Sierpenski Triangle Area

[DIVIDER]
The Sierpenski Triangle is a fractal, much like the Cantor Set, which is generated by starting with a Triangle and iteratively taking the middle fourth out of all of the triangles.
To the right are the first four iterations of the Sierpenski Triangle.
Iterations of the Sierpenski Triangle
A triangle:[A trianglen]
1. [First Iteration] 2. [Second Iteration]
3. [Third Iteration] 4. [Forth Iteration]
Now, there are points which are guarunteed never to be taken out of the set - specifically, any point located at the corners of any of the triangles. However, the area of the Sierpenski triangle converges to zero. That's right! This two dimensional figure has no area! The following is one way that it can be proved.

Let us create an iteration and name it Arean, standing for the Area at the nth iteration. We know that:
Area0 = c
Arean = (3/4)*Arean-1, n>0
Arean = c(3/4)n
 So:

IterationArea
01.00000*c
10.75000*c
20.56250*c
30.42187*c
40.31641*c
50.23730*c
......
100.05631*c
......
1003.207*10-13*c
......
Furthermore, since we know that the limit of xn as n goes to infinity (-1 < x < 1) is 0, we get:
limn->inf (c(3/4)n) = c*0 = 0

-David

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