The Sierpinsky Carpet
In the two previous sections we have
talked about the fractal dimension of a convoluted line... the
outline of a plane object. In this part we consider the 'fractalizing'
of an area. Consider a square where each side is 3 units long.
The area of the square is 9 units2:
Now let us remove a square of one unit side from the middle.
The amount removed is 1 unit2; and what remains is
8 units2:
Now we remove 8 square each with sides 1/3 unit. The amount removed
is 8 * (1/3)2 = 8/9 unit2; and what remains
is 7.1111 units2:
Now we remove 64 squares of side 1/9 unit. The amount removed
is 64 * (1/9)2; and what remains is 6.3210 unit2:
Now we remove 512 squares of side 1/27 unit. The amount removed
is 512 * (1/27)2; and what remains is 5.6187 unit2:
We observe that this process can, in theory, be continued forever;
and that there is no need to keep track of every pixel in each
image as the process continues, because this is a recursively
defined 'object'. The notion of 'self similarity' is also exhibited.
The product of the process is a fractal. We note that the the
total perimeter of all the the holes continues to increase as
the remaining area decreases. Eventually, presumably, we will
have something of no area but infinite perimeter of all the holes?
The operational definition for the dimension
of such a fractal is:
2 - (log(rate of change of residual area)/log(rate of change of resolution))
By resolution we are referring to the
size of the holes. Plotting this information on a log/log graph
and finding the slope (0.107), allows us to state that the fractal
dimension of the Sierpinsky Carpet is 1.893.
