An Overview of the Fractal Dimension

(Appendix B)

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 of iteration, on each round we will get a value that we will feed back into the system. Now, if we analyze the values that we obtained at each iteration, we may see patterns where the values are diverging, converging, or constant. Then we reach a zone where the results provide chaotic values without any traceable pattern. With the help of a computer, we can divide the chaotic zone into much smaller increments and plot the values. 

The result of this plot 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 we observe that the elements are creating an emerging order of self-similar patterns, where in different scales they display the same degree of order. The most surprising quality of fractals is that they are created by repeating basic operational rules which signifies a deep simplicity at the core of this complex universe. 

Fractal mathematics is geometry of the real world, which has rough or irregular surfaces. In contrast, Euclidean geometry is based on the assumption that surfaces are smooth (planes have only two dimensions). Euclidean 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. Fractal mathematics is one of the greatest achievements of the human mind in the digital age.