If you break a line of length 1 into self similar bits of length $\frac{1}{m}$ 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 $\frac{1}{m}$ 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 $\frac{1}{m}$ 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):

${\displaystyle \mathrm{number}\mathrm{of}\mathrm{self}\mathrm{similar}\mathrm{bits}=(\mathrm{magnification}\mathrm{factor}{)}^d.}$

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