If you break a line of length 1 into self similar bits of length ¹/_m there are m¹ bits and we say 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 we say 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 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:

number of bits = (magnification factor)^d

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