{"id":2143,"date":"2017-11-14T22:14:32","date_gmt":"2017-11-15T03:14:32","guid":{"rendered":"https:\/\/geekhaus.com\/math103_fall2017\/?p=2143"},"modified":"2017-12-16T08:25:10","modified_gmt":"2017-12-16T13:25:10","slug":"funky-8-carpet-fractal","status":"publish","type":"post","link":"https:\/\/geekhaus.com\/math103_fall2017\/2017\/11\/14\/funky-8-carpet-fractal\/","title":{"rendered":"Funky 8 Carpet Fractal"},"content":{"rendered":"

For my fractal, I removed the coordinates [0,0], [2,1], [1,2], and [3,3].\u00a0 This created a shape that is symmetrical and that is why I like it.\u00a0 If you turn it a little, it looks like the number ‘8’ but with straight lines instead of curved ones.<\/p>\n

\"\"<\/p>\n

Area Calculations<\/h3>\n

Level 1:
\nFor this fractal, we assumed the area 16 units.\u00a0 Each large square we divided it into has the area of 1 unit.\u00a0 I removed 4 squares so the area for my level 1 fractal is<\/p>\n

(area of the original carpet fractal) – [(area of each large square removed) x (number of large squares removed)]
\nor
\n16 – (1×4) = 12.<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Level 2:
\nFor the area of level 2, I took the area of level 1 ([16 – (1 x 4)] or 12) and subtracted the area of a medium sized square (1\/16) times the number of medium sized squares removed (48).<\/p>\n

(area of the level 1) –\u00a0[(area of each medium square removed) x (number of medium squares removed)]
\nor
\n12 – [(1\/16) x 48] = 9<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Level 3:
\nFor the area of my level 3 fractal, I took the area of level 2 (12 – (1\/16 x 48) or 9) and subtracted the area of a small sized square (1\/256) times the number of small sized squares removed (576).<\/p>\n

(area of level 2) –\u00a0[(area of each small square removed) x (number of small squares removed)]
\nor
\n9 – [(1\/256) x 576] = 6.75<\/p>\n

Infinite Geometric Series Calculations<\/h3>\n

To calculate the geometric series, we use this formula:
\n\"\"
\nTo use it, I need to know what ‘a’ and ‘r’ are.
\nFirst, I will use this to help understand where ‘a’ and ‘r’ are coming from:<\/p>\n

\"\"
\nNow I will write out the area calculations all together like this:
\n\"\"
\nNext, I re-wrote it so that all the terms with a power were together like this:
\n\"\"
\nNow we can see which number is repeated and which number has the power like this one:
\n\"\"
\nWe can see that 4 is the number that is repeated and 12\/16 is the number to the power so:
\na = 4
\nr = 12\/16<\/p>\n

Now I can use the formula
\n\"\"
\nand plug in 4 for a and 12\/16 for r to get
\n\"\"
\nwhich can me simplified to
\n\"\"
\nWe solve and end up with 4 x 4 which is 16 which is surprising because that is the area of our original carpet fractal.<\/p>\n

Dimension Calculations<\/h3>\n

To find the dimensions, we need to look at the linear scaling factor and how many of the scaled down versions of the fractal make up the whole fractal.\u00a0 The linear scaling factor is how many pieces the bottom line is divided into. \u00a0In this case, the linear scaling factor is \u00bc.<\/p>\n

\"\"<\/p>\n

The other number we need is how many smaller copies make up the larger whole fractal.\u00a0 In this case that number is 12.
\n\"\"<\/p>\n

Then, I take the formula for finding dimension:<\/p>\n

\"\"<\/p>\n

S is the scale down factor, D is dimension, and N is the number of smaller copies that make up the larger fractal.\u00a0 I plug in the numbers and get<\/p>\n

\"\"<\/p>\n

Then, I have to solve for dimension (D).\u00a0 For this I use Log and get:<\/span><\/p>\n

\"\"<\/p>\n

I plugged that into my calculator and got<\/span><\/p>\n

\u00a0\u00a0 D = 1.79<\/span><\/span><\/p>\n

so the dimension of my fractal is 1.79, neither 1 nor 2 dimensions.\u00a0 <\/span><\/span><\/p>\n

Sharing:
\nHere is a link to my fractal on Thingiverse.\u00a0<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

For my fractal, I removed the coordinates [0,0], [2,1], [1,2], and [3,3].\u00a0 This created a shape that is symmetrical and that is why I like it.\u00a0 If you turn it a little, it looks like the number ‘8’ but with straight lines instead of curved ones. For this fractal, we assumed the area 16 units.\u00a0 Each large […]<\/a><\/p>\n<\/div>","protected":false},"author":8,"featured_media":2320,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[44],"tags":[],"coauthors":[41],"_links":{"self":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2143"}],"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\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/comments?post=2143"}],"version-history":[{"count":12,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2143\/revisions"}],"predecessor-version":[{"id":2920,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/posts\/2143\/revisions\/2920"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media\/2320"}],"wp:attachment":[{"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/media?parent=2143"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/categories?post=2143"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/tags?post=2143"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/geekhaus.com\/math103_fall2017\/wp-json\/wp\/v2\/coauthors?post=2143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}