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=> Sierpiński Sieve
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Ziyaretçi defteri
 

Sierpiński Carpet

SierpinskiCarpet

A fractal which is 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

 {0->[0 0 0; 0 0 0; 0 0 0],1->[1 1 1; 1 0 1; 1 1 1]}.
(1)

Let N_n be the number of black boxes, L_n the length of a side of a white box, and A_n the fractional area of black boxes after the nth iteration. Then

N_n = 8^n
(2)
L_n = 3^(-n)
(3)
A_n = L_n^2N_n
(4)
= (8/9)^n.
(5)

The numbers of black cells after n=0, 1, 2, ... iterations are therefore 1, 8, 64, 512, 4096, 32768, 262144, ... (Sloane's A001018). The capacity dimension is therefore

d_(cap) = -lim_(n->infty)(lnN_n)/(lnL_n)
(6)
= log_38
(7)
= (3ln2)/(ln3)
(8)
= 1.892789260...
(9)

(Sloane's A113210).


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