A fractal which is constructed analogously to the Sierpiński sieve, but using squares instead of triangles. It can be constructed using string rewriting beginning with a cell [1] and iterating the rules

Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration. Then

			(2)
			(3)
			(4)
			(5)

The numbers of black cells after , 1, 2, ... iterations are therefore 1, 8, 64, 512, 4096, 32768, 262144, ... (Sloane's A001018). The capacity dimension is therefore

			(6)
			(7)
			(8)
			(9)

(Sloane's A113210).