SOME FRACTALS APPLICATIONS IN GENERAL TOPOLOGICAL SPACES

Abstract

Nature is full of figures, which repeat themselves, but with different sizes, for example, the small rock is similar to the mountain from which it was broken, and any part of a tree is similar to the tree as a whole. Also, there exist many things that have the same level of regularity that we feel when we become more near from it and concentrate on small areas, a famous example is the coast liens which have non-smooth paths and can not be noticed from far. The famous paper that discussed by Mandelbrot in 1967 (How long is the coastline of Great Britain) and some paradoxes when we try to measure the coastline on a map. At the first sight this question may seem trivial but if we have a map one can sit down with a ruler of length S to measure the coast soon with a value for L for the coast length. The problem is that when repeating the operation with a large scale map yield a greater estimate of the length as in the figure 1.