Fractal Geometry

 Prof. Dr. Saeed Naif Turki - Department of physics

eps.saeedn.turkisntr2006@uoanbar.edu.iq

The author's official website

The roots of fractal geometry go back to the 19th century, when mathematicians started to challenge Euclid’s principles. Euclidean geometry gives a first approximation to the structure of physical objects. It cannot easily describe non-linear shapes and non-integral systems. Then, there is a need to a new geometry to describe these systems. Mandelbrot introduced the term "fractal" to describe objects that are irregular and have many of the seemingly complex shapes.  A geometric object whose dimension is fractional is called fractal. Fractals are self-similar or self-affine. For self-similar fractals, any small part of a fractal can be magnified to get the original fractal. The common examples of this type of fractals are the Koch curve, the Koch snowflake and the Seirpinski triangle, as show in Fig. (1). While, in self-affine fractals, a smaller piece of the whole appears to have undergone different scale reductions in the longitudinal and transverse directions. Examples of this latter type of fractals are shown in Fig. (2).

 (a)

 (b)

  (c)

Fig. (1) Common self-similar fractals: (a) Seirpinski triangle, (b) Koch curve and (c).Koch snowflake

(a)This fractal is self-affine instead of self-similar because the  pieces are scaled by different amounts in the x- and y-directions. The   coloring of the  pieces on the right emphasizes this

(b)  The Wiener Brownian motion (WBM)

 Fig (2) Common self-affine fractals