For this fractal, I removed 3 squares out of the 16 that we started with. I removed a corner square, a square in the middle and one on the bottom row. The coordinates removed are (0,1), (1,2), and (3,3) To be able to calculate the area, we had to use a specific formula and apply it to the number of boxes removed in the fractal. The formula was 16- total # of big boxes- total #of a medium box- total # of a small box. The first level had 3 boxes taken out so I subtracted 16 minus 3 which gave me 13. In the second level, the medium box area was 1/16, so I multiplied the total of level one (13) by 3 and then by 1/16 and got 2.4375. Then applying the formula, I subtracted 16 by 3 and by 2.4375 and got the medium box total of 10.5625. For the third level, the small box area was 1/16^2 which is equal to 1/256. For this calculation, I had to multiply 13 (13)(3)(1/16^2). The answer I received was 1.98046875. When I applied the formula and did 16-3-2.4375-1.98046875, the total of the small box came out to be 8.58203125. These are the calculations of my fractal’s surface area:

As more boxes are added to each level, the lesser amount of surface area there will be after each iteration.

**Infinite Geometric Series Calculations**

These calculations were very difficult to calculate but when I figured it out, the calculations made sense. With the geometric series formula, a was equal to 3 and r is equal to 13/16. In my calculations below I learned that the answer to the formula a(1/1-r) with my calculations was equal to 16. By applying the formula, I multiplied 3 by 1/(1-13/16), which was then simplified to 3 times 1/(3/16) which gave me an answer of 16. I later subtracted 16 by 16 and got zero, which proves that my series is infinite through my calculations below.

**Fractal Dimension Calculations**

For the equation of the fractal dimensions, I had to solve a logarithm. The squares in my fractal were scaled down to 1/4. When looking at a smaller copy of the fractal, the larger one is 13 copies of the original fractal so the little square is 1/13 of the whole fractal. My calculations below explain the dimensions of my fractal. I used a logarithm to calculate the dimensions of the fractal which resulted in the answer of 1.85022. It starts out with 1/4^D equals 1/13. I then simplify it to 4^D equals 13. To make it a logarithm, the equation then becomes log4^13 equals D, resulting in D being 1.85022

**Level 1 Fractal**

The surface area of the fractal is 13.

**Level 2 Fractal**

The surface area for the second level of this fractal is 10.5625.

**Level 3 Fractal**

The surface area of the third level of this fractal is 8.58203125.

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