{"id":2113,"date":"2017-11-13T19:33:14","date_gmt":"2017-11-14T00:33:14","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2113"},"modified":"2017-12-16T12:12:48","modified_gmt":"2017-12-16T17:12:48","slug":"plus-sign-carpet-fractal","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/13\/plus-sign-carpet-fractal\/","title":{"rendered":"Plus Sign Carpet Fractal"},"content":{"rendered":"

\"\"<\/p>\n

Level 1\"\"<\/h3>\n

To start with our carpet fractal, level 1 has 1 square removed from each of the corners creating a plus sign shape. When repeated for a level two, it creates a plus sign made of plus signs, and so on.<\/p>\n

To find the area, I took the area of the original square and subtracted the area of the corner squares I removed. This gives us the area of our level one.<\/p>\n

 <\/p>\n

16-4=12<\/p>\n

 <\/p>\n

Level 2\"\"<\/h3>\n

 <\/p>\n

 <\/p>\n

Level 3<\/h2>\n

Next I removed 4 corner squares from each of the 12 remaining squares of level 1<\/strong>. That leaves us with our level 1<\/strong> area minus 4 corners of the 12 squares in the level 2<\/strong>. This means we subtract the 4 corners 12 times, once for each for the remaining squares. We do this to subtract the 4 corner squares of the 12 left from level 1 but the squares we remove are smaller relative to the remaining squares from level 1.<\/p>\n

(16-4)-(12*4)(1\/16)<\/p>\n

 <\/p>\n

Basically imagine that one square from level one is another, smaller, level one, which is why we must multiply the area we subtract by 1\/16.<\/p>\n

\"\"<\/p>\n

Infinite Geometric Series Calculations<\/h3>\n

To find the infinite geometric series I used the geometric series equation a+ar^2+ar^2+ar^3\u2026ar^n I calculated the infinite area of my fractal. Because r > 1 I used it in ax1\/(1-r). Plugging the numbers in I got 4 x 1\/(1-(12\/16))=0 which you can further see simplified below. We take that answer and subtract it from the total area of the original square (16) so 16-16=0.<\/p>\n

 <\/p>\n

16 – [(4*1)+(4*12)(1\/16)+(4)(12\/16^2)+(4)(12\/16^3)…..]<\/p>\n

a\/1-r<\/p>\n

4 x 1\/(1-(12\/16))<\/p>\n

4\/(4\/16)<\/p>\n

4\/(1\/4)<\/p>\n

4*4<\/p>\n

16<\/p>\n

 <\/p>\n

The infinite series of the\u00a0 fractal is 0 meaning there is no area but still a possible perimeter.<\/p>\n

 <\/p>\n

Fractal Dimension<\/h3>\n

The fractal carpet is neither 1-dimensional nor 2-dimensional, as it is infinite. The fractal lies in between the 1st and 2nd dimension and must be calculated. We use the formula\u00a0(scale-down factor)^(dimension) = 1\/(number of copies)<\/p>\n

To start every level is 1\/4th for each next level. The amount of copies is the amount of squares remaining from a new iteration, look at the first level to see this. We then\u00a0use the logarithm, log base 4 of 12.<\/p>\n

The dimension of this fractal is 1.79248125.<\/p>\n","protected":false},"excerpt":{"rendered":"

To start with our carpet fractal, level 1 has 1 square removed from each of the corners creating a plus sign shape. When repeated for a level two, it creates a plus sign made of plus signs, and so on. To find the area, I took the area of the original […]<\/a><\/p>\n<\/div>","protected":false},"author":17,"featured_media":2257,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[28],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2113"}],"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\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2113"}],"version-history":[{"count":12,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2113\/revisions"}],"predecessor-version":[{"id":2929,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2113\/revisions\/2929"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2257"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2113"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2113"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2113"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}