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]}.](/images/equations/CantorDust/NumberedEquation1.svg) |
(1)
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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
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(2)
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(3)
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(4)
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(5)
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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
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(6)
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(7)
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(8)
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(9)
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