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 we say 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 we say 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. So the formula is:

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

where d is the dimension, i.e. $n=m^d$.