Imagine breaking a cube into 64 identical small cubes. The length of the edge
of the big cube is 4 times the length of the edge of a small cube and we say
that the magnification factor is 4. As64=43 we see that the number of
small self similar pieces is equal to the magnification factor cubed. The
number 3 is called the dimension of the cube.
If you break a line of length 1 into self similar bits of length
1/m there are m1 bits and the dimension of the
line is 1.
If you break up a square of side 1 into self similar squares with
edge 1/m then there are m2 smaller squares and the
dimension is 2.
If you break up a cube of side 1 into self similar cubes with edge
1/m then there are m3 smaller cubes and the
dimension is 3.
In each case we say the magnification factor is m meaning that we
have to scale the lengths by a factor of m to produce the original
shape. The formula for dimension is: n = md where n is the number of
self similar bits, and d is the dimension.
We can generalise what we know about 1, 2 and 3 dimensions to the non integer
dimensions of fractals using the formula (where d is the dimension):
number of self similar bits = (magnification factor)d.
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This is the program that draws the squareflake.
to flake :side :stage
repeat 4[line :side :stage rt 90]
end
to line :side :stage
if :stage = 0 [fd :side stop]
line :side /4 :stage - 1 lt 90
line :side /4 :stage - 1 rt 90
line :side /4 :stage - 1 rt 90
repeat 2 [line :side /4 :stage - 1] lt 90
line :side /4 :stage - 1 lt 90
line :side /4 :stage - 1 rt 90
line :side /4 :stage - 1
end