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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?

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    You should look into Hausdorff measure and dimension. The Hausdorff dimension of the carpet is higher than 1 and less than 2. So the area is 0 and the length is infinite.2015-06-12

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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.

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    @Muencher, if you feel that Henry's answer is satisfactory, please click on the check mark on the left of Henry's answer.2011-11-20