Box-Counting Dimension of a (Filled-In) Square
Covering a filled-in square with smaller and smaller boxes we see the pattern shown in the table on the right.
 
N(1)
= 1
N(1/2)
= 4 =
1/(1/4)
=
(1/(1/2))
2
N(1/4)
= 16 =
1/(1/16)
=
(1/(1/4))
2
N(1/8)
= 64 =
1/(1/64)
=
(1/(1/8))
2
and so in general
N(r)
=
(1/r)
2
.
Return to
Box-Counting Dimension
.