A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions.
Examples of fractals
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object.
Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.
Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by
L = M*l^(1-D)
where M is some positive constant and D is the fractal dimension.
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension