I was playing around with Sierpinski's triangle and derived this equation for the total area for all the down triangles with an infinite number of iterations. (the area of the triangle as a whole =1)

$\displaystyle

A\downarrow=\lim_{n\rightarrow \infty}\sum_{n=1}^{\infty}(\frac{1}{4})^n\cdot 3=1$

(Edit:I realized the sum was kind of useless... But I will keep it there considering it looks cooler and the equation means the same thing with or without )

So the opposite must also be true.

$\displaystyle

A\uparrow=\lim_{n\rightarrow \infty}1-(\frac{3}{4})^n=0$

I am not sure if this is right, and I would like someone to check it please