Cantor dust is a fractal that can be constructed using string rewriting beginning with a cell
[0] and iterating the rules
(1)
The Wolfram Language as CantorMesh n ,
2].
Let area
of black boxes after the
The number of black squares after A000302 ). The capacity
dimension is therefore
See also Box Fractal ,
Cantor Set ,
Cantor Square Fractal ,
Haferman
Carpet ,
Sierpiński Carpet ,
Sierpiński
Sieve
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References Broden, J.; Espinosa, M.; Nazareth, N.; and Voth, N. "Knots Inside Fractals." 5 Sep 2024. https://arxiv.org/abs/2409.03639 . Dickau,
R. M. "Cantor Dust." http://mathforum.org/advanced/robertd/cantor.html . Mandelbrot,
B. B. The
Fractal Geometry of Nature. Ott,
E. Chaos
in Dynamical Systems. Sloane, N. J. A. Sequences A000302 /M3518
and A100831 in "The On-Line Encyclopedia
of Integer Sequences." Referenced on Wolfram|Alpha Cantor Dust
Cite this as:
Weisstein, Eric W. "Cantor Dust." From
MathWorld https://mathworld.wolfram.com/CantorDust.html
Subject classifications
![{0->[0 0 0; 0 0 0; 0 0 0],1->[1 0 1; 0 0 0; 1 0 1]}.](/images/equations/CantorDust/NumberedEquation1.svg)
th iteration of Cantor dust is implemented
in the Wolfram Language as CantorMesh[n,
2].
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
, 1, 2, ... iterations is therefore 1, 4, 16, 64, 256, 1024,
4096, 16384, ... (OEIS A000302). The capacity
dimension is therefore
