The Haferman carpet is the beautiful fractal constructed using string rewriting beginning with a cell [1] and iterating the rules
(1)
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(Allouche and Shallit 2003, p. 407).
![Haferman carpet](/images/gifs/HafermanCarpet.jpg)
Taking five iterations gives the beautiful pattern illustrated above.
![](/images/covers/0521823323.jpg)
This fractal also appears on the cover of Allouche and Shallit (2003).
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)
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(3)
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The numbers of black cells after , 1, 2, ... iterations are therefore 1, 4, 61, 424, 4441,
36844, ... (OEIS A118005). The capacity
dimension is therefore
(4)
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(5)
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