Why Fractal Geometry?
Fractals provide scientists with a new vocabulary to read the book of nature. Galileo's circles and triangles are insufficient to describe nature in all its rugged complexity. In addition, the fact that natural objects are commonly self-similar, makes fractals ideal models for many of those objects. Fractal geometry also provides scientists with a new way of looking and experimenting with old problems using a different perspective. What is most exciting about fractals is that they successfully bring geometry to where it did not appear to belong, an idea reminiscent of general relativity, which is based on the introduction of geometry to understand the cosmos.
What is Fractal Geometry Good For?
Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, we have seen how an original simple object can be transformed into an enourmoulsy complex one by adding ever increasing detail to it, at the same time preserving affinity between the whole and the parts, or scale invariance. Just think of a big oak tree in winter. Its branches are naked so itis easy to distinguish the way in which a twig splits and becomes tow which then split again, to become four; in much the same way in which the trunk first split into slender branches which split agina and then again, and again. The self-similarity is evident, the whole looks just like its parts, yet not exactly. Nature has slightly altered the construction rule, introducing some degree of randomness which will make one oak slightly different from any other oak tree in the world.
Now, imagine packing all the information required by the tree to become a beautiful large oak tree into the smallest possible space, with the greatest economy of means. It would appear logical that rather than encoding all the unique, intricate complex branching of a mature oak in its seed, all the details of its evolving shape, nature simply encodes the splitting rule, and the urge to repeat it, to iterate. This, plus a little randomness during growth that changes the number of splits or their place in a branch is enough to create a unique oak tree. In fact a whole computer data compression industry is based on similar ideas that permit coding and compressing large files to be quickly transmitted through the Internet.
Thus, there are many examples of self-similarity, power laws and fractals in nature and in everyday life. Shapes with fractal dimension greater than the Euclidean dimension by a factor of 0.2 to 0.3 seem favored in nature; for instance, coastlines have fractal dimension D~1.2; for landscapes D~2.2 and for clouds D~3.3. In terms of the underlying dynamics, what does this mean? Why are natural quantities related in such organized manner? Furthermore, we have seen that a fractal model is usually every effective at simulating natural shapes; and if the fractal dimension is in the appropriate range, the model mimics nature very closely.
Lightning is a good example of a fractal object appearing in nature.