0,0), (1,1), (3,1), (1,3), and (3,3). The surface area for level one is 11 because we started out with 16 pieces total, and I removed 5 pieces. The area for level two […]<\/a><\/p>\n<\/div>","protected":false},"author":16,"featured_media":2396,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[39],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2395"}],"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\/16"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2395"}],"version-history":[{"count":10,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2395\/revisions"}],"predecessor-version":[{"id":2894,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2395\/revisions\/2894"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2396"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2395"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2395"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2395"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}