The standard definition of dimension in linear algebra is based on the notion of vector spaces, forcing the dimension to be a non-negative number. For example, a line segment would have dimension 1 and an area dimension 2.

However, this definition is not very useful for fractals. To illustrate this, let us consider the following object:

This fractal can be recursively defined over the following procedure:

1. Start with a square of edge length 1.
2. Divide the square into 9 equal size sub-squares.
3. Remove the central square.
4. Recurse steps 2.-4. on each of the remaining 8 squares.

The area of this object is zero, since each iteration decreases the total area by a factor of 8/9, giving zero in the limes of infinitely many iterations. However, the general appearance of the fractal is rather space filling and two-dimensional.

To quantify this observation further, let us define above fractal by the bottom-up lattice-based procedure used for creating the image shown:

1. Start with a square of size 1.
2. Make 8 copies of this square and form a 3 times larger square with empty center.
3. Recurse and go to 2.

We find that after k iterations the diameter of the fractal is L=3^k, and the number of unit squares contained in it is N=8^k. With other words, a fractal of this type with diameter L contains N(L) = L^log₃⁸ = L^1.893 unit squares.

Note that the exponent in this formula is a fractionate number between 1 and 2. Since for line segments the exponent would be 1 and for areas 2, both equal to the dimension, we define the exponent 1.893 seen here as the fractal dimension of above fractal [1].

In general, to find the dimension d of a lattice-based fractal, place a square box of diameter L over it and count the number of unit elements N(L) contained in that box. The fractal dimension d is then given by:

1. Ott, E., Chaos in Dynamical Systems. 1993, Cambridge, NY: Cambridge University Press, ch. 3.