I always liked the Sierpinski triangle, and happened upon the related article about the Sierpinski carpet.
The article is pretty sparse, and states the area of the carpet is zero (in standard Lebesgue measure).
Is there a proper proof or book in which a proof may be found?
If after $n$ iterations for the square the area of the carpet is $A_n$, then you have the recursion $A_{n+1}=\dfrac{8}{9} A_n$, so $A_{n}=\left(\frac{8}{9}\right)^n A_0$ and the limit of the area is $0$ as $n$ increases.