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 square and subtracted the area of the corner squares I removed. This gives us the area of our level one.
Next I removed 4 corner squares from each of the 12 remaining squares of level 1. That leaves us with our level 1 area minus 4 corners of the 12 squares in the level 2. 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.
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.
Infinite Geometric Series Calculations
To find the infinite geometric series I used the geometric series equation a+ar^2+ar^2+ar^3…ar^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.
16 – [(4*1)+(4*12)(1/16)+(4)(12/16^2)+(4)(12/16^3)…..]
4 x 1/(1-(12/16))
The infinite series of the fractal is 0 meaning there is no area but still a possible perimeter.
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 (scale-down factor)^(dimension) = 1/(number of copies)
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 use the logarithm, log base 4 of 12.
The dimension of this fractal is 1.79248125.