Cantor Dust

Cantor dust in two dimensions is the Cartesian product , where is the Cantor set (Broden et al. 2024). This fractal is the limiting set of the construction using string rewriting beginning with a black unit square labeled [1] and iterating the rules

(1)

Each replacement subdivides a square into nine congruent squares, each with one third of its side length.

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

Let be the number of black squares, their side length, and their total area after iterations, where is a nonnegative integer and the initial unit square has . 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 of the limiting set is therefore

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

(OEIS A100831).