Haferman Carpet
المؤلف:
Allouche, J.-P. and Shallit, J
المصدر:
Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.
الجزء والصفحة:
...
19-9-2021
1604
Haferman Carpet
The Haferman carpet is the beautiful fractal constructed using string rewriting beginning with a cell [1] and iterating the rules
![<span style=]() {0->[1 1 1; 1 1 1; 1 1 1],1->[0 1 0; 1 0 1; 0 1 0]} " src="https://mathworld.wolfram.com/images/equations/HafermanCarpet/NumberedEquation1.gif" style="height:54px; width:188px" /> |
(1)
|
(Allouche and Shallit 2003, p. 407).
Taking five iterations gives the beautiful pattern illustrated above.
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
The numbers of black cells after 
, 1, 2, ... iterations are therefore 1, 4, 61, 424, 4441, 36844, ... (OEIS A118005). The capacity dimension is therefore
REFERENCES:
Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.
Sloane, N. J. A. Sequence A118005 in "The On-Line Encyclopedia of Integer Sequences."
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