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. As
$64=4^3$ 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
${1\over m}$ there are $m^1$ bits and the dimension of the line is
1.
If you break up a square of side 1 into self similar squares with
edge ${1\over m}$ then there are $m^2$ smaller squares and the
dimension is 2.
If you break up a cube of side 1 into self similar cubes with edge
${1\over m}$ then there are $m^3$ 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: $n = m^d$ 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): $$\rm{number of self similar bits} =
\rm{(magnification factor})^d.$$
You can download a free copy of MSW Logo from
http://www.softronix.com/logo.html
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