Thus the limiting curve is of infinite length and indeed the length between any two points of the curve is infinite. Considering the intuiitve notion of 1 dimensional shapes, although this object appears to be a curve with one starting point and one end point, it is not possible to uniquely specify any position along the curve with one number as we expect to be able to do with Euclidean curves which are 1 dimensional. Although the method of creating this curve is straightforward, there is no algebraic formula the describes the points on the curve. Some of the major differences between fractal and Euclidean geometry are outlined in the following table. Firstly the recognition of fractal is very modern, they have only formally been studied in the last 10 years compared to Euclidean geometry which goes back over 2000 years.
Fractal shapes are self similar and independent of size or scaling.
Third, Euclidean geometry provides a good description of man made objects whereas fractals are required for a representation of naturally occuring geoemtries. It is likely that this limitation of our traditional language of shape is responsible for the stiking difference between mass produced objects and natural shapes. Finally, Euclidean geometries are defined by algebraic formulae, for example defines a sphere. Fractals are normally the result of a iterative or recursive construction or algorithm. A brief description of an 0L system will be presented here but for a more complete description the user should consult the literature.