The Sierpiński carpet is the fractal illustrated above which may be 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, ... (OEIS A001018). The capacity
dimension is therefore
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Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge
University Press, pp. 405-407, 2003.Broden, J.; Espinosa, M.; Nazareth,
N.; and Voth, N. "Knots Inside Fractals." 5 Sep 2024. https://arxiv.org/abs/2409.03639.Dickau,
R. M. "The Sierpinski Carpet." http://mathforum.org/advanced/robertd/carpet.html.Gleick,
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B. B. The
Fractal Geometry of Nature. New York: W. H. Freeman, p. 144, 1983.Peitgen,
H.-O.; Jürgens, H.; and Saupe, D. Chaos
and Fractals: New Frontiers of Science. New York: Springer-Verlag, p. 144,
1992.Reiter, C. A. "Sierpiński Fractals and GCDs."
Computers and Graphics18, 885-891, 1994.Sierpiński,
W. "On Curves Which Contain the Image of Any Given Curve." Mat. Sbornik30,
267-287, 1916. Reprinted in Oeuvres Choisies, Vol. 2, pp. 107-119.Sloane,
N. J. A. Sequences A001018 and A113210 in "The On-Line Encyclopedia of Integer
Sequences."
![{0->[0 0 0; 0 0 0; 0 0 0],1->[1 1 1; 1 0 1; 1 1 1]}.](/images/equations/SierpinskiCarpet/NumberedEquation1.svg)
th
iteration of the Sierpiński carpet is implemented in the Wolfram
Language as MengerMesh[n].
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
, 1, 2, ... iterations are therefore 1, 8, 64, 512, 4096,
32768, 262144, ... (OEIS A001018). The capacity
dimension is therefore
