Fractal geometry is the geometry of chaos. If we take a nonlinear function and assume initial values for constants and variables for the first round, then start applying the iteration process, on each round we get a value that we will feed back to the system. Now, if we consider the values that we obtained at each round of iteration, we see that the patterns are diverging, converging, or constant. But then we reach a zone where the results provide chaotic values without any traceable pattern. With the help of a computer, we can divide that chaotic zone into thousands of points and plot the values.
The result reveals self-similar patterns in which any magnified part looks just like the larger section from which it was taken. Even though we see chaos on the surface, at deeper levels of observation, we can see the elements are creating an emerging order of self-similar patterns, of which different scales all display the same degree of order. The most surprising quality of fractals is that they are created by repeating basic operational rules. That signifies a deep simplicity at the core of this complex universe. Self-similarity is a powerful property of nonlinear systems.
Fractal mathematics is geometry of the real world, which has rough or irregular surfaces. In contrast, Euclidian geometry is based on the assumption that surfaces are smooth (planes have only two dimensions). Euclidian geometry assumes that structures exist only in whole dimensions (1, 2, 3), but fractal geometry allows us to explain the fractional dimensions that exist in the real world (e.g., a snowflake’s dimension is 4/3). Fractal geometry describes chaotic transition and demonstrates the self-similarities that can be found within chaos. Fractal mathematics is one of the greatest achievements of the human mind in the digital age. It has enabled us to formulate a mathematics discipline that is capable of demonstrating real-world phenomena.