Cantor dust is a fractal that can be constructed using string rewriting beginning with a cell [0] and iterating the rules

{0->[0 0 0; 0 0 0; 0 0 0],1->[1 0 1; 0 0 0; 1 0 1]}. " src="https://mathworld.wolfram.com/images/equations/CantorDust/NumberedEquation1.gif" style="height:54px; width:192px" />

(1)

The th iteration of Cantor dust is implemented in the Wolfram Language as CantorMesh[n, 2].

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

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

The number of black squares after , 1, 2, ... iterations is therefore 1, 4, 16, 64, 256, 1024, 4096, 16384, ... (OEIS A000302). The capacity dimension is therefore

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

REFERENCES:

Dickau, R. M. "Cantor Dust." http://mathforum.org/advanced/robertd/cantor.html.

Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, p. 80, 1983.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103-104, 1993.

Sloane, N. J. A. Sequences A000302/M3518 and A100831 in "The On-Line Encyclopedia of Integer Sequences."